Geometric pluripotential theory on K\"ahler manifolds

TL;DR

This survey explores the development of finite energy pluripotential theory on Kähler manifolds, highlighting its metric geometry and variational approach to solving complex Monge-Ampère equations.

Contribution

It provides a comprehensive overview of recent advances in the metric and variational aspects of pluripotential theory on Kähler manifolds.

Findings

Potential spaces have rich metric structures.

Variational methods lead to existence results for Monge-Ampère equations.

Infinite dimensional convex optimization frameworks are effective.

Abstract

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Amp\`ere equations. The purpose of this survey is to describe these developments from basic principles.

Equations

$$\mathrm{Ric}\tilde{\omega }=\lambda \tilde{\omega }.$$

$$c_1(X)=\lambda [\omega ].$$

$$(\omega +i\partial \bar{\partial }u{)}^n=e^{-\lambda u+f_0}{\omega }^n,$$

$$\mathcal{K}:{\mathcal{H}}_{\omega }\to \mathbb{R}\text{ and }\mathcal{F}:{\mathcal{H}}_{\omega }\to \mathbb{R}$$

$$\chi (0)=0,1\in \partial \chi (1).$$

$${\chi }^{\ast }(h)=\underset{l\in \mathbb{R}}{\sup }(lh-\chi (l)).$$

$$\chi (a)+{\chi }^{\ast }({\chi }^{\prime }(a))=a{\chi }^{\prime }(a),\chi (a)+{\chi }^{\ast }(b)\ge ab,a,b\in \mathbb{R},{\chi }^{\prime }(a)\in \partial \chi (a),$$

$$L^{\chi}(\mu)=\Big{\{}f:\Omega\to\mathbb{R}\cup\{\infty,-\infty\}:\ \exists r>0\textup{ s.t. }\int_{\Omega}\chi(rf)d\mu<\infty\Big{\}}.$$

$$\|f\|_{\chi,\mu}=\inf\Big{\{}r>0:\ \int_{\Omega}\chi\Big{(}\frac{f}{r}\Big{)}d\mu\leq\chi(1)\Big{\}}.$$

$$\displaystyle\int_{\Omega}\chi\Big{(}\frac{f+g}{r_{1}+r_{2}}\Big{)}d\mu\leq\frac{r_{1}}{r_{1}+r_{2}}\int_{\Omega}\chi\Big{(}\frac{f}{r_{1}}\Big{)}d\mu+\frac{r_{2}}{r_{1}+r_{2}}\int_{\Omega}\chi\Big{(}\frac{g}{r_{2}}\Big{)}d\mu\leq\chi(1).$$

$$l\le \chi (1)+l\le \chi (1+l)\text{ for all }l\ge 0.$$

$$\int_{\{n|f|>1\}}\Big{(}|f|-\frac{1}{n}\Big{)}d\mu\leq\int_{\{n|f|>1\}}\frac{\chi(n|f|)}{n}d\mu\leq\int_{\Omega}\frac{\chi(nf)}{n}d\mu\leq\frac{\chi(1)}{n}.$$

$${\int }_{\Omega }fgd\mu \le \Vert f{\Vert }_{\chi ,\mu }\Vert g{\Vert }_{{\chi }^{\ast },\mu },f\in L^{\chi }(\mu ),g\in L^{{\chi }^{\ast }}(\mu ).$$

$$\int_{\Omega}\frac{fg}{r_{1}r_{2}}d\mu\leq\int_{\Omega}\chi\Big{(}\frac{f}{r_{1}}\Big{)}d\mu+\int_{\Omega}\chi^{*}\Big{(}\frac{g}{r_{2}}\Big{)}d\mu\leq\chi(1)+\chi^{*}(1)=1.\vspace{-0.2in}$$

$$l{\chi }^{\prime }(l)\le p\chi (l),l>0,$$

$${\epsilon }^p\chi (l)\le \chi (\epsilon l)\le \epsilon \chi (l),l>0.$$

$${\epsilon }^p{\chi }_{\delta }(l)\le {\chi }_{\delta }(\epsilon l).$$

$$\|f\|_{\chi,\mu}=\alpha>0\textup{ if and only if }\int_{\Omega}\chi\Big{(}\frac{f}{\alpha}\Big{)}d\mu=\chi(1).$$

$$M_p(l)=\max \{l,l^p\},m_p(l)=\min \{l,l^p\},$$

$$m_p(\Vert f{\Vert }_{\chi ,\mu })\le \frac{{\int }_{\Omega }\chi (f)d\mu }{\chi (1)}\le M_p(\Vert f{\Vert }_{\chi ,\mu }).$$

$$m_{1/p}\Big{(}\frac{\int_{\Omega}\chi(f)d\mu}{\chi(1)}\Big{)}\leq\|f\|_{\chi,\mu}\leq M_{1/p}\Big{(}\frac{\int_{\Omega}\chi(f)d\mu}{\chi(1)}\Big{)}.$$

$$\underset{k\to +\infty }{\lim }\Vert f{\Vert }_{{\chi }_k,\mu }=\Vert f{\Vert }_{\chi ,\mu }.$$

$$\int_{X}\chi\Big{(}\frac{f}{(1+\varepsilon)N}\Big{)}d\mu<\chi(1)-2\delta<\chi(1)<\chi(1)+2\delta<\int_{X}\chi\Big{(}\frac{f}{(1-\varepsilon)N}\Big{)}d\mu.$$

$$\int_{X}\chi_{k}\Big{(}\frac{f}{(1+\varepsilon)N}\Big{)}d\mu<\chi_{k}(1)-\delta<\chi_{k}(1)<\chi_{k}(1)+\delta<\int_{X}\chi_{k}\Big{(}\frac{f}{(1-\varepsilon)N}\Big{)}d\mu.$$

$${\tilde{\chi }}_k(l)=({\delta }_k\star \chi )(h_{k}l)-({\delta }_k\star \chi )(0),l\in \mathbb{R},k\in {\mathbb{N}}^{\ast }.$$

$$l{\chi }_k^{\prime }(l)\le p_k{\chi }_k(l),l>0,$$

$$\text{mathbbm}{1}_{V}i\partial \bar{\partial }{\varphi }^1\wedge \dots \wedge i\partial \bar{\partial }{\varphi }^n=\text{mathbbm}{1}_{V}i\partial \bar{\partial }{\psi }^1\wedge \dots \wedge i\partial \bar{\partial }{\psi }^n.$$

$$\text{PSH}(X,\omega )=\{u\text{ is quasi-plurisubharmonic on }X\text{, and }{\omega }_u:=\omega +i\partial \bar{\partial }u\ge 0\text{ as currents}\}.$$

$$\text{mathbbm}{1}_{{\cap }_j\{u^j>v\}}{\omega }_{\max (u^1,v)}\wedge \dots \wedge {\omega }_{\max (u^n,v)}=\text{mathbbm}{1}_{{\cap }_j\{u^j>v\}}{\omega }_{u^1}\wedge \dots \wedge {\omega }_{u^n}.$$

$$\text{mathbbm}{1}_{{\cap }_j\{v^j>-h_1\}}{\omega }_{v_{h_1}^1}\wedge \dots \wedge {\omega }_{v_{h_1}^n}=\text{mathbbm}{1}_{{\cap }_j\{v^j>-h_1\}}{\omega }_{v_{h_2}^1}\wedge \dots \wedge {\omega }_{v_{h_2}^n}\le \text{mathbbm}{1}_{{\cap }_j\{v^j>-h_2\}}{\omega }_{v_{h_2}^1}\wedge \dots \wedge {\omega }_{v_{h_2}^n},$$

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Geometric pluripotential theory on Kähler manifolds

Tamás Darvas

(UNIVERSITY OF MARYLAND)

Abstract

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge–Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge–Ampère equations. The purpose of this survey is to describe these developments from basic principles.

Contents

1. Preface

2. 1 A primer on Orlicz spaces

3. 2 Finite energy pluripotential theory on Kähler manifolds

4. 2.1 Full mass $\omega$-psh functions

5. 2.2 Finite energy classes

6. 2.3 Examples and singularity type of finite energy potentials

7. 2.4 Envelopes of finite energy classes

8. 3 The Finsler geometry of the space of Kähler potentials

9. 3.1 Riemannian geometry of the space of Kähler potentials

10. 3.2 The Orlicz geometry of the space of Kähler potentials

11. 3.3 The weak geodesic segments of $\textup{PSH}(X,\omega)$

12. 3.4 Extension of the $L^{p}$ metric structure to finite energy spaces

13. 3.5 The Pythagorean formula and applications

14. 3.6 The complete metric spaces $(\mathcal{E}_{p}(X,\omega),d_{p})$

15. 3.7 Special features of the $L^{1}$ Finsler geometry

16. 3.8 Relation to classical notions of convergence

17. 4 Applications to Kähler–Einstein metrics

18. 4.1 The action of the automorphism group

19. 4.2 The existence/properness principle and relation to Tian’s conjectures

20. 4.3 Continuity and compactness properties of the Ding functional

21. 4.4 Convexity of the Ding functional

22. 4.5 Uniqueness of KE metrics and reductivity of the automorphism group

23. 4.6 Regularity of weak minimizers of the Ding functional

24. 4.7 Properness of the K–energy and existence of KE metrics

25. A Appendix

26. A.1 Basic formulas of Kähler geometry

27. A.2 Approximation of $\omega$–psh functions on Kähler manifolds

28. A.3 Regularity of envelopes of $\omega$–psh functions

29. A.4 Cartan type decompositions of Lie groups

30. Bibliography

Preface

A circle of problems, going back to Calabi [30], asks to find Kähler metrics with special curvature properties on a compact Kähler manifold $(X,\omega)$. Of special interest are the Kähler–Einstein (KE) metrics $\tilde{\omega}$, that are cohomologous to $\omega$, and whose Ricci curvature is proportional to the metric tensor, i.e.,

[TABLE]

Existence of such metrics on $(X,\omega)$ is only possible under cohomological restrictions, in particular the first Chern class needs to be a scalar multiple of the Kähler class $[\omega]$:

[TABLE]

When $\lambda\leq 0$, by the work of Aubin and Yau it is always possible to find a unique KE metric on $(X,\omega)$ [1, 112]. The case of Fano manifolds, structures that satisfy ($\star$ ‣ Preface) with $\lambda>0$, is much more intricate. In particular, the problem of finding KE metrics in this case is equivalent with solving the following global scalar equation of complex Monge–Ampère type on $X$:

[TABLE]

where $f_{0}$ is a fixed smooth function on $X$. The solution $u$ belongs to $\mathcal{H}_{\omega}$, the set of smooth functions (potentials) that satisfy $\omega+i\partial\bar{\partial}u>0$, which is an open subset of $C^{\infty}(X)$. As is well known, this equation does not always admit a solution, and our desire is to characterize Fano manifolds that admit KE metrics.

Switching point of view, the KE problem has a very rich variational theory as well. Indeed, Mabuchi and Ding [87, 63] introduced functionals

[TABLE]

whose minimizers are exactly the KE potentials, the solutions of ($\star\star$ ‣ Preface) (see (4.6) and (4.48) for precise definition of these functionals). As a result, KE metrics exist if and only if the minimizer set of $\mathcal{K}$ (or $\mathcal{F}$) is non–empty. Along these lines we ask ourselves: what conditions guarantee existence/uniqueness of minimizers? Our source of inspiration will be the following elementary finite–dimensional result, which will allow to turn the variational approach into a problem of infinite–dimensional convex optimization:

Theorem**.**

*Suppose $F:\mathbb{R}^{n}\to\mathbb{R}$ is a strictly convex functional. If $F$ has a minimizer it has to be unique. Regarding the existence, the following are equivalent:

(i) $F$ has a (unique) minimizer $x_{0}\in\mathbb{R}^{n}$.

(ii) $F$ is proper, i.e., there exists $C,D>0$ such that $F(x)\geq C|x|-D,\ x\in\mathbb{R}^{n}$.*

We give a sketch of the elementary proof. Uniqueness of minimizers is a consequence of strict convexity. That properness of $F$ implies existence of a minimizer follows from the fact that a bounded $F$–minimizing sequence subconverges to some $x_{0}\in\mathbb{R}$, and that convex functions are continuous. That existence of a (unique) minimizer $x_{0}\in\mathbb{R}$ implies properness of $F$ follows from the fact that the unit sphere $\mathbb{S}^{n-1}(x_{0},1)$ is compact, hence $C:=\inf_{\mathbb{S}^{n-1}(x_{0},1)}(F(x)-F(x_{0}))=F(y)-F(x_{0})$ for some $y\in\mathbb{S}^{n-1}(x_{0},1)$. Uniqueness of minimizers implies that $C>0$. Using convexity of $F$, one concludes that $F(x)\geq C|x|-D$ for some $D>0$.

The above simple argument already sheds light on what needs to be accomplished in our infinite dimensional setting to obtain an analogous result for existence/uniqueness of KE metrics. First, we need to understand the convexity of $\mathcal{K}$ and $\mathcal{F}$. Second, even in the above short argument we have used twice that $\mathbb{R}^{n}$ is complete. Consequently, an adequate metric structure needs to be chosen on $\mathcal{H}_{\omega}$, and its completion needs to be understood. Third, (pre)compactness of spheres/balls in this new metric geometry needs to be explored.

Regarding convexity, unfortunately $\mathcal{K}$ and $\mathcal{F}$ are not convex along the straight line segments of $\mathcal{H}_{\omega}$. In order to address this, Mabuchi, Semmes and Donaldson independently introduced a non-positively curved Riemannian $L^{2}$ type metric on $\mathcal{H}_{\omega}$ that produces geodesics along which $\mathcal{K}$ and $\mathcal{F}$ are indeed convex [87, 98, 66]. Inspired from this, a careful analysis of infinite dimensional spaces led Bando–Mabuchi to prove uniqueness of KE metrics [2], and later Berman–Berndtsson to discover even more general uniqueness results [8].

On the other hand, there is strong evidence to suggest that the $L^{2}$ geometry of Mabuchi–Semmes–Donaldson alluded to above does not have the right compactness properties to allow for a characterization of existence of KE metrics. In order to address this, one needs to introduce more general $L^{p}$ type Finsler metrics on $\mathcal{H}_{\omega}$ and compute the metric completion of the related path length metric spaces $(\mathcal{H}_{\omega},d_{p})$ [47, 48].

After sufficient metric theory is developed, it is apparent that the $L^{1}$ geometry of $\mathcal{H}_{\omega}$ will be the one we should focus on. Indeed, sublevel sets of $\mathcal{K}$ restricted to spheres/balls are $d_{1}$–precompact, allowing to establish an equivalence between existence of KE metrics and $d_{1}$–properness of $\mathcal{K}$ and $\mathcal{F}$. Lastly, $d_{1}$–properness can be expressed using simple analytic means. This allowed the author and Y.A. Rubinstein to verify numerous related conjectures of Tian [57] going back to the nineties [106, 107].

Structure of the survey.

The aim of this work is to give a self contained introduction to special Kähler metrics using pluripotential theory/infinite–dimensional geometry.

In Chapter 1, we give a very brief introduction to Orlicz spaces that are generalizations of the classical $L^{p}$ spaces. Our treatment will be rather minimalistic and we refer to [97] for a complete treatment.

In Chapter 2, we develop some background in finite energy pluripotential theory, necessary for later developments, closely following the original treatises of Guedj–Zeriahi and collaborators [72, 73, 11, 12], that were inspired by work of Cegrell [34] in the local case. We refer to these works for a comprehensive treatment, as well as the recent excellent textbook [74].

Chapter 3 contains the main technical machinery presented in this work. Here we introduce the $L^{p}$ Finsler geometry of the space of Kähler potentials $\mathcal{H}_{\omega}$, and compute the metric completion of this space with respect to the corresponding path length metrics $d_{p}$. The $d_{p}$–completions of $\mathcal{H}_{\omega}$ will be identified with $\mathcal{E}_{p}(X,\omega)$, the finite energy spaces of Guedj–Zeriahi described in the previous chapter (Theorem 3.36). In particular, we can endow these spaces with a rich metric geometry, inspiring the title of this work.

In Chapter 4 we discuss applications to existence/uniqueness of KE metrics on Fano manifolds. First we describe an abstract properness/existence principle (Theorem 4.7) that adapts the above finite–dimensional Theorem to our infinite–dimensional setting. As we verify the assumptions of this principle, we will present self contained proofs of the Bando–Mabuchi uniqueness theorem (Theorem 4.23) [2] and the Matsushima theorem about reductivity of the automorphism group of a KE manifold (Proposition 4.25) [88]. After this, in Theorem 4.11 and Theorem 4.36 we resolve different versions of Tian’s conjectures [106, 107] characterizing existence of KE metrics in terms of energy properness, following [57].

Prerequisites.

An effort has been made to keep prerequisites at a minimum. However due to size constraints, such requirements on part of the reader are inevitable. We assume that our reader is familiar with the basics of Bedford–Taylor theory of the complex Monge–Ampère operator. Mastery of [20, Chapters I-III] or [60, Chapter I and Chapter III.1-3] is more then sufficient, and for a thorough treatment we highly recommend the recent textbook [74]. We also assume that our reader is familiar with the basics of Kähler geometry, though we devote a section in the appendix to introduce our terminology, and recall some of the essentials. For a comprehensive introduction into Kähler geometry we refer the reader to the recent textbook [102], as well as [60, 113].

Though our main focus is the pluripotential theoretic point of view, some results in this survey rest on important regularity theorems regarding equations of complex Monge–Ampère type. Due to space constraints we cannot present a detailed proof of these theorems, but we will isolate their statements and keep them at a minimum, while providing precise references at all times.

Relation to other works.

As stressed above, our focus in this work is on self–contained treatment of the chosen topics. On the down side, we could not devote enough space to the vast historical developments of the subject, and for such a treatment we refer to the survey [95], that discusses similar topics using a more chronological approach.

Without a doubt the choice of topics represent our bias and limitations, and many important recent developments could not be surveyed. In particular, recent breakthroughs on K–stability (the work of Chen–Donaldson–Sun [39], Tian [108], Chen–Wang–Sun [41], Berman–Boucksom–Jonsson [10]) could not be presented, and we refer to [68, 109] for recent surveys on this topic. For results about the quantization of the geometry of the space of Kähler metrics, we refer to the original papers [18, 43, 67, 90, 91, 55], as well as the survey [92] along with references therein. Geometric flows could not be discussed either, and we refer to [27, 12, 13, 52, 100, 101] for work on the Ricci and Calabi flows that uses the theoretical machinery described in this survey.

The relation with constant scalar curvature Kähler (csck) metrics is also not elaborated. A preliminary version of this memoir appeared on the website of the author in the early months of 2017. Since then a number of important works have appeared building on the topics presented in this work: Chen–Cheng cracked the PDE theory of the csck equation [37, 38], allowing to fully prove a converse of a theorem by Berman–Darvas–Lu [14]. Very recently He–Li pointed out that the contents of this survey generalize to Sasakian manifolds [78], paving the way to existence theorems for canonical metrics in that context as well.

Acknowledgments.

I thank P. Gupta, W. He, L. Lempert, J. Li, C.H. Lu, Y.A. Rubinstein, K. Smith, V. Tosatti and the anonymous referees for their suggested corrections, careful remarks, and precisions. Also, I thank the students of MATH868D at the University of Maryland for their intriguing questions and relentless interest throughout the Fall of 2016. Chapter 3 is partly based on work done as a graduate student at Purdue University, and I am indebted to L. Lempert for encouragement and guidance. Chapter 4 surveys to some extent joint work with Y. Rubinstein, and I am grateful for his mentorship over the years. This research was partially supported by NSF grant DMS-1610202 and BSF grant 2012236.

Chapter 1 A primer on Orlicz spaces

Plainly speaking, Orlicz spaces are generalizations of $L^{p}$ Banach spaces. As we will see in our later study, we will prefer working with Orlicz norms over $L^{p}$ norms, since a careful choice of weight makes Orlicz norms smooth away from the origin. The same cannot be said about $L^{p}$ norms. We give here a brief and self-contained introduction, only touching on aspects that will be needed later. For a more thorough treatment we refer to [97].

Suppose $(\Omega,\Sigma,\mu)$ is a measure space with $\mu(\Omega)=1$ and $(\chi,\chi^{*})$ is a complementary pair of Young weights. This means that $\chi:\mathbb{R}\to\mathbb{R}^{+}\cup\{\infty\}$ is convex, even, lower semi-continuous (lsc) and satisfies the normalizing conditions

[TABLE]

Recall that $\partial\chi(l)\subset\mathbb{R}$ is the set of subgradients to $\chi$ at $l$, i.e., $v\in\partial\chi(l)$ if and only if $\chi(l)+vh\leq\chi(l+h),\ h\in\mathbb{R}$. As described, $\chi$ is simply a normalized Young weight. The complement $\chi^{*}$ is the Legendre transform of $\chi$:

[TABLE]

Using convexity of $\chi$ and the above identity, one can verify that $\chi^{*}$ is also a normalized Young weight. Additionally $(\chi,\chi^{*})$ satisfies the Young identity and inequality:

[TABLE]

in particular, due to our normalization: $\chi(1)+\chi^{*}(1)=1$.

The most typical example to keep in mind is the pair $\chi_{p}(l)=|l|^{p}/p$ and $\chi^{*}_{p}(l)=|l|^{q}/q$, where $p,q>1$ and $1/p+1/q=1$. Let $L^{\chi}(\mu)$ be the following space of measurable functions:

[TABLE]

One can introduce the following norm on $L^{\chi}(\mu)$:

[TABLE]

The set $\{f\in L^{\chi}(\mu):\int_{\Omega}\chi(f)d\mu\leq\chi(1)\}$ is convex and symmetric in $L^{\chi}(\mu)$, hence $\|\cdot\|_{\chi,\mu}$ is nothing but the Minkowski seminorm of this set. This is the content of the following lemma:

Lemma 1.1**.**

Suppose $f,g\in L^{\chi}(\mu)$. Then $\|f+g\|_{\chi,\mu}\leq\|f\|_{\chi,\mu}+\|g\|_{\chi,\mu}.$

Proof.

Suppose $\int_{\Omega}\chi(f/r_{1})d\mu\leq\chi(1),\ \int_{\Omega}\chi(g/r_{2})d\mu\leq\chi(1)$ for some $r_{1},r_{2}>0$. Convexity of $\chi$ implies that

[TABLE]

Hence $\|f+g\|_{\chi,\mu}\leq r_{1}+r_{2}$, finishing the argument. ∎

Together with the previous one, the next lemma implies that $(L^{\chi}(\mu),\|\cdot\|_{\chi,\mu})$ is a normed space:

Lemma 1.2**.**

Suppose $f\in L^{\chi}(\mu)$. Then $\|f\|_{\chi,\mu}=0$ implies that $f=0$ a.e. with respect to $\mu$.

Proof.

As $1\in\partial\chi(1)$ and $\chi\geq 0$, it follows that

[TABLE]

As a consequence of this inequality, $L^{\chi}(\mu)\subset L^{1}(\mu)$. Also, since $\|f\|_{\chi,\mu}=0$, it follows that $\int_{\Omega}\chi(nf)d\mu\leq\chi(1)$ for all $n\in\mathbb{N}$. By (1.5) we can write:

[TABLE]

Applying the dominated convergence theorem to this inequality gives $\int_{|f|>0}|f|d\mu=0$, finishing the proof. ∎

Though we will not make use of it, one can also show that $(L^{\chi}(\mu),\|\cdot\|_{\chi,\mu})$ is complete, hence it is a Banach space (see [97, Theorem 3.3.10]). The reason we work with a complementary pair of Young weights is because in this setting the Hölder inequality holds:

Proposition 1.3**.**

For $f\in L^{\chi}(\mu)$ and $g\in L^{\chi^{*}}(\mu)$ we have

[TABLE]

Proof.

Let $r_{1}>\|f\|_{\chi,\mu}$ and $r_{2}>\|g\|_{\chi^{*},\mu}$. Using both the Young inequality and the identity (1.2), (1.6) follows in the following manner:

[TABLE]

∎

Orlicz spaces can be quite general and in our study we will be interested in spaces whose normalized Young weight is finite and satisfies the growth estimate

[TABLE]

for some $p\geq 1$. To clarify, $\chi^{\prime}(l)$ is just an arbitrary subgradient of $\chi$ at $l$. For such weights we write $\chi\in\mathcal{W}^{+}_{p}$, following the notation of [73]. As it turns out, weights $\chi$ that satisfy (1.7) can be thought of as distant cousins of the homogeneous $L^{p}$ weight $|l|^{p}/p\in\mathcal{W}^{+}_{p}$:

Proposition 1.4**.**

For $\chi\in\mathcal{W}^{+}_{p},\ p\geq 1$ and $0<\varepsilon<1$ we have

[TABLE]

Proof.

The second estimate follows from convexity of $\chi$. For the first estimate we notice that for any $\delta>0$ the weight $\chi_{\delta}(l):=\chi(l)+\delta|l|$ also satisfies (1.7). As $\chi_{\delta}(h)>0$ for $h>0$, we can integrate $\chi^{\prime}_{\delta}(h)/\chi_{\delta}(h)\leq p/h$ from $\varepsilon l$ to $l$ to obtain:

[TABLE]

Letting $\delta\to 0$ the desired estimate follows. ∎

Estimate (1.8) immediately implies that for $f\in L^{\chi}(\mu)$ the function $l\to\int_{\Omega}\chi(lf)d\mu$ is continuous, hence we have

[TABLE]

To simplify future notation, we introduce the increasing functions

[TABLE]

for $p>0,l\geq 0$. Observe that $M_{p}\circ m_{1/p}(l)=m_{p}\circ M_{1/p}(l)=l$. As a consequence of (1.8) and (1.9), we immediately obtain the following estimates, characterizing Orlicz norm convergence:

Proposition 1.5**.**

If $\chi\in\mathcal{W}_{p}^{+}$ and $f\in L^{\chi}(\mu)$ then

[TABLE]

[TABLE]

As a result, for a sequence $\{f_{j}\}_{j\in\mathbb{N}}$ we have $\|f_{j}\|_{\chi,\mu}\to 0$ if and only if $\int_{\Omega}\chi(f_{j})d\mu\to 0$. Also, $\|f_{j}\|_{\chi,\mu}\to N$ for $N>0$ if and only if $\int_{\Omega}\chi(f_{j}/N)d\mu\to\chi(1)$.

Later in this survey we will need to approximate certain Orlicz norms with Orlicz norms having smooth $\mathcal{W}^{+}_{p}$-weights. The following two approximation results, which are by no means optimal, will be useful in our treatment:

Proposition 1.6**.**

Suppose $\chi\in\mathcal{W}^{+}_{p}$ and $\{\chi_{k}\}_{k\in\mathbb{N}}$ is a sequence of normalized Young weights that converges uniformly on compacts to $\chi.$ Let $f$ be a bounded $\mu$–measurable function on $X$. Then $f\in L^{\chi}(\mu),L^{\chi_{k}}(\mu),\ k\in\mathbb{N}$ and we have that

[TABLE]

Proof.

Suppose $N=\|f\|_{\chi,\mu}$. If $N=0$, then $f=0$ a.e. with respect to $\mu$ implying that $\|f\|_{\chi_{k},\mu}=\|f\|_{\chi,\mu}=0$. So we assume that $N>0$. As $\chi\in\mathcal{W}^{+}_{p}$, by (1.9), for any $\varepsilon>0$ there exists $\delta>0$ such that

[TABLE]

As $\chi_{k}$ tends uniformly on compacts to $\chi$, $f$ is bounded and $\mu(X)=1$, it follows from the dominated convergence theorem that for $k$ big enough we have

[TABLE]

This implies that $(1-\varepsilon)N\leq\|f\|_{\chi_{k},\mu}\leq(1+\varepsilon)N$, from which the conclusion follows. ∎

Proposition 1.7**.**

Given $\chi\in\mathcal{W}^{+}_{p}$, there exists $\chi_{k}\in\mathcal{W}^{+}_{p_{k}}\cap C^{\infty}(\mathbb{R}),$ $k\in\mathbb{N}$, with $\{p_{k}\}_{k}$ possibly unbounded, such that $\chi_{k}\to\chi$ uniformly on compacts.

Proof.

There is great freedom in constructing the sequence $\chi_{k}$. First we smoothen and normalize $\chi$, introducing the sequence $\tilde{\chi}_{k}$ in the process:

[TABLE]

Here $\delta$ is a typical choice of bump function that is smooth, even, has support in $(-1,1)$, and $\int_{\mathbb{R}}\delta(t)dt=1$. Accordingly, $\delta_{k}(\cdot):=k\delta(k(\cdot))$, and $h_{k}>0$ is chosen in such a way that $\tilde{\chi}_{k}$ becomes normalized ($\tilde{\chi}_{k}(0)=0$ and $\tilde{\chi}_{k}^{\prime}(1)=1$). As $\chi$ is normalized it follows that $\tilde{\chi}^{\prime}_{k}(1-\frac{1}{k})\leq 1\leq\tilde{\chi}^{\prime}_{k}(1+\frac{1}{k})$. In particular, $1-1/k\leq h_{k}\leq 1+1/k$, hence the $\tilde{\chi}_{k}$ are normalized Young weights that converge to $\chi$ uniformly on compacts.

As $\chi$ is strictly increasing on $\mathbb{R}^{+}$, so is each $\tilde{\chi}_{k}$, but our construction does not seem to guarantee that $\tilde{\chi}_{k}\in\mathcal{W}^{+}_{p_{k}}$ for some $p_{k}\geq 1$. This can be fixed by changing smoothly the values of $\tilde{\chi}_{k}$ on the sets $|x|<1/k$ and $|x|>k$ in such a manner that the altered weights $\chi_{k}$ satisfy the estimate

[TABLE]

for some $p_{k}\geq 1$. Since $\tilde{\chi}_{k}^{\prime}(t)>0,\ t>0$ and $\tilde{\chi}^{\prime}_{k}(0)=0$, this can be done using elementary calculus. The sequence $\{\chi_{k}\}_{k}$ thus obtained satisfies the required properties. ∎

Chapter 2 Finite energy pluripotential theory on Kähler manifolds

In this section we introduce the basics of finite energy pluripotential theory on Kähler manifolds. Our treatment is by no means comprehensive and we refer the readers to [74, Chapter 8-10] for a more elaborate study. Our short introduction closely follows [73] and [47]. More precisely, Sections 2.1–2.3 are based on [73], and Section 2.4 is based on [47].

The plurifine topology of an open set $U\subset\mathbb{C}^{n}$ is the coarsest topology making all plurisubharmonic (psh) functions on $U$ continuous. Clearly, the resulting topology on $U$ is finer then the Euclidean one. The motivating result behind this notion is that of Bedford–Taylor [3], according to which the complex Monge–Ampère operator is local with respect to the plurifine topology, i.e., if $\phi^{j},\psi^{j}\in\textup{PSH}\cap L^{\infty}_{\textup{loc}}(U)$ and $V\subset U$ is plurifine open such that $\phi^{j}|_{V}=\psi^{j}|_{V}$, then

[TABLE]

When dealing with a $n$ dimensional Kähler manifold $(X,\omega)$, the largest class of potentials one can consider is that of $\omega$-psh functions:

[TABLE]

In concrete terms, $u\in\textup{PSH}(X,\omega)$ means that $g+u|_{U}\in\textup{PSH}(U)$, for all open sets $U\subset X$ on which the metric $\omega$ can be written as $\omega=i\partial\bar{\partial}g,\ g\in C^{\infty}(U)$. We immediately get that $u$ is usc and $u\in L^{1}(X)$.

Given the local nature of (2.1), it generalizes to Kähler manifolds in a straightforward manner, and in particular we have the following identity for $u^{j},v\in\textup{PSH}(X,\omega)\cap L^{\infty}$ that we will use numerous times below:

[TABLE]

2.1 Full mass $\omega$-psh functions

One of the cornerstones of Bedford–Taylor theory is associating a complex Monge–Ampère measure to bounded psh functions. Their construction adapts to elements of $\textup{PSH}(X,\omega)\cap L^{\infty}$ in a straightforward manner.

As it turns out, it is possible to generalize the Bedford–Taylor construction to all elements of $\textup{PSH}(X,\omega)$, as we describe now. First we consider the case of products. Let $v^{j}\in\textup{PSH}(X,\omega),\ j\in\{1,\ldots,n\}$ and by $v^{j}_{h}=\max(v^{j},-h)\in\textup{PSH}(X,\omega)\cap L^{\infty},\ h\in\mathbb{R}$ we denote the canonical cutoffs of $v^{j}$. If $h_{1}<h_{2}$ then (2.2) implies that

[TABLE]

hence $\{\mathbbm{1}_{\cap_{j}\{v^{j}>-h\}}\omega_{v^{1}_{h}}\wedge\ldots\wedge\omega_{v^{n}_{h}}\}_{h}$ is an increasing sequence of Borel measures on $X$. This leads to the notion of the non-pluripolar product $\omega_{v^{1}}\wedge\ldots\wedge\omega_{v^{n}}$, that generalizes the corresponding Bedford–Taylor construction:

[TABLE]

When $v=v^{1}=\ldots=v^{n}$, the above construction yields $\omega^{n}_{u}:=\omega_{u}\wedge\ldots\wedge\omega_{u}$, the non-pluripolar complex Monge-Ampere measure of $v$:

[TABLE]

By the above definition $\int_{B}\omega_{v}^{n}=\lim_{h\to\infty}\int_{B}\mathbbm{1}_{\{v>-h\}}\omega_{v_{h}}^{n}$ for all Borel sets $B\subset X$. This is stronger than simply saying that $\omega_{v}^{n}$ is the weak limit of $\mathbbm{1}_{\{v>-h\}}\omega_{v_{h}}^{n}$.

Before we examine this construction more closely, let us recall a few facts about smooth elements of $\textup{PSH}(X,\omega)$. Of great importance in this work is the set of smooth Kähler potentials:

[TABLE]

Clearly, $\mathcal{H}_{\omega}\subset\textup{PSH}(X,\omega)$ and in fact any element of $\textup{PSH}(X,\omega)$ can be approximated by a decreasing sequence in $\mathcal{H}_{\omega}$:

Theorem 2.1** ([24]).**

Given $u\in\textup{PSH}(X,\omega)$, there exists a decreasing sequence $\{u_{k}\}_{k}\subset\mathcal{H}_{\omega}$ such that $u_{k}\searrow u$.

We give a proof of this result in Appendix A.2. Among other things, this theorem implies that the total volume of $X$ is the same for all currents $\omega_{u}$ with bounded potential:

Lemma 2.2**.**

if $v\in\textup{PSH}(X,\omega)\cap L^{\infty}$, then $\int_{X}\omega_{v}^{n}=\int_{X}\omega^{n}=:\textup{Vol}(X)$.

Proof.

By an application of Stokes theorem, the statement holds for $v\in\mathcal{H}_{\omega}$. The general result follows after we approximate $v\in\textup{PSH}(X,\omega)\cap L^{\infty}$ with a decreasing sequence $v_{k}\in\mathcal{H}_{\omega}$ (Theorem 2.1), and we use Bedford–Taylor theory ([20, Theorem 2.2.5]) to conclude that $\omega_{v_{k}}^{n}\to\omega_{v}^{n}$ weakly. ∎

In contrast with the above, given our definition (2.3), it is clear that we only have $\int_{X}\omega_{v}^{n}\leq\textup{Vol}(X)$ for $v\in\textup{PSH}(X,\omega)$. This leads to the natural definition of $\omega$-psh functions of full mass:

[TABLE]

Though $\textup{PSH}(X,\omega)\cap L^{\infty}\subsetneq\mathcal{E}(X,\omega)$ (see the examples of Section 2.3), many of the properties that hold for bounded $\omega$-psh functions, still hold for (unbounded) elements of $\mathcal{E}(X,\omega)$ as well, like the comparison principle:

Proposition 2.3**.**

[73, Theorem 1.5]* Suppose $u,v\in\mathcal{E}(X,\omega)$. Then*

[TABLE]

Proof.

First we show (2.5) for $u,v$ bounded. Using (2.2) we can write:

[TABLE]

Using Lemma 2.2 and (2.2) we can continue to write:

[TABLE]

Noticing that $\{v<u\}=\cup_{\varepsilon>0}\{v+\varepsilon<u\}=\cup_{\varepsilon>0}\{v+\varepsilon\leq u\}$, we can replace $v$ with $v+\varepsilon$ in the above inequality and let $\varepsilon\to 0$ to obtain (2.5) for bounded potentials.

In general, let $u_{l}=\max(u,-l),v_{k}=\max(u,-k),\ l,k\in\mathbb{N}$ be the canonical cutoffs of $u,v$. For these, by the above we have $\int_{\{v_{l}<u_{k}\}}\omega_{u_{k}}^{n}\leq\int_{\{v_{l}<u_{k}\}}\omega_{v_{l}}^{n}$. This inequality together with the inclusions $\{v_{l}<u\}\subset\{v_{l}<u_{k}\}\subset\{v<u_{k}\}$ gives

[TABLE]

We can let $l\to\infty$, and using the definition of $\omega_{v_{l}}^{n}$ from (2.3) it follows that $\int_{\{v<u\}}\omega_{u_{k}}^{n}\leq\int_{\{v<u_{k}\}}\omega_{v}^{n}.$ Now letting $k\to\infty$ and using again (2.3) we obtain

[TABLE]

∎

Of great importance in what follows will be the monotonicity property of $\mathcal{E}(X,\omega)$:

Proposition 2.4**.**

[73, Proposition 1.6]* Suppose $u\in\mathcal{E}(X,\omega)$ and $v\in\textup{PSH}(X,\omega)$. If $u\leq v$ then $v\in\mathcal{E}(X,\omega)$.*

Proof.

Set $\psi:=v/2$. We can assume without loss of generality that $u\leq v<-2$, hence $\psi<-1$. This choice of normalization allows to write the following inclusions for the canonical cutoffs $u_{j},v_{j},\psi_{j}$:

[TABLE]

Using these inclusions and the comparison principle (Proposition 2.3) we conclude:

[TABLE]

Using Lemma 2.2 and (2.2) we get $\int_{\{u_{2j}\leq-j\}}\omega_{u_{2j}}^{n}=\int_{\{u_{j}\leq-j\}}\omega_{u_{j}}^{n}=\int_{\{u\leq-j\}}\omega_{u_{j}}^{n}$, hence $\int_{\{\psi_{j}\leq-j\}}\omega_{\psi_{j}}^{n}\to 0$ as $j\to\infty$, implying that $\psi=v/2\in\mathcal{E}(X,\omega)$.

Now we argue that in fact $v\in\mathcal{E}(X,\omega)$. For the canonical cutoffs we have the identity $v_{2j}/2=\psi_{j},\ j\in\mathbb{N}$, hence we get the estimate $\omega_{\psi_{j}}=\omega/2+\omega_{v_{2j}}/2\geq\omega_{v_{2j}}/2$. This allows to conclude that

[TABLE]

hence the left hand integral converges to zero as $j\to\infty$, since $\psi\in\mathcal{E}(X,\omega)$. Using (2.3), this implies that $\int_{X}\omega_{v}^{n}=\textup{Vol}(X)$, i.e., $v\in\mathcal{E}(X,\omega)$ as desired. ∎

Lastly we note that formula (2.2) generalizes to full mass $\omega$-psh functions as well:

Lemma 2.5**.**

For $u,v\in\mathcal{E}(X,\omega)$ we have $\mathbbm{1}_{\{u>v\}}\omega_{\max(u,v)}^{n}=\mathbbm{1}_{\{u>v\}}\omega_{u}^{n}$.

Proof.

Proposition 2.4 implies that $\alpha:=\max(u,v)\in\mathcal{E}(X,\omega)$. For the canonical cutoffs we observe that $\max(u_{j},v_{j+1})=\max(u,v,-j)=\alpha_{j}$. As the cutoffs are bounded we have

[TABLE]

The definition of $\omega_{u}^{n}$ and $\omega_{\alpha}^{n}$ (2.3) implies that $\mathbbm{1}_{\{u>v\}}\omega_{u_{j}}^{n}\to\mathbbm{1}_{\{u>v\}}\omega_{u}^{n}$ and $\mathbbm{1}_{\{u>v\}}\omega_{\alpha_{j}}^{n}\to\mathbbm{1}_{\{u>v\}}\omega_{\alpha}^{n}$. Since $\{u>v\}\subset\{u_{j}>v_{j+1}\}$ and $\{u_{j}>v_{j+1}\}\setminus\{u>v\}\subset\{u\leq-j\}$, it follows that

[TABLE]

Since $\{u_{j}>v_{j+1}\}\setminus\{u>v\}\subset\{\max(u,v)\leq-j\}$ we also obtain that $0\leq(\mathbbm{1}_{\{u>v\}}-\mathbbm{1}_{\{u_{j}>v_{j+1}\}})\omega_{\alpha_{j}}^{n}\leq\mathbbm{1}_{\{\alpha\leq-j\}}\omega_{\alpha_{j}}^{n}\to 0.$ Combining these last facts with (2.6), and taking the limit, we arrive at the desired result. ∎

2.2 Finite energy classes

By considering weights $\chi\in\mathcal{W}^{+}_{p}$, one can introduce various finite energy subclasses of $\mathcal{E}(X,\omega)$, important in our later geometric study:

[TABLE]

where $E_{\chi}$ is the $\chi$–energy defined by the expression:

[TABLE]

Recall from Chapter 1 that the condition $E_{\chi}(u)<\infty$ is equivalent to $u\in L^{\chi}(\omega_{u}^{n})$. Of special importance are the weights $\chi_{p}(t):=|t|^{p}/p$ and the associated finite energy classes

[TABLE]

By the next result, to test membership in $\mathcal{E}_{\chi}(X,\omega)$ it is enough to test the finiteness condition $E_{\chi}(u)<\infty$ on the canonical cutoffs:

Proposition 2.6**.**

[73, Proposition 1.4]* Suppose $u\in\mathcal{E}(X,\omega)$ with canonical cutoffs $\{u_{k}\}_{k\in\mathbb{N}}$. If $h:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is continuous and increasing then*

[TABLE]

Additionally, if the above conditions hold, then $\int_{X}h(|u|)\omega_{u}^{n}=\lim_{k\to\infty}\int_{X}h(|u_{k}|)\omega_{u_{k}}^{n}.$

Proof.

If $\limsup_{k}\int_{X}h(|u_{k}|)\omega_{u_{k}}^{n}<C$ then the family of measures $\{h(\max(0,|u_{k}|))\omega_{u_{k}}^{n}\}_{k}$ is precompact, hence one can extract a weakly converging subsequence of measures $\{h(\max(0,|u_{k_{j}}|))\omega_{u_{k_{j}}}^{n}\}_{k_{j}}\to\mu$ with $\mu(X)<C$. By the definition of $\omega_{u}^{n}$ (2.3) it follows that $\omega_{u_{k_{j}}}^{n}\to\omega_{u}^{n}$ weakly. Since $\{h(\max(0,|u_{k_{j}}|))\}_{k_{j}}$ is a sequence of lsc functions increasing to $h(\max(0,|u|))$, a standard measure theoretic lemma implies that $\int_{X}h(\max(0,|u|))\omega_{u}^{n}\leq\mu(X)$ (see [20, Lemma A2.2]). This implies that $\int_{X}h(|u|)\omega_{u}^{n}$ is finite.

Now assume that $\int_{X}h(|u|)\omega_{u}^{n}\leq C$. By (2.2) and the definition of $\omega_{u}^{n}$ it follows that $\mathbbm{1}_{\{u>-k\}}\omega_{u}^{n}=\mathbbm{1}_{\{u>-k\}}\omega_{u_{k}}^{n}$, implying that $\int_{\{u\leq-k\}}\omega_{u_{k}}^{n}=\int_{\{u\leq-k\}}\omega_{u}^{n}$. As a consequence, we can write the following:

[TABLE]

Since $\int_{\{u=-\infty\}}h(|u|)\omega_{u}^{n}=0$, it follows that $\int_{X}h(|u_{k}|)\omega_{u_{k}}^{n}$ is bounded above and moreover $\int_{X}h(|u_{k}|)\omega_{u_{k}}^{n}\to\int_{X}h(|u|)\omega_{u}^{n}$. ∎

With the aid of the previous proposition, we can prove our next result, sometimes called the “fundamental estimate”:

Proposition 2.7**.**

[73, Lemma 3.5]* Suppose $\chi\in\mathcal{W}^{+}_{p}$ and $u,v\in\mathcal{E}_{\chi}(X,\omega)$ satisfies $u\leq v\leq 0$. Then*

[TABLE]

Proof.

First assume that $u,v\in\mathcal{H}_{\omega}$. We first show the following estimate:

[TABLE]

To show this estimate, integration by parts gives

[TABLE]

The first integral on the right hand side is bounded above by $\int_{X}\chi(u)\omega_{v}^{j}\wedge\omega_{u}^{n-j}$. Indeed, as $u\leq 0$ we have $\int_{X}i\chi^{\prime}(u)\partial u\wedge\bar{\partial}u\wedge\omega_{v}^{j}\wedge\omega_{u}^{n-j-1}\leq 0$.

Concerning the second integral on the right hand side of (2.11) we notice that $i\partial\bar{\partial}\chi(u)=i\chi^{\prime\prime}(u)\partial u\wedge\bar{\partial}u+\chi^{\prime}(u)i\partial\bar{\partial}u\geq\chi^{\prime}(u)\omega_{u}$, as currents. Indeed, this is straitforward for smooth $\chi$, and in general it is possible to use approximation.

Since $v\leq 0$ we can write

[TABLE]

where in the last inequality we have used that $\chi\in\mathcal{W}^{+}_{p}$. Combining the above with (2.11) yields (2.10). Iterating (2.10) $n$ times and using the fact that $u\leq v$ gives (2.9) for $u,v\in\mathcal{H}_{\omega}$.

In case $u,v\in\textup{PSH}(X,\omega)\cap L^{\infty}$, by adding small constants, we can assume that $u<v<0$. We get $E_{\chi}(v_{k})\leq(p+1)^{n}E_{\chi}(u_{k})$, where $u_{k},v_{k}\in\mathcal{H}_{\omega}$ are decreasing smooth approximants of $u,v$, given by Theorem 2.1, satisfying $u_{k}\leq v_{k}\leq 0$. An application of [74, Theorem 4.26] now gives the result for $u,v$ bounded.

In case $u,v\in\mathcal{E}_{\chi}(X,\omega)$, we know that $E_{\chi}(u_{k})\leq(p+1)^{n}E_{\chi}(v_{k})$ for the canonical cutoffs $u_{k}:=\max(u,-k),v_{k}:=\max(v,-k)$. Proposition 2.6 allows to take the limit $k\to\infty$, to obtain (2.9). ∎

As a corollary of this result, we obtain the monotonicity property for $\mathcal{E}_{\chi}(X,\omega)$:

Corollary 2.8**.**

Suppose $u\in\mathcal{E}_{\chi}(X,\omega)$ and $v\in\textup{PSH}(X,\omega)$. If $u\leq v$ then $v\in\mathcal{E}_{\chi}(X,\omega)$.

Proof.

We can assume without loss of generality that $u\leq v\leq 0$. Proposition 2.4 implies that $v\in\mathcal{E}(X,\omega)$. Also for the canonical cutoffs $v_{k}$ we have $u\leq v_{k}$, hence $E_{\chi}(v_{k})\leq(p+1)^{n}E_{\chi}(u)$ for all $k\in\mathbb{N}$. Proposition 2.6 now gives that $E_{\chi}(v)\leq(p+1)^{n}E_{\chi}(u)$, finishing the proof. ∎

The next result says that if $u,v\in\mathcal{E}_{\chi}(X,\omega)$, then in fact $u\in L^{\chi}(\omega_{v}^{n})$:

Proposition 2.9**.**

[73, Proposition 3.6]* Suppose $u,v\in\mathcal{E}_{\chi}(X,\omega),\ \chi\in\mathcal{W}^{+}_{p}$. If $u,v\leq 0$ then*

[TABLE]

Proof.

We first show that $\chi^{\prime}(2t)\leq p2^{p-1}\chi^{\prime}(t),\ t>0$. For this, after possibly adding $\delta|t|$ to $\chi(t)$, we can momentarily assume that $\chi^{\prime}(t),\chi(t)>0$ for any $t>0$.

By convexity we have $\chi(t)/t\leq\chi^{\prime}(t)$, and (1.8) gives $\chi(2t)/\chi(t)\leq 2^{p}$, hence we can write:

[TABLE]

Consequently we have the following sequence of inequalities:

[TABLE]

Noticing that $\{u<-2t\}\subset\{u<-t+v\}\cup\{v<-t\}$ we can continue to write:

[TABLE]

where in the second line we have used Proposition 2.3 and in the last line we have used that $\{u<-t+v\}\subset\{u<-t\}$. ∎

We note the following result about decreasing approximants and the class $\mathcal{E}_{\chi}(X,\omega)$:

Lemma 2.10**.**

[73, Proposition 5.6]* Suppose $\chi\in\mathcal{W}^{+}_{p}$ and $\{u_{k}\}_{k\in\mathbb{N}}\subset\mathcal{E}_{\chi}(X,\omega)$ is a sequence decreasing to $u\in\textup{PSH}(X,\omega)$. If $\sup_{k}E_{\chi}(u_{k})<\infty$ then $u\in\mathcal{E}_{\chi}(X,\omega)$ and*

[TABLE]

Proof.

Let $u^{l}_{k}=\max(u_{k},-l)$ and $u^{l}=\max(u,-l)$ be the canonical cutoffs. As $-l\leq u^{l}\leq u^{k}_{l}$, by Proposition 2.7 and [74, Theorem 4.26] we get that $E_{\chi}(u^{l})=\lim_{k}E_{\chi}(u^{l}_{k})\leq(p+1)^{n}\limsup_{k}E_{\chi}(u_{k}).$ Finally, Proposition 2.6 gives that $u\in\mathcal{E}_{\chi}(X,\omega)$, along with the desired estimate. ∎

2.3 Examples and singularity type of finite energy potentials

It is clear that $\textup{PSH}(X,\omega)\cap L^{\infty}\subset\mathcal{E}_{p}(X,\omega)$ for all $p\geq 1$. In this short section we will show that this inclusion is always strict, as one can construct unbounded elements of $\mathcal{E}_{p}(X,\omega)$. However the content of our first result is that the singularity type of full mass potentials is always mild, even when they are unbounded.

Given $u\in\textup{PSH}(X,\omega)$ and $x_{0}\in X$ it is possible to measure the local singularity of $u$ at $x_{0}$ using the Lelong number $\mathcal{L}(u,x_{0})$, whose definition we now recall:

[TABLE]

where $U_{r}\subset\mathbb{C}^{n}$ is some coordinate neighborhood of $x_{0}$ that identifies $x_{0}$ with $0\in\mathbb{C}^{n}$. Roughly speaking, $\mathcal{L}(u,x_{0})$ measures the extent to which the singularity of $u$ at $x_{0}$ is logarithmic. For an extensive treatment of Lelong numbers we refer to [74, Section 2.3].

To begin, we observe that elements of $\mathcal{E}(X,\omega)$ have singularity so mild that Lelong numbers can not detect them:

Proposition 2.11**.**

[73, Corollary 1.8]* If $u\in\mathcal{E}(X,\omega)$ then all Lelong numbers of $u$ are zero.*

Proof.

Let $x_{0}\in X$ and $U\subset X$ a coordinate neighborhood of $x_{0}$, identifying $x_{0}$ with $0\in\mathbb{C}^{n}$ via a biholomorphism $\varphi:B(0,2)\to U$. Let $v\in\textup{PSH}(X,\omega)$ such that $v$ is smooth on $X\setminus\{x_{0}\}$ and $v\circ\varphi|_{B(0,1)}=c\log|x|$ for some $c>0$. Using a partition of unity, such $v$ can be easily constructed.

If $\mathcal{L}(u,x_{0})>0$, then for some $\varepsilon>0$ we will have $u\leq\varepsilon v+1/\varepsilon$. The monotonicity property (Proposition 2.4) now implies that $\varepsilon v\in\mathcal{E}(X,\omega)$. However this cannot happen, yielding a contradiction. Indeed, by the definition of $\omega_{\varepsilon v}^{n}$ (see (2.3)) we have that

[TABLE]

where in the last line we have used that \big{(}i\partial\bar{\partial}\max(\log|z|,\log r)\big{)}^{n}=(2\pi)^{n}d\sigma_{\partial B(0,r)}, where $d\sigma_{\partial B(0,r)}$ is the Euclidean probability surface measure of $\partial B(0,r)$ (see the exercise following [20, Corollary 2.2.7]). ∎

The purpose of our next proposition is to give a flexible construction for unbounded elements of $\mathcal{E}_{p}(X,\omega)$:

Proposition 2.12**.**

[73, Example 2.14], [21, Proposition 5]* Suppose that $u\in\textup{PSH}(X,\omega)$, $u<-1$ and $\alpha\in(0,1/2)$. Then $v=-(-u)^{\alpha}\in\mathcal{E}_{p}(X,\omega)$ for all $p\in[1,(1-\alpha)/\alpha)$.*

Proof.

First assume that $u\in\mathcal{H}_{\omega}$ and $u\leq-1$. For $v=-(-u)^{\alpha}$ we have

[TABLE]

Consequently $v\in\mathcal{H}_{\omega}$, and for some $C:=C(\alpha)>1$ we have

[TABLE]

For arbitrary $a>0$, integration by parts gives the following estimates

[TABLE]

where in the last inequality we have used that $(-u)^{-a}\leq 1$. This estimate and (2.14) implies that for any $b\in(0,1-\alpha)$ we have

[TABLE]

Returning to the general case $u\in\textup{PSH}(X,\omega)$, let $\{u_{k}\}_{k}\subset\mathcal{H}_{\omega}$ be a sequence of smooth potentials decreasing to $u$ (Theorem 2.1). We can assume that $u_{k}<-1$, hence the above inequality implies that $\int_{X}|v_{k}|^{b/\alpha}\omega_{v_{k}}^{n}$ is uniformly bounded for $v_{k}:=-(-u_{k})^{\alpha}$. Consequently, Lemma 2.10 implies that $v\in\mathcal{E}_{p}(X,\omega)$ for any $p\in(0,(1-\alpha)/\alpha)$. ∎

2.4 Envelopes of finite energy classes

Envelope constructions are ubiquitous throughout pluripotential theory. In our setting, given an usc function $f:X\to[-\infty,\infty)$, the simplest envelope one can consider is

[TABLE]

As we know, the supremum of a family of $\omega$-psh functions may not be $\omega$-psh, as $P(f)$ may not be usc to begin with. However by [20, Theorem 1.2.3] the usc regularization $P(f)^{*}$ is indeed $\omega$-psh. As $f$ is usc, we obtain that $P(f)^{*}\leq f^{*}=f$, immediately giving that $P(f)^{*}$ is a candidate in the definition of $P(f)$, hence $P(f)=P(f)^{*}$, i.e., $P(f)\in\textup{PSH}(X,\omega)$.

Slightly generalizing the above concept, for usc functions $\{f_{1},f_{2},\ldots,f_{k}\}$ we introduce the rooftop envelope

[TABLE]

When $f$ is smooth (or just continuous), as $P(f)$ is usc, we obtain that the non-contact set $\{f>P(f)\}\subset X$ is open. A classical Perron type argument (see [3, Corollary 9.2] or [20, Proposition 1.4.10]). yields that

[TABLE]

This observation suggests that $P(f)$ can have at most bounded but not continuous second derivatives. This is mostly confirmed by the next result, whose proof is provided in the appendix:

Theorem 2.13** (Theorem A.12, [56, Theorem 2.5]).**

Given $f_{1},...,f_{k}\in C^{\infty}(X)$, then $P(f_{1},f_{2},...,f_{k})\in C^{1,\alpha}(X),\ \alpha\in(0,1)$. More precisely, the following estimate holds:

[TABLE]

By a bound on the $C^{1,\bar{1}}$ norm of $P(f)$ we mean a uniform bound on all mixed second order derivatives $\partial^{2}P(f)/\partial z_{j}\bar{\partial}z_{k}$. Since $P(f)$ is $\omega$-psh, this is equivalent to saying that $\Delta^{\omega}P(f)$ is bounded, and by the Calderon–Zygmund estimate [75, Chapter 9, Lemma 9.9], we automatically obtain that $P(f_{1},f_{2},\ldots,f_{k})\in C^{1,\alpha}(X),\alpha<1$. We introduce the following subspace of $\textup{PSH}(X,\omega)$:

[TABLE]

When $k=1$, the above result was first proved in [15] using the Kiselman technique for attenuation of singularities. An independent “PDE proof” has been given by Berman [6], and we present this in the appendix (see Theorem A.7). The proof of the general case $k\geq 1$ was given in the paper [56], that provided a detailed regularity analysis for the rooftop envelopes introduced above. By reduction to the case $k=1$, we prove this more general result in the appendix as well (see Theorem A.12). Lastly, we mention that very recently it was shown by Tosatti [111] and Chu-Zhou [44] that in fact it is possible to bound the full Hessian of $P(f_{1},\ldots,f_{k})$ in terms of the Hessians of $f_{1},\ldots,f_{k}$.

With the above introduced notation, we will use Theorem 2.13 in the following form:

Corollary 2.14**.**

If $u_{0},u_{1},\ldots,u_{k}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ then $P(u_{0},u_{1},\ldots,u_{k})\in\mathcal{H}_{\omega}^{1,\bar{1}}$.

Proof.

Let $f_{j}^{i}\in C^{\infty}(X)$ be such that $f_{j}^{i}\to u_{j}$ uniformly, and the mixed second derivatives of $f_{j}^{i}$ are uniformly bounded. By the previous theorem, the mixed second derivatives of $P(f^{i}_{0},f^{i}_{1},\ldots,f^{i}_{k})$ are uniformly bounded as well. Since $P(f^{i}_{0},f^{i}_{1},\ldots,f^{i}_{k})\to P(u_{0},u_{1},\ldots,u_{k})$ uniformly, the result follows. ∎

As a consequence of the above corollary, we get a volume partition formula for $\omega_{P(u_{0},u_{1})}^{n}$:

Proposition 2.15**.**

[47, Proposition 2.2]* For $u_{0},u_{1}\in\mathcal{H}_{\omega}^{1,\bar{1}}$, we introduce the contact sets $\Lambda_{u_{0}}=\{P(u_{0},u_{1})=u_{0}\}$ and $\Lambda_{u_{1}}=\{P(u_{0},u_{1})=u_{1}\}$. Then the following partition formula holds:*

[TABLE]

Proof.

As pointed out in (2.16), $\omega_{P(u_{0},u_{1})}^{n}$ is concentrated on the coincidence set $\Lambda_{u_{0}}\cup\Lambda_{u_{1}}.$ Having bounded Laplacian implies that all second order partials of $P(u_{0},u_{1})$ are in any $L^{p}(X),\ p<\infty$ [75, Chapter 9, Lemma 9.9]. It follows from [75, Chapter 7, Lemma 7.7] that on $\Lambda_{u_{0}}$ all the second order partials of $P(u_{0},u_{1})$ and $u_{0}$ agree a.e., and the analogous statement holds on $\Lambda_{u_{1}}$. Hence, using [20, Proposition 2.1.6] one can write:

[TABLE]

finishing the proof. ∎

The partition formula (2.18) is at the core of many theorems presented later in this survey. Interestingly, it fails to hold evein in the slightly more general case of Lipschitz potentials

[TABLE]

For a counterexample, suppose $\dim X=1$ and $g_{x}$ is the $\omega-$Green function with pole at $x\in X$. Such function is characterized by the property $\int_{X}g_{x}\omega=0$ and $\omega+i\partial\bar{\partial}g_{x}=\delta_{x}$. We choose $u_{0}=\max\{g_{x},0\}$ and $u_{1}=0$. In this case $P(u_{0},u_{1})=0,\ \Lambda_{u_{0}}=\{g_{x}\leq 0\}$ and $\Lambda_{u_{1}}\setminus\Lambda_{u_{0}}=X\setminus\Lambda_{u_{0}}\neq\emptyset$. As $\textup{Vol}(X)=\int_{\Lambda_{u_{0}}}\omega_{u_{0}}^{n}=\int_{X}\omega_{P(u_{0},u_{1})}^{n},$ it is seen that the right hand side of (2.18) has total integral greater then the left hand side, hence they can not equal.

As $\int_{X}\omega_{u_{1}}^{n}$ is finite, it follows that $\omega_{u_{1}}^{n}(\{u_{0}=u_{1}+\tau\})>0$ only for a countable number of values $\tau\in\mathbb{R}$. Consequently, (2.18) implies the following observation:

Remark 2.16**.**

Given $u_{0},u_{1}\in\mathcal{H}_{\omega}^{1,\bar{1}}$, for any $\tau\in\mathbb{R}$ outside a countable set we have:

[TABLE]

In particular, $\textup{Vol}(X)=\int_{\{P(u_{0},u_{1}+\tau)=u_{0}\}}\omega_{u_{0}}^{n}+\int_{\{P(u_{0},u_{1}+\tau)=u_{1}+\tau\}}\omega_{u_{1}}^{n}$.

Corollary 2.14 simply says that the operation $(u_{0},u_{1})\to P(u_{0},u_{1})$ is closed inside the class $\mathcal{H}_{\omega}^{1,\bar{1}}$. The next proposition tells that the same holds inside the finite energy classes as well:

Proposition 2.17**.**

[47, Lemma 3.4]* Suppose $\chi\in\mathcal{W}^{+}_{p}$ and ${u_{0}},{u_{1}}\in\mathcal{E}_{\chi}(X,\omega)$. Then $P(u_{0},u_{1})\in\mathcal{E}_{\chi}(X,\omega)$, and if $u_{0},u_{1}\leq 0$ then following estimate holds:*

[TABLE]

Proof.

As $P(u_{0}-c,u_{1}-c)=P(u_{0},u_{1})-c$ for $c\in\mathbb{R}$, it follows that without loss of generality we can assume that $u_{0},u_{1}<0$.

By Theorem 2.1, it is possible to find negative potentials $u^{j}_{0},u^{j}_{1}\in\mathcal{H}_{\omega}$ that decrease to $u_{0},u_{1}$. Furthermore, by Proposition 2.15 we can also assume that $P(u^{j}_{0},u^{j}_{1})\in\mathcal{H}_{\omega}^{1,\bar{1}}$ satisfies

[TABLE]

Using this formula we can write:

[TABLE]

where in the last line we have used Proposition 2.7. As $P(u^{j}_{0},u^{j}_{1})$ decreases to $P(u_{0},u_{1})$, by Lemma 2.10 we have $P(u_{0},u_{1})\in\mathcal{E}_{\chi}(X,\omega)$, and (2.19) holds. ∎

Observe that $tu_{0}+(1-t)u_{1}\geq P(u_{0},u_{1})$ for any $t\in[0,1]$, hence as a consequence of the previous proposition and the monotonicity property of $\mathcal{E}_{\chi}(X,\omega)$ (Corollary 2.8) we obtain that $\mathcal{E}_{\chi}(X,\omega)$ is convex:

Corollary 2.18**.**

If $u_{0},u_{1}\in\mathcal{E}_{\chi}(X,\omega)$ then $tu_{0}+(1-t)u_{1}\in\mathcal{E}_{\chi}(X,\omega)$ for any $t\in[0,1]$.

Before we can establish the continuity property of the Monge–Ampère operator along monotonic sequences of potentials, we need to establish the following auxilliary result, which states that if $u\in\mathcal{E}_{\chi}(X,\omega)$, then it is possible to find a $\tilde{\chi}$ with bigger growth than $\chi$ such that $u\in\mathcal{E}_{\tilde{\chi}}(X,\omega)$ still holds:

Lemma 2.19**.**

Suppose $u\in\mathcal{E}_{\chi}(X,\omega),\ \chi\in\mathcal{W}^{+}_{p}$. Then there exists $\tilde{\chi}\in\mathcal{W}^{+}_{2p+1}$ such that $\chi(t)\leq\tilde{\chi}(t)$, $\chi(t)/\tilde{\chi}(t)\searrow 0$ as $t\to\infty$, and $u\in\mathcal{E}_{\tilde{\chi}}(X,\omega)$.

Proof.

The weight ${\tilde{\chi}}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ will be constructed as an increasing limit of weights ${\tilde{\chi}}^{j}\in\mathcal{W}^{+}_{2p+1},j\in\mathbb{N}$ constructed below. We set ${\tilde{\chi}}_{0}(t)=\chi$. Let $t_{0}\in\mathbb{R}^{+}$, to be specified later. We define ${\tilde{\chi}}_{1}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ by the formula

[TABLE]

Notice that ${\tilde{\chi}}_{1}$ satisfies $\chi\leq\tilde{\chi}_{1}$ and the following also hold

[TABLE]

[TABLE]

We can choose $t_{1}$ to be big enough such that $E_{{\tilde{\chi}}_{1}}(u)<E_{\chi}(u)+1$.

Now pick $t_{2}>t_{1}$, again specified later. One defines ${\tilde{\chi}}_{2}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ in a similar manner:

[TABLE]

As $\eqref{CompWeightEst2}$ holds for ${\tilde{\chi}}_{1}$, it is possible to choose $t_{2}>t_{1}$ big enough so that the ${\tilde{\chi}}_{2}$-analogs of (2.21),(2.22) are satisfied and $E_{{\tilde{\chi}}_{2}}(u)<E_{\chi}(u)+1$.

We define ${\tilde{\chi}}_{k},k\in\mathbb{N}$, following the above procedure. As $\lim_{t\to\infty}{\tilde{\chi}}_{k}(t)/\chi(t)=2^{k}$, the limit weight ${\tilde{\chi}}(t)=\lim_{k\to\infty}{\tilde{\chi}}_{k}(t)$ is seen to satisfy the requirements of the lemma. ∎

The following result, allowing to take weak limits of certain measures, will be used in many different contexts throughout the survey:

Proposition 2.20**.**

*Assume that $\{\phi_{k}\}_{k\in\mathbb{N}},\{\psi_{k}\}_{k\in\mathbb{N}},\{v^{j}_{k}\}_{k\in\mathbb{N}}\subset\mathcal{E}_{\chi}(X,\omega)$ decrease (increase a.e.) to $\phi,\psi,v^{j}\in\mathcal{E}_{\chi}(X,\omega)$ respectively, $j\in\{1,\ldots,n\}$. Suppose that

(i) $\psi_{k}\leq\phi_{k}$;

(ii) $h:\mathbb{R}\to\mathbb{R}$ is continuous with $\limsup_{|l|\to\infty}\frac{|h(l)|}{\chi(l)}\leq C$ for some $C>0$;

Then $h(\phi_{k}-\psi_{k})\omega_{v^{1}_{k}}\wedge\ldots\wedge\omega_{v^{n}_{k}}\to h(\phi-\psi)\omega_{v^{1}}\wedge\ldots\wedge\omega_{v^{n}}$ weakly.*

We will apply this porposition mostly for $h=\chi$ and $h=1$. In the latter case this proposition simply tells that the non-pluripolar products, as defined in (2.3), converge weakly along monotonic sequences of $\mathcal{E}_{\chi}(X,\omega)$.

Remark. Though in this work we will only deal with $\chi\in\mathcal{W}^{+}_{p}$, the same argument yields the result for concave weights $\chi\in\mathcal{W}^{-}$ (introduced in [73]) as well.

Proof.

Let $\gamma\in C^{\infty}(X).$ We can suppose without loss of generality that all the potentials involved are negative. First we suppose that there exists $L>1$ such that $-L<\phi,\phi_{k},\psi,\psi_{k},v,v_{k}<0$, and prove the theorem under this assumption. [74, Theorem 4.26] gives this immediately, but we give a more elementary argument instead. Given $\varepsilon>0$ one can find an open $O\subset X$ such that $\textup{Cap}_{X}(O)<\varepsilon$ and $\phi,\phi_{k},\psi,\psi_{k},v^{j},v^{j}_{k}$ are all continuous on $X\setminus O$ ([20, Theorem 2.2] or [72, Definition 2.4, Corollary 2.8]). We have

[TABLE]

The integral on $O$ is bounded by $\varepsilon,$ where $C:=C(L,\gamma)$. The second integral tends to [math] as on the closed set $X\setminus O$ we have $\phi_{k}\to\phi$ and $\psi_{k}\to\psi$ uniformly. We also have

[TABLE]

as the function $h({\phi-\psi})$ is quasi–continuous and bounded. Indeed, quasi–continuity and boundedness implies again that for all $\varepsilon>0$ one can find an open $\tilde{O}\subset X$ such that $\textup{Cap}_{X}(\tilde{O})<\varepsilon$ and $h({\phi-\psi})$ is continuous on $X\setminus\tilde{O}$. Furthermore, by Tietze’s extension theorem we can extend $h({\phi-\psi})|_{X\setminus\tilde{O}}$ to a continuous function $\alpha$ on $X$. As $\textup{Cap}_{X}(\tilde{O})<\varepsilon$, we have that $\int_{X}|h({\phi-\psi})-\alpha|\omega_{v^{1}_{k}}\wedge\ldots\wedge\omega_{v^{n}_{k}}\leq C\varepsilon$ and $\int_{X}|h({\phi-\psi})-\alpha|\omega_{v^{1}}\wedge\ldots\wedge\omega_{v^{n}}\leq C\varepsilon$. On the other hand $\int_{X}\alpha\omega_{v^{1}_{k}}\wedge\ldots\wedge\omega_{v^{n}_{k}}\to\int_{X}\alpha\omega_{v^{1}}\wedge\ldots\wedge\omega_{v^{n}}$ by Bedford-Taylor theory (see [20, Theorem 2.2.5]). Putting these facts together we get (2.24). Finally, (2.4) and (2.24) together give the proposition for bounded potentials.

Now we argue that the result also holds when $\phi,\phi_{k},\psi,\psi_{k},v,v_{k}$ are unbounded. For this we only need to show that

[TABLE]

[TABLE]

as $L\to\infty$, uniformly with respect to $k$, where $v^{jL}=\max(v^{j},-L)$ and $v^{jL}_{k},\psi^{L}_{k},\psi^{L},\phi^{L}_{k},\phi^{L}$ are defined similarly.

We only argue (2.25) as the proof of (2.26) is identical. By Proposition 2.17 there exists $\beta\in\mathcal{E}_{\chi}(X,\omega)$ such that $\beta\leq\phi,\phi_{k},\psi,\psi_{k},v^{j},v^{j}_{k}$ for any $k$, due to the monotonicity of our sequences. For example, in case all sequences are decreasing, we can take $\beta:=P(\phi,\psi,v_{1},,\ldots,v_{n})$. The other cases are treated similarly.

By (2.1) the non-pluripolar product is local in the plurifine topology, hence both integrands in (2.25) are the same on $\{\beta>-L\}$. As a result we can write

[TABLE]

As a result, to argue (2.25) it is enough to show that both terms in the above expression converge to zero. We now focus on the second expression. As $0\geq\psi^{L}_{k}-\phi^{L}_{k}\geq\beta$ we can write:

[TABLE]

where in the last line we have used (1.8). To continue, we introduce the potentials $\eta:=({v^{1}_{k}+\ldots+v^{n}_{k}+\beta})/({n+1})$ and $\eta^{L}:=({v^{1L}_{k}+\ldots+v^{nL}_{k}+\beta})/({n+1})$. These potentials are in $\mathcal{E}_{\chi}(X,\omega)$ by Corollary 2.18. We also pick ${\tilde{\chi}}\in\mathcal{W}^{+}_{2p+1}$ as in the previous lemma, for $\eta\in\mathcal{E}_{\chi}(X,\omega)$. We can continue the above estimates in the following manner:

[TABLE]

where in the last step we used Proposition 2.7. As $\chi(L)/\tilde{\chi}(L)\to 0$, this implies that the second term in (2.25) converges to zero.

Since

[TABLE]

the same argument gives that the first term in (2.4) converges to zero as well, finishing the proof of (2.25). ∎

Lastly, we prove the domination principle for the class $\mathcal{E}_{1}(X,\omega)$ (recall (2.8)). We mention that these results also hold more generally for the class $\mathcal{E}(X,\omega)$, by a theorem of S. Dinew [65, 25]. The short proof below was pointed out to us by C.H. Lu, and it is based on the arguments of [51].

Proposition 2.21**.**

Let $\phi,\psi\in\mathcal{E}_{1}(X,\omega)$. If $\psi\leq\phi$ almost everywhere with respect to $\omega_{\phi}^{n}$ then $\psi\leq\phi$.

Since $\mathcal{E}_{p}(X,\omega)\subset\mathcal{E}_{1}(X,\omega),\ p\geq 1$, we obtain that the domination principle trivially holds for $\phi,\psi\in\mathcal{E}_{p}(X,\omega),\ p\geq 1$ as well.

Proof.

We can assume without loss of generality that $\phi,\psi<0$. As $\phi,\psi\in\textup{PSH}(X,\omega)$, it suffices to prove that $\psi\leq\phi$ a.e. with respect to $\omega^{n}$. This will then imply that $\psi\leq\phi$ globally.

Suppose that $\omega^{n}(\{\phi<\psi\})>0$. By the next lemma, this implies existence of $u\in\mathcal{E}_{1}(X,\omega)$ such that $u\leq\psi$ and $\omega_{u}^{n}(\{\phi<\psi\})>0$.

By Corollary 2.18 we have that $tu+(1-t)\psi\in\mathcal{E}_{1}(X,\omega)$ for any $t\in[0,1]$. By the comparison principle (Proposition 2.3) we can write:

[TABLE]

We conclude that $0=\omega^{n}_{u}(\{\phi<tu+(1-t)\psi\})\nearrow\omega^{n}_{u}(\{\phi<\psi\})$, as $t\searrow 0$. As a result, $\omega^{n}_{u}(\{\phi<\psi\})=0$, a contradiction. ∎

Lemma 2.22**.**

Suppose that $U\subset X$ is a Borel set with non–zero Lebesgue measure and $\psi\in\mathcal{E}_{1}(X,\omega)$. Then there exists $u\in\mathcal{E}_{1}(X,\omega)$ such that $u\leq\psi$ and $\omega_{u}^{n}(U)>0$.

Proof.

We can assume that $\psi<0$. We fix $C>0$ and let $\psi_{C}:=P(\psi+C,0)$. Let $u_{j}\in\mathcal{H}_{\omega}$ be a decreasing approximating sequence of $\psi$, that exists by Theorem 2.1. We can assume that $u_{j}<0$. The partition formula of Proposition 2.15 gives:

[TABLE]

Notice that $P(u_{j}+C,0)\searrow\psi_{C}$ as $j\to\infty$. Taking the limit, Proposition 2.20 gives us the estimate of measures $\omega_{\psi_{C}}^{n}\leq-\frac{\psi}{C}\omega_{\psi}^{n}+\omega^{n}$. Using this we can write:

[TABLE]

As $\omega_{\psi_{C}}^{n}(X)=\omega^{n}(X)$, for $C$ big enough we get that $\omega_{\psi_{C}}^{n}(U)>0$. The choice $u:=\psi_{C}-C$ satisfies the requirements of the lemma. ∎

Chapter 3 The Finsler geometry of the space of Kähler potentials

As follows from the definition (2.4), the space of Kähler potentials $\mathcal{H}_{\omega}$ is a convex open subset of $C^{\infty}(X)$, hence one can think of it as a trivial Fréchet manifold. As such, one can introduce on $\mathcal{H}_{\omega}$ a collection of $L^{p}$ type Finsler metrics whose geometry we will study in this section. If $u\in\mathcal{H}_{\omega}$ and $\xi\in T_{u}\mathcal{H}_{\omega}\simeq C^{\infty}(X)$, then the $L^{p}$-length of $\xi$ is given by the following expression:

[TABLE]

where $\textup{Vol}(X)=\int_{X}\omega^{n}.$ In case $p=2$, we recover the Riemannian geometry of Mabuchi [87] (independently discovered by Semmes [98] and Donaldson [66]). Though these metrics will be our primary object of study, unfortunately the associated weights $\chi_{p}(l)=l^{p}/p$ are not twice differentiable for $1\leq p<2$. This will cause problems as we shall see, and an approximation with more regular weights will be necessary to carry out even the most basic geometric arguments. For this reason, one needs to first work with even more general Finsler metrics on $\mathcal{H}_{\omega}$ with smooth weights in $\mathcal{W}^{+}_{p}$ (see (1.7)) and then use approximation via Propositions 1.6 and 1.7 to return to the $L^{p}$ metrics in the end.

Following our discussion above, we introduce the Orlicz–Finsler length of $\xi$ for any weight $\chi\in\mathcal{W}^{+}_{p}$:

[TABLE]

The above expression introduces a norm on each fiber of $T\mathcal{H}_{\omega}$, and the length of a smooth curve $[0,1]\ni t\to\alpha_{t}\in\mathcal{H}_{\omega}$ is computed by the usual formula:

[TABLE]

To clarify, smoothness of $t\to\alpha_{t}$ simply means that the map $\alpha(t,x)=\alpha_{t}(x)$ is smooth as a map from $[0,1]\times X$ to $\mathbb{R}$. Furthermore, the distance $d_{\chi}({u_{0}},{u_{1}})$ between ${u_{0}},{u_{1}}\in\mathcal{H}_{\omega}$ is the infimum of the $l_{\chi}$-length of smooth curves joining ${u_{0}}$ and ${u_{1}}$:

[TABLE]

The distance $d_{\chi}$ is a pseudo–metric (the triangle inequality holds), but $d_{\chi}(u,v)=0$ may not imply $u=v$, as our setting is infinite–dimensional. We will see in Theorem 3.6 below that $d_{\chi}$ is a bona fide metric, but this will require a careful analysis of our Finsler structures.

When dealing with the $L^{p}$ metric structures (3.1), the associated curve length and path length metric will be denoted by $l_{p}$ and $d_{p}$ respectively.

As we will see, the different weights in $\mathcal{W}^{+}_{p}$ induce different geometries on $\mathcal{H}_{\omega}$, however the equation for the shortest length curves between points of $\mathcal{H}_{\omega}$, the so called geodesics, will be essentially the same. Motivated by this, in the next section we will focus on the $L^{2}$ Riemannian geometry first, in which case we can get an explicit equation for these geodesics.

3.1 Riemannian geometry of the space of Kähler potentials

When $p=2$ the metric of (3.1) is induced by the following non-degenerate inner product:

[TABLE]

This Riemannian structure was first studied by Mabuchi [87] and later independently by Semmes [98] and Donaldson [66]. For another introductory survey on the Mabuchi geometry of $\mathcal{H}_{\omega}$ we refer to [23].

Let us compute the Levi–Civita connection of this metric. For this we choose a smooth curve $[0,1]\ni t\to u_{t}\in\mathcal{H}_{\omega}$ and $[0,1]\ni t\to\phi_{t},\psi_{t}\in C^{\infty}(X)$, two vector fields along $t\to u_{t}$. In the future, when working with time derivatives, we will use the notation $\dot{u}_{t}=du_{t}/dt$, $\ddot{u}_{t}=d^{2}u_{t}/dt^{2}$, etc.

We will identify the Levi–Civita connection $\nabla_{(\cdot)}(\cdot)$, using the fact that it is torsion free and satisfies the following product rule:

[TABLE]

Using the identities $\frac{d}{dt}\omega_{u_{t}}^{n}=ni\partial\bar{\partial}\dot{u}_{t}\wedge\omega_{u_{t}}^{n-1}=\frac{1}{2}\Delta^{\omega_{u_{t}}}\dot{u}_{t}\omega_{u_{t}}^{n}$ and $-\int_{X}\langle\nabla^{\omega_{u_{t}}}f,\nabla^{\omega_{u_{t}}}g\rangle\omega_{u_{t}}^{n}=\int_{X}f\Delta^{\omega_{u_{t}}}g\omega_{u_{t}}^{n}$ (see the discussion following (A.4) in the appendix), we can start writing:

[TABLE]

Comparing with (3.6), this line of calculation suggests the following formula for the Levi–Civita connection, and it is easy to see that the resulting connection is indeed torsion free:

[TABLE]

This immidiately implies that $t\to u_{t}$ is a geodesic if and only if $\nabla_{\dot{u}_{t}}\dot{u}_{t}=0$, or equivalently

[TABLE]

As discovered independently by Semmes [98] and Donaldson [66], the above equation can be understood as a complex Monge–Ampère equation. For this one has to introduce the trivial complexification $u\in C^{\infty}(S\times X)$, using the formula

[TABLE]

where $S=\{0<\textup{Re s}<1\}\subset\mathbb{C}$ is the unit strip. We pick a coordinate patch $U\subset X$, where the metric $\omega_{u}$ has a potential $g_{u}\in C^{\infty}(X)$, i.e., $\omega_{u}=i\partial\bar{\partial}g_{u}=i{g_{u}}_{j\bar{k}}dz_{j}\wedge d\bar{z}_{k}$. Then on $[0,1]\times U$ the geodesic equation (3.8) is seen to be equivalent to $\ddot{u}-g_{u}^{j\bar{k}}\dot{u}_{j}\dot{u}_{\bar{k}}=0$, where $g_{u}^{j\bar{k}}$ is the inverse of ${g_{u}}_{j\bar{k}}$. By involving the complexified variable $s$, this identity is further seen to be equivalent to $u_{s\bar{s}}-g_{u}^{j\bar{k}}u_{j\bar{s}}u_{s\bar{k}}=0$ on $S\times U$. After multiplying with $\det({g_{u}}_{j\bar{k}})$, this last equation can be written globally on $S\times X$ as:

[TABLE]

where $\pi:S\times X\to X$ is the projection map to the second component. Consequently, the problem of joining the potentials $u_{0},u_{1}\in\mathcal{H}_{\omega}$ with a smooth geodesic equates to finding a smooth solution $u\in\mathcal{C}^{\infty}({S}\times X)$ to the following boundary value problem:

[TABLE]

To be precise, here $\lim_{s\to 0,1}u(s,\cdot)=u_{0,1}$ simply means that $u(s,\cdot)$ converges uniformly to $u_{0,1}$ as $s\to 0,1$. Unfortunately, as detailed below, this boundary value problem does not usually have smooth solutions, but a unique weak solution (in the sense of Bedford–Taylor) does exist. Instead, one replaces (3.8) with the following equation for $\varepsilon$-geodesics:

[TABLE]

By an elementary calculation, similar to the one giving (3.9), the associated boundary value problem for this equation becomes:

[TABLE]

Since $\omega_{u_{0}},\omega_{u_{1}}>0$, we see that $\pi^{*}\omega+i\partial\overline{\partial}u^{\varepsilon}>0$ on $S\times X$. As a result the condition $\omega+i\partial\overline{\partial}u|_{\{s\}\times X}>0,\ s\in S$ is automatically satisfied. In contrast with (3.10), this Dirichlet problem is elliptic and its solutions are smooth, moreover we have the following regularity result due to X.X. Chen [35] (with complements by Blocki [23]):

Theorem 3.1**.**

The boundary value problem (3.12) admits a unique smooth solution $u^{\varepsilon}\in C^{\infty}(\overline{S}\times X)$ with the following bounds that are independent of $\varepsilon>0$:

[TABLE]

[TABLE]

Recall from our discussion preceding (2.17) that having a bound on $\Delta u$ is equivalent to bounding all mixed second order complex derivatives of $u$ on $S\times X$. We refer to [23, Theorem 12] for an elaborate treatment of Theorem 3.1 (see also the survey paper [26]).

The estimate for $\dot{u}^{\varepsilon}$ is argued in the last step of the proof of [23, Lemma 16], as a consequence of the comparison principle. Let us however mention that a much more precise estimate for $\nabla u^{\varepsilon}$ is argued in [22, Theorem 1].

In addition, recently Chu–Tosatti–Weinkove have showed that one can more generally bound the Hessian of $u^{\varepsilon}$ [42] independently of $\varepsilon$. Additionally, it was shown by Berman–Demailly [15] and He [76] that one can in fact bound each $\Delta^{\omega}u_{t}^{\varepsilon},\ t\in[0,1],$ using bounds on $\Delta^{\omega}u_{0}$ and $\Delta^{\omega}u_{1}$.

Using the Bedford–Taylor interpretation of $(\pi^{*}\omega+i\partial\overline{\partial}u^{\varepsilon})^{n+1}$ as a Borel measure, the boundary value problems (3.10) and (3.12) can be stated for $u,u^{\varepsilon}\in\textup{PSH}(S\times X,\pi^{*}\omega)$ that are only bounded and not necessarily smooth. Additionally, after pulling back by the $\log$ function, we can equivalently state (3.10) and (3.12) as boundary value problems with circle–invariant solutions on $A\times X$, where $A$ is the annulus $\{e^{0}<|z|<e^{1}\}\subset\mathbb{C}$. Consequently, the next result (whose proof closely follows [23, Theorem 21]) will assure that uniqueness of solutions to (3.10) and (3.12) holds not only for smooth solutions, but also for solutions that are merely in $\textup{PSH}(S\times X,\pi^{*}\omega)\cap L^{\infty}$:

Theorem 3.2**.**

Suppose $M$ is a $k$ dimensional compact complex manifold with smooth boundary and Kähler form $\eta$. If $u,v\in\textup{PSH}(M,\eta)\cap L^{\infty}$ with $\liminf_{x\to\partial M}(u-v)(x)\geq 0$ and $(\eta+i\partial\bar{\partial}v)^{k}\geq(\eta+i\partial\bar{\partial}u)^{k}$, then $u\geq v$ on $M$.

Since $\pi^{*}\omega$ is only non-negative on $S\times X$, the above result is not directly applicable to our situation. This small inconvenience can be fixed by taking $\eta:=\pi^{*}\omega+i\partial\overline{\partial}g$, where $g$ is a smooth function on $S$, such that $i\partial\overline{\partial}g>0$, $g(t+ir)=g(t)$, and $g(ir)=g(1+ir)=0$.

Proof.

Let $\delta>0$ and $v_{\delta}:=\max(u,v-\delta)\in\textup{PSH}(M,\eta)\cap L^{\infty}$. Then $v_{\delta}=u$ near $\partial M$. To conclude the proof, it is enough to show that $v_{\delta}=u$ on $M$.

From Bedford–Taylor theory (see [20, Theorem 2.2.10]) it follows that

[TABLE]

As $v_{\delta}$ and $u$ agree near the boundary and $v_{\delta}\geq u$, we can integrate by parts [20, Theorem 1.3.4] and write:

[TABLE]

As each of the summands above is non-negative, it follows that

[TABLE]

The equality $v_{\delta}=u$ will follow, if we can show that $\int_{M}i\partial(u-v_{\delta})\wedge\bar{\partial}(u-v_{\delta})\wedge\eta^{k-1}=0$. Indeed, this would imply that all Sobolev derivatives of $u-v_{\delta}$ are zero. This last identity will be established as the last step in an inductive argument showing that

[TABLE]

This identity holds for $j=k-1$ by the above. As all the steps are carried out similarly, we only show that

[TABLE]

Denote $f=u-v_{\delta}$. Using (3.15) and integration by parts we can write:

[TABLE]

Using the Cauchy–Schwarz inequality of Bedford–Taylor theory [20] we can write

[TABLE]

hence by (3.14), it follows that the first term in (3.1) is zero. It can be shown similarly that the second term in (3.1) is zero as well, implying (3.16) and finishing the proof. ∎

Now we return to (3.10) and (3.12). As each $u^{\varepsilon}$ is invariant in the $i\mathbb{R}$ direction, plurisubharmonicity of $u^{\varepsilon}$ implies that $t\to u^{\varepsilon}(t,x)$ is convex for each $x\in X$. As each $u^{\varepsilon}$ solves the boundary value problem (3.12), we additionally obtain that

[TABLE]

On top of proving uniqueness of general solutions to (3.10) and (3.12), the above theorem also shows that the family $\{u^{\varepsilon}\}_{\varepsilon}$ increases as $\varepsilon\searrow 0$. Using this, (3.18) and the continuity of the complex Monge–Ampère operator along increasing sequences [20, Theorem 2.2.5] it follows that solutions of (3.12) approximate solutions of (3.10), i.e.,

[TABLE]

From here, an application of the Arzela–Ascoli compactness theorem yields that the estimates of (3.13) also hold for solutions of (3.10):

Theorem 3.3**.**

For the unique solution $u\in\textup{PSH}(S\times X,\pi^{*}\omega)$ of (3.10) the following estimates hold:

[TABLE]

By the examples of [84, 53] this regularity result is essentially optimal, as no higher order estimates are possible for weak solutions. Often we will refer to the unique solution of (3.10) as the weak $C^{1,\bar{1}}$–geodesic joining $u_{0},u_{1}\in\mathcal{H}_{\omega}$, and it will be denoted as

[TABLE]

Moving on, as follows from standard elliptic PDE theory, given a smooth family of boundary data for (3.12) one obtains a corresponding smooth family of solutions. As another simple consequence of the uniqueness theorem, we obtain a uniform bound on the rate of change of this family as the parameter changes:

Corollary 3.4** ([35]).**

Suppose $[0,1]\ni\rho\to u_{0}(\cdot,\rho),u_{1}(\cdot,\rho)\in\mathcal{H}_{\omega}$ is smooth family of boundary data for (3.12). For the smooth family of solutions $[0,1]\ni\rho\to u^{\varepsilon}(\cdot,\cdot,\rho)\in C^{\infty}(\overline{S}\times X)$ we have the following estimate for all $\varepsilon>0$:

[TABLE]

Proof.

Let $\rho_{0}\in[0,1]$ and $C>\max(\|{du_{0}}/{d\rho}\|_{C^{0}(X\times[0,1])},\|{du_{1}}/{d\rho}\|_{C^{0}(X\times[0,1])}).$ Then Theorem 3.2 implies that

[TABLE]

for all $\delta\in\mathbb{R}$ such that $\rho_{0}+\delta\in[0,1]$. This gives that ${|u^{\varepsilon}(s,x,\rho_{0}+\delta)-u^{\varepsilon}(s,x,\rho_{0})|}/{\delta}\leq C,$ implying the desired estimate. ∎

As a last consequence of the uniqueness theorem, we note the following concrete estimate for the tangent vectors of weak $C^{1,\bar{1}}$-geodesics:

Lemma 3.5**.**

Given $u_{0},u_{1}\in\mathcal{H}_{\omega}$, let $[0,1]\ni t\to u_{t}\in\mathcal{H}_{\omega}$ be the weak $C^{1,\bar{1}}$–geodesic joining $u_{0},u_{1}$. Then the following estimate holds:

[TABLE]

Proof.

Let $C:=\sup_{X}|u_{0}-u_{1}|$. Convexity in the $t$ variable implies that $\dot{u}_{0}\leq\dot{u}_{t}\leq\dot{u}_{1}$. Hence we only need to show that $-C\leq\dot{u}_{0}$ and $\dot{u}_{1}\leq C$.

Examining (3.8), it is clear that the curve $[0,1]\in t\to v_{t}:=u_{0}-Ct\in\mathcal{H}_{\omega}$ is a smooth geodesic connecting $u_{0}$ and $u_{0}-C$. Hence its complexification $v$ is a solution to (3.10).

As $u_{0}-C\leq u_{1}$, Theorem 3.2 gives that that $v_{t}\leq u_{t},\ t\in[0,1]$, implying that $-C\leq\dot{u}_{0}$.

The same trick applied to the “reverse” $C^{1,\bar{1}}$–geodesic $t\to u_{1-t}$ gives $-C\leq-\dot{u}_{1}$, finishing the proof. ∎

3.2 The Orlicz geometry of the space of Kähler potentials

As discussed in the beginning of the present chapter, in our study of $L^{p}$ Finsler metrics on $\mathcal{H}_{\omega}$, we need to return to the full generality of Orlicz–Finsler metrics (3.2) on $\mathcal{H}_{\omega}$ with weight in $\mathcal{W}^{+}_{p}$. We will do this in this section, and our main theorem connects the $d_{\chi}$ pseudo–distance (see (3.4)) with the weak $C^{1,\bar{1}}$ geodesic of the $L^{2}$ Mabuchi geometry (see (3.21)), in the process showing that $d_{\chi}$ is indeed a bona fide metric:

Theorem 3.6**.**

[48, Theorem 1]* If $\chi\in\mathcal{W}^{+}_{p},\ p\geq 1$ then $(\mathcal{H}_{\omega},d_{\chi})$ is a metric space and for any $u_{0},u_{1}\in\mathcal{H}_{\omega}$ the weak $C^{1,\bar{1}}$ geodesic $t\to u_{t}$ connecting $u_{0},u_{1}$ satisfies:*

[TABLE]

Although the metric spaces $(\mathcal{H}_{\omega},d_{\chi})$ are not quasi isometric for different $\chi$ (as we will see, they have different metric completions), it is remarkable that the same weak $C^{1\bar{1}}$–geodesic is “length minimzing” for all $d_{\chi}$ metric structures. Additionally, as we will see in the next sections, this same curve is an honest metric geodesic in the completion of each space $(\mathcal{H}_{\omega},d_{\chi})$.

In the proof of Theorem 3.6, we first show the result for Finsler metrics with smooth weight $\chi$, and afterwards use approximation via Proposition 1.6 and Proposition 1.7 to establish the result for all metrics with weight in $\mathcal{W}^{+}_{p}$, which includes as particular case the $L^{p}$ metrics.

In the particular case of the $L^{2}$ metric, Theorem 3.6 was obtained by X.X. Chen [35] and our proof in case of Finsler metrics with smooth weight follows his ideas and a careful differential analysis of Orlicz norms.

To ease the technical nature of future calculations, for the rest of this section we assume the normalizing condition

[TABLE]

This can always be achieved by rescaling the Kähler metric $\omega$. Given a differentiable normalized Young weight $\chi$, the differentiability of the associated norm $\|\cdot\|_{\chi}$ (see (1.3)) is well understood (see [97, Chapter VII]). As a first step in proving Theorem 3.6, we adapt [97, Theorem VII.2.3] to our setting:

Proposition 3.7**.**

[48, Proposition 3.1]* Suppose $\chi\in\mathcal{W}^{+}_{p}\cap C^{\infty}(\mathbb{R})$. Given a smooth curve $(0,1)\ni t\to u_{t}\in\mathcal{H}_{\omega}$, and a vector field $(0,1)\ni t\to f_{t}\in C^{\infty}(X)$ along this curve with $f_{t}\not\equiv 0,\ t\in(0,1)$, the following formula holds:*

[TABLE]

where $\nabla_{(\cdot)}(\cdot)$ is the covariant derivative from (3.7).

Proof.

We introduce the smooth function $F:\mathbb{R}^{+}\times(0,1)\to\mathbb{R}$ given by

[TABLE]

As $\chi\in\mathcal{W}^{+}_{p}$, by (1.8) we have $\chi^{\prime}(l)>0,\ l>0$ and $\chi^{\prime}(l)<0,\ l<0$. As $t\to f_{t}$ is non-vanishing, it follows that

[TABLE]

for all $r>0,\ t\in(0,1)$. Using the fact that $F(\|f_{t}\|_{\chi,u_{t}},t)=\chi(1)$ (see (1.9)), an application of the implicit function theorem yields that the map $t\to\|f_{t}\|_{\chi,u_{t}}$ is differentiable and the following formula holds:

[TABLE]

Recalling the formula for the covariant derivative (3.7), an integration by parts yields (3.25). ∎

The estimate for $\varepsilon$–geodesics (see (3.11)) from the following technical lemma will be of great use in our later study:

Lemma 3.8**.**

Suppose $\chi\in\mathcal{W}^{+}_{p}$ and $u_{0},u_{1}\in\mathcal{H}_{\omega}$. Then the $\varepsilon$–geodesic $[0,1]\ni t\to u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ connecting $u_{0},u_{1}$ satisfies the following estimate:

[TABLE]

for all $t\in[0,1]$, where $R:=R(\chi,\|u_{0}\|_{C^{2}},\|u_{1}\|_{C^{2}})$.

Proof.

As $t\to u^{\varepsilon}_{t}(x)$ is convex for any $x\in X$, on the set $\{u_{0}\geq u_{1}\}$ the estimate $\dot{u}^{\varepsilon}_{0}\leq u_{1}-u_{0}\leq 0$ holds, hence

[TABLE]

We can similarly deduce that

[TABLE]

Let us assume momentarily that $\chi$ is smooth. For $t\in[0,1]$, using the Riemannian connection $\nabla_{(\cdot)}(\cdot)$ (see (3.7)) and the fact that $t\to u^{\varepsilon}_{t}$ is an $\varepsilon$–geodesic (see (3.11)), we can write:

[TABLE]

where in the last estimate we have used that $\dot{u}^{\varepsilon}_{t}$ is uniformly bounded in terms of $\|u_{0}\|_{C^{2}},\|u_{1}\|_{C^{2}}$ (Theorem 3.1). For general $\chi\in\mathcal{W}^{+}_{p}$, since $\chi$ is convex, an approximation argument yields the same estimate.

After putting together the last three estimates, (3.26) follows. ∎

As a consequence of the previous two results we obtain the following corollary:

Corollary 3.9**.**

Suppose $\chi\in\mathcal{W}^{+}_{p}\cap C^{\infty}(\mathbb{R})$ and $u_{0},u_{1}\in\mathcal{H}_{\omega},\ u_{0}\neq u_{1}$. Then there exists $\varepsilon_{0}>0$ dependent on upper bounds for $\|u_{0}\|_{C^{2}(X)},\|u_{1}\|_{C^{2}(X)}$ and lower bounds for $\|\chi(u_{1}-u_{0})\|_{L^{1}(\omega^{n})},\omega_{u_{0}}^{n}/\omega^{n}$ and $\omega_{u_{1}}^{n}/\omega^{n}$, such that for all $\varepsilon\in(0,\varepsilon_{0})$ the $\varepsilon$–geodesic $[0,1]\ni t\to u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ of (3.12), connecting $u_{0},u_{1}$ satisfies:

[TABLE]

Proof.

This is a simple application of the formula of Proposition 3.7 for the $\varepsilon$–geodesic $t\to u^{\varepsilon}_{t}$ and the vector field $t\to f_{t}:=\dot{u}_{t}$. Indeed, by the previous lemma, one can choose $\varepsilon_{0}>0$ as indicated, so that $\dot{u}_{t}$ is non-vanishing for all $t\in[0,1]$, hence the assumptions of Proposition 3.7 are satisfied. ∎

Continuing to focus on smooth weights $\chi$, we establish concrete bounds for the $\chi$–length of tangent vectors along the $\varepsilon$–geodesics and their derivatives. This is the analog of [23, Lemma 13] in our more general setting:

Proposition 3.10**.**

*Suppose $\chi\in\mathcal{W}^{+}_{p}$ and $u_{0},u_{1}\in\mathcal{H}_{\omega},\ u_{0}\neq u_{1}$. Then there exists $\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ the $\varepsilon$–geodesic $[0,1]\ni t\to u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ connecting $u_{0},u_{1}$ satisfies:

(i) $\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}>R_{0},\ t\in[0,1]$.

(ii) \big{|}\frac{d}{dt}\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}\big{|}\leq\varepsilon R_{1},\ t\in[0,1],

where $\varepsilon_{0}>0,R_{0}>0,R_{1}>0$ depend on upper bounds for $\|u_{0}\|_{C^{2}(X)}$, $\|u_{1}\|_{C^{2}(X)}$ and lower bounds for $\|\chi(u_{1}-u_{0})\|_{L^{1}(\omega^{n})},\omega_{u_{0}}^{n}/\omega^{n}$ and $\omega_{u_{1}}^{n}/\omega^{n}$.*

Proof.

The estimate of (i) follows from (3.26) and Proposition 1.5. To establish (ii) we assume momentarily that $\chi\in C^{\infty}(\mathbb{R})$, and we shrink $\varepsilon_{0}$ enough to satisfy the requirements of Corollary 3.9. Using the Young identity (1.2) we can write:

[TABLE]

Using (i) and the fact that $\dot{u}^{\varepsilon}_{t}$ is uniformly bounded in terms of $\|u_{0}\|_{C^{2}},\|u_{1}\|_{C^{2}}$ (Theorem 3.1) the estimate of (ii) follows for smooth $\chi$. For general $\chi$ we simply use the approximation result Proposition 1.6 to conclude that (3.2) still holds, where $\frac{d}{dt}$ is interpreted as a Lipschitz derivative. ∎

We can now establish the main geometric estimate for $\varepsilon$–geodesics, which generalizes the corresponding statement for the $L^{2}$–metric ([23, Theorem 14]):

Proposition 3.11** ([48]).**

Suppose $\chi\in\mathcal{W}^{+}_{p}\cap C^{\infty}(\mathbb{R})$, $[0,1]\ni s\to\psi_{s}\in\mathcal{H}_{\omega}$ is a smooth curve, $\phi\in\mathcal{H}_{\omega}\setminus\psi([0,1])$ and $\varepsilon>0$. We denote by $u^{\varepsilon}\in C^{\infty}([0,1]\times[0,1]\times X)$ the smooth function for which $[0,1]\ni t\to u^{\varepsilon}_{t}(\cdot,s):=u^{\varepsilon}(t,\cdot,s)\in\mathcal{H}_{\omega}$ is the $\varepsilon$–geodesic connecting $\phi$ and $\psi_{s}$. There exists $\varepsilon_{0}(\phi,\psi)>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ the following holds:

[TABLE]

for some $R(\phi,\psi,\chi,\varepsilon_{0})>0$ independent of $\varepsilon>0$.

Proof.

Whenever it will not cause confusion, we will drop sub/superscripts when addressing the $\varepsilon$–geodesic $t\to u^{\varepsilon}_{t}(\cdot,s)$. To avoid cumbersome notation, derivatives in the $t$–direction will be denoted by dots, derivatives in the $s$–direction will be denoted by $d/ds$, and sometimes we also omit dependence on $(t,s)$.

Fix $s\in[0,1]$. By Proposition 3.7 and Proposition 3.10(i) there exists $\varepsilon_{0}(\phi,\psi)>0$ such that for $\varepsilon\in(0,\varepsilon_{0})$ the following holds:

[TABLE]

Using the Young identity (1.2) and the fact that $\nabla_{(\cdot)}(\cdot)$ is a Riemannian connection, we can continue:

[TABLE]

We make the following side computation:

[TABLE]

After possibly further shrinking $\varepsilon_{0}(\phi,\psi)>0$, from Proposition 3.10(i)(ii) and (3.11) it follows that $\|\dot{u}\|_{\chi,u}$ is uniformly bounded away from zero and both $\nabla_{\dot{u}}\dot{u}\omega_{u}^{n}$ and $\frac{d}{dt}\|\dot{u}\|_{\chi,u}$ are of the form $\varepsilon R$, where $R$ is an uniformly bounded quantity for $\varepsilon<\varepsilon_{0}(\phi,\psi)$.

Furthermore, it follows from Theorem 3.1 that $\dot{u}$ is uniformly bounded independently of $\varepsilon$. All of this implies that the quantity of (3.30) is also of the form $\varepsilon R$. Lastly, Corollary 3.4 implies that $du/ds$ is uniformly bounded as well, hence putting the above together it follows that the second term in the numerator of (3.2) is also of the form $\varepsilon R$, and we can continue to write:

[TABLE]

As $\chi^{*}$ is the Legendre transform of $\chi$, it follows that ${\chi^{*}}^{\prime}(\chi^{\prime}(l))=l,\ l\in\mathbb{R}$. Using this, our prior observations and the chain rule, we obtain that the expression

[TABLE]

is again of magnitude $\varepsilon R$, hence in our sequence of calculations we can write

[TABLE]

where in the last line we have used the Young inequality (1.2) in the following manner:

[TABLE]

Integrating estimate (3.2) with respect to $s$ yields the desired inequality. ∎

With the previous result established, there is no more need to differentiate expressions involving the $l_{\chi}$ length of curves, hence we can return to general Finsler metrics on $\mathcal{H}_{\omega}$, with possibly non–smooth weight $\chi\in\mathcal{W}^{+}_{p}$, and prove Theorem 3.6 in the process:

Proof of Theorem 3.6.

First we show the following identity for the weak $C^{1,\bar{1}}$–geodesic joining $u_{0},u_{1}$:

[TABLE]

We can assume that $u_{0}\neq u_{1}$. By Theorem 3.1 and (3.19),(3.20), the smooth $\varepsilon$–geodesics $u^{\varepsilon}$ connecting $u_{0},u_{1}$ $C^{1,\alpha}$–converge to the weak $C^{1,\bar{1}}$ geodesic $u$, connecting $u_{0},u_{1}$. As a result, $\dot{u}^{\varepsilon}_{t}$ converges uniformly to $\dot{u}_{t}$. Next we argue that the lengths of these tangent vectors converge as well:

Claim 3.12**.**

$\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}\to\|\dot{u}_{t}\|_{\chi,u_{t}}$* as $\varepsilon\to 0$.*

From Proposition 3.10(i) and Theorem 3.1 it follows that there exists $C_{2}>C_{1}>0$ such that for small enough $\varepsilon>0$ we have

[TABLE]

In particular, we only have to argue that all cluster points of the set $\{\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}\}_{\varepsilon}$ are equal to $\|\dot{u}_{t}\|_{\chi,u_{t}}$. Let $N$ be such a cluster point, and after taking a subsequence, we can assume that $\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}\to N$ as $\varepsilon\to 0$. By uniform convergence of tangent vectors, we obtain that $\dot{u}^{\varepsilon}_{t}/\|\dot{u}^{\varepsilon}_{t}\|_{\chi,u^{\varepsilon}_{t}}$ converges to $\dot{u}_{t}/N$ uniformly as well. Since $\omega_{u^{\varepsilon}_{t}}^{n}\to\omega_{u_{t}}^{n}$ weakly, this allows to conclude that

[TABLE]

Using (1.9) we get that $N=\|\dot{u}_{t}\|_{\chi,u_{t}}$, finishing the proof of the claim.

Using the claim, we can apply the dominated convergence theorem to conclude that

[TABLE]

hence $d_{\chi}(u_{0},u_{1})\leq l_{\chi}(u_{t})$. To prove the reverse inequality, and with that establishing (3.32), we assume first that $\chi\in\mathcal{W}^{+}_{p}\cap C^{\infty}$. We have to prove that

[TABLE]

for all smooth curves $[0,1]\ni t\to\phi_{t}\in\mathcal{H}$ connecting $u_{0},u_{1}$. We can assume that $u_{1}\not\in\phi[0,1)$ and let $h\in[0,1)$. Letting $\varepsilon\to 0$ in the estimate of the previous result, by (3.33) we obtain that

[TABLE]

where $[0,1]\ni t\to u_{1-t},w^{h}_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ are the weak $C^{1,\bar{1}}$–geodesic segments joining $u_{1},u_{0}$ and $u_{1},\phi_{h}$ respectively. As $h\to 1$, by Lemma 3.5 we have $l_{\chi}(w^{h}_{t})\to 0$ and we obtain (3.34) for smooth weights $\chi$.

For general $\chi\in\mathcal{W}^{+}_{p}$ by Proposition 1.7 there exists a sequence $\chi_{k}\in\mathcal{W}^{+}_{p_{k}}\cap C^{\infty}(\mathbb{R})$ such that $\chi_{k}$ converges to $\chi$ uniformly on compacts. From what we just proved it follows that

[TABLE]

Using Proposition 1.6 and the dominated convergence theorem ($\dot{\phi}_{t},\dot{u}_{t}$ are uniformly bounded), we can take the limit in this last estimate to conclude (3.34), which gives (3.32). Formula (3.23) follows now from the fact that $l_{\chi}(u_{t})=\int_{0}^{1}\|\dot{u}_{l}\|_{\chi,u_{l}}dl$ and Lemma 3.13 below.

Finally, if $u_{0}\neq u_{1}$ then after taking the limit $\varepsilon\to 0$ in the estimate of Lemma 3.8 we obtain that $\dot{u}_{0}\not\equiv 0$, hence $d_{\chi}(u_{0},u_{1})=\|\dot{u}_{0}\|_{\chi,u_{0}}>0$. This implies that $(\mathcal{H},d_{\chi})$ is a metric space, as claimed. ∎

According to the last lemma of this section, the $\chi$–length of tangent vectors along a $C^{1,\bar{1}}$–geodesic is always constant. This parallels a similar result of Berndtsson [18, Proposition 2.2].

Lemma 3.13**.**

*Given $u_{0},u_{1}\in\mathcal{H}_{\omega}$, let $[0,1]\ni t\to u_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ be the weak $C^{1,\bar{1}}$–geodesic connecting $u_{0},u_{1}$. Then for any $\chi\in\mathcal{W}^{+}_{p}$ and $t_{0},t_{1}\in[0,1]$ the following hold:

[TABLE]

Proof.

As we already argued in the proof of the previous result, for the $\varepsilon$–geodesics $t\to u^{\varepsilon}_{t}$ joining $u_{0},u_{1}$ we have $\|\dot{u}^{\varepsilon}_{t_{0}}\|_{\chi,u^{\varepsilon}_{t_{0}}}\to\|\dot{u}_{t_{0}}\|_{\chi,u_{t_{0}}}$ and $\|\dot{u}^{\varepsilon}_{t_{1}}\|_{\chi,u^{\varepsilon}_{t_{1}}}\to\|\dot{u}_{t_{1}}\|_{\chi,u_{t_{1}}}$ as $\varepsilon\to 0$. Furthermore Proposition 3.10(ii) implies that $|\|\dot{u}^{\varepsilon}_{t_{1}}\|_{\chi,u^{\varepsilon}_{t_{1}}}-\|\dot{u}^{\varepsilon}_{t_{0}}\|_{\chi,u^{\varepsilon}_{t_{0}}}|\leq|t_{1}-t_{0}|\varepsilon R_{1}$. Putting all of this together, and letting $\varepsilon\to 0$, (3.35) follows. ∎

3.3 The weak geodesic segments of $\textup{PSH}(X,\omega)$

As noted after Theorem 3.3, it is not possible to join smooth potentials $u_{0},u_{1}$ with a geodesic staying inside $\mathcal{H}_{\omega}$. The main point of the present section is to show that a similar phenomenon does not occur if $u_{0},u_{1}$ is allowed to be more singular. Indeed, as we will see, if $u_{0},u_{1}$ are bounded, or they are from a finite energy space, then it is possible to define a weak geodesic connecting them that is also bounded or stays inside the finite energy space respectively. Our study will connect properties of weak geodesics with that of envelopes, and we will make good use of the results of Section 2.4.

When considering the boundary value problem (3.10), we constructed the (weak) solution $u$ as the limit of solutions to the family of elliptic problems (3.12). Moving away from this idea, as noted by Berndtsson [19, Section 2.1], it possible to describe the (weak) solution $u$ in another way, using a slight generalization of the classical Perron–Bremmerman envelope from the local theory.

The advantage of this approach is that one can consider very general boundary data in (3.10). Indeed, to begin, let $u_{0},u_{1}\in\textup{PSH}(X,\omega)$. In the future, we will refer to $i\mathbb{R}$–invariant elements of $\textup{PSH}(S\times X,\pi^{*}\omega)$ as weak subgeodesics (recall that $S=\{0<\textup{Re }z<1\}\subset\mathbb{C}$). This name is justified by the following formula:

[TABLE]

where $\mathcal{S}$ is the following family of weak subgeodesics:

[TABLE]

As we know, the supremum of a family of $\pi^{*}\omega$-psh functions may not be $\pi^{*}\omega$-psh, and the first step is to show that $u$, as defined in (3.36), is $\pi^{*}\omega$-psh nonetheless. Indeed, by convexity in the $t$ variable, each member of $w_{t}\in\mathcal{S}$ satisfies $w_{t}\leq(1-t)u_{0}+tu_{1}$, hence this also holds for the supremum $u$:

[TABLE]

By taking the usc regularization of the above inequality, we conclude that the same inequality holds with $u^{*}$ in place of $u$:

[TABLE]

We obtain that $u^{*}\in\mathcal{S}$, hence $u^{*}\leq u$ by (3.36). Trivially $u\leq u^{*}$, and this implies that $u=u^{*}\in\textup{PSH}(S\times X,\pi^{*}\omega)$.

We will call the curve $[0,1]\ni t\to u_{t}\in\textup{PSH}(X,\omega)$ resulting from the construction of (3.36) the weak geodesic connecting $u_{0},u_{1}$. This terminology is justified by the following result, which says that (3.36) gives the unique solution to (3.10) for boundary data that is merely bounded:

Lemma 3.14**.**

When $u_{0},u_{1}\in\textup{PSH}(X,\omega)\cap L^{\infty}$ then the unique bounded $\pi^{*}\omega$-psh solution of (3.10) is given by (3.36).

As a result of this lemma, for $u_{0},u_{1}\in\textup{PSH}(X,\omega)\cap L^{\infty}$, we will call the weak geodesic $t\to u_{t}$ connecting $u_{0},u_{1}$ a bounded geodesic.

Proof.

Let $C:=\|u_{1}-u_{0}\|_{L^{\infty}}$. It is easy to see that $t\to(u_{0}-Ct)$ and $t\to(u_{1}-C(1-t))$ are (sub)geodesics that are both members of $\mathcal{S}$, hence so is their maximum $v_{t}:=\max(u_{0}-Ct,u_{1}-C(1-t))\in\mathcal{S}$. This and (3.37) gives

[TABLE]

Consequently, $u\in\textup{PSH}(S\times X,\pi^{*}\omega)\cap L^{\infty}$ and $\lim_{t\to 0,1}u_{t}=u_{0,1}$. The classical Perron–Bremmerman argument can now be adapted to this setting to give $(\pi^{*}\omega+i\partial\bar{\partial}u)^{n+1}=0$. Lastly, uniqueness of $u$ is a consequence of Theorem 3.2. ∎

Turning back to non-bounded endpoints $u_{0},u_{1}$, it turns out that even the very general weak geodesic segment $t\to u_{t}$ connecting $u_{0},u_{1}$ exhibits some structure, as we will see in the next two results:

Proposition 3.15** ([47]).**

*Suppose $u^{k}_{0},u_{0},u^{k}_{1},u_{1}\in\textup{PSH}(X,\omega)$ are such that $u^{k}_{0}\searrow u_{0}$ and $u^{k}_{1}\searrow u_{1}$. Let $[0,1]\ni t\to u^{k}_{t},u_{t}\in\textup{PSH}(X,\omega)$ be the weak geodesics connecting $u^{k}_{0},u^{k}_{1}$ and $u_{0},u_{1}$ respectively. Then the following hold:

(i) $u^{k}_{t}\searrow u_{t},\ t\in[0,1]$.

(ii) For any $t_{1},t_{2}\in[0,1]$ we have that $[0,1]\ni l\to u_{(1-l)t_{1}+lt_{2}}\in\textup{PSH}(X,\omega)$ is the weak geodesic joining $u_{t_{1}}$ and $u_{t_{2}}$.*

Proof.

By the definition of $u^{k}\in\textup{PSH}(S\times X,\pi^{*}\omega)$ (3.36) it is clear that $u^{k}$ is decreasing in $k$ and $v:=\lim_{k}u_{k}\in\textup{PSH}(S\times X,\pi^{*}\omega)$. As $u$ is a candidate in the definition of each $u_{k}$, it follows that $u\leq u^{k}$, hence also $u\leq v$. For the other direction, by (3.37) we have that $u^{k}_{t}\leq(1-t)u^{k}_{0}+tu^{k}_{1}$ hence we can take the limit to obtain that

[TABLE]

Consequently $v$ is a candidate for $u$, giving that $v\leq u$, finishing the proof of (i).

Now we turn to proving (ii). Let $u^{k}_{0}=\max(u_{0},-k)$ and $u^{k}_{1}=\max(u_{1},-k)$ be the canonical cutoffs and let $t\to u^{k}_{t}$ be the bounded geodesics joining $u^{k}_{0}$ and $u^{k}_{1}$. Part (i) implies that $u^{k}_{t_{1}}\searrow u_{t_{1}}$ and $u^{k}_{t_{2}}\searrow u_{t_{2}}$. Hence, applying (i) again, it is enough to prove that $[0,1]\ni l\to u^{k}_{(1-l)t_{1}+lt_{2}}\in\textup{PSH}(X,\omega)$ is the bounded/weak geodesic joining $u^{k}_{t_{1}},u^{k}_{t_{2}}$.

Now (3.38) implies that each $t\to u^{k}_{t}$ is Lipschitz continuous in the $t$ variable, hence $u^{k}_{t}\to u^{k}_{t_{1,2}}$ uniformly as ${t\to t_{1,2}}$. By Lemma 3.14, $[0,1]\ni l\to u^{k}_{(1-l)t_{1}+lt_{2}}\in\textup{PSH}(X,\omega)$ is indeed the unique bounded/weak geodesic joining $u^{k}_{t_{1}}$ and $u^{k}_{t_{2}}$. ∎

The next result connects weak geodesics to the rooftop envelopes of Section 3.3:

Lemma 3.16** ([47]).**

Suppose $u_{0},u_{1}\in\textup{PSH}(X,\omega)$ and $t\to u_{t}$ is the weak geodesic connecting $u_{0},u_{1}$. Then the following holds:

[TABLE]

Proof.

Notice that $t\to v_{t}:=u_{t}-\tau t$ is the weak geodesic connecting $u_{0},u_{1}-\tau$, hence the proof of the general case reduces to the particlar case $\tau=0$.

By definition we have $P(u_{0},u_{1})\leq u_{0},u_{1}$. As a result, for the constant (sub)geodesic $t\to h_{t}:=P(u_{0},u_{1})$ we have $h\in\mathcal{S}$. This trivially gives $h_{t}\leq u_{t},\ t\in[0,1]$, hence $P(u_{0},u_{1})\leq\inf_{t\in(0,1)}u_{t}$.

For the reverse inequality, we use the Kiselman minimum principle [60, Chapter I, Theorem 7.5], which guarantees that $w:=\inf_{t\in(0,1)}u_{t}\in\textup{PSH}(X,\omega)$. Using this and (3.37) we obtain that $w\leq u_{0},u_{1}$ hence $w$ is a candidate for $P(u_{0},u_{1})$, i.e., $w\leq P(u_{0},u_{1})$, finishing the proof. ∎

We can now relate the super–level sets of tangent vectors along weak geodesics to contact sets of rooftop envelopes:

Lemma 3.17** ([47]).**

Suppose $u_{0},u_{1}\in\textup{PSH}(X,\omega)$. Let $t\to u_{t}$ be the weak geodesic joining $u_{0},u_{1}$. Then for any $\tau\in\mathbb{R}$ we have

[TABLE]

Proof.

By the previous result we have $\inf_{t\in[0,1]}(u_{t}-t\tau)=P(u_{0},u_{1}-\tau).$ Given $x\in X$, it follows that $P(u_{0},u_{1}-\tau t)(x)=u_{0}(x)$ if and only if $\inf_{t\in[0,1]}(u_{t}(x)-\tau t)=u_{0}(x)$. Convexity in the $t$ variable implies that this last identity is equivalent to $\dot{u}_{0}(x)\geq\tau$. ∎

With the aid of Lemma 3.16, we can show that weak geodesics with endpoints in finite energy classes stay inside finite energy classes:

Proposition 3.18** ([47]).**

Suppose $u_{0},u_{1}\in\mathcal{E}_{\chi}(X,\omega),\ \chi\in\mathcal{W}^{+}_{p},\ p\geq 1$. Then for the weak geodesic $t\to u_{t}$ connecting $u_{0},u_{1}$ we have that $u_{t}\in\mathcal{E}_{\chi}(X,\omega)$ for all $t\in[0,1]$. In case $u_{0},u_{1}\leq 0$ the following estimate holds:

[TABLE]

As a consequence of this proposition, for $u_{0},u_{1}\in\mathcal{E}_{\chi}(X,\omega)$ we will call the the curve $[0,1]\ni t\to u_{t}\in\mathcal{E}_{\chi}(X,\omega)$ the finite energy geodesic connecting $u_{0},u_{1}$.

Proof.

To start, Proposition 2.17 implies that $P(u_{0},u_{1})\in\mathcal{E}_{\chi}(X,\omega)$. By Lemma 3.16, we have that $P(u_{0},u_{1})\leq u_{t}$, hence by the monotonicity property (Corollary 2.8) it follows that $u_{t}\in\mathcal{E}_{\chi}(X,\omega),\ t\in[0,1]$.

When $u_{0},u_{1}\leq 0$, (3.37) implies that $u_{t}\leq 0,\ t\in[0,1]$. To finish the proof, by Proposition 2.17 and Proposition 2.7 we have the following estimates:

[TABLE]

∎

3.4 Extension of the $L^{p}$ metric structure to finite energy spaces

For the rest of this chapter we will focus only various $L^{p}$ Finsler geometries of $\mathcal{H}_{\omega}$. Most of the results we present also have analogs for the more general Orlicz Finsler structures discussed in the previous sections (see [48]). Having later applications in mind, we do not seek the greatest generality, and we leave it to the interested reader to adapt our argument to more general metrics. As done it previously, we will assume the volume normalization condition (3.24) throughout this section as well.

Given $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$, by Theorem 2.1 there exists decreasing sequences $u^{k}_{0},u^{k}_{1}\in\mathcal{H}_{\omega}$ such that $u^{k}_{0}\searrow u_{0}$ and $u^{k}_{1}\searrow u_{1}$. We propose to define the distance $d_{p}(u_{0},u_{1})$ by the formula:

[TABLE]

We will show that the above limit exists and it is also independent of the approximating sequences. Developing this further, our main result of this section is the following:

Theorem 3.19** ([48]).**

$(\mathcal{E}_{p}(X,\omega),d_{p})$* is a geodesic pseudo–metric space that extends $(\mathcal{H}_{\omega},d_{p})$. Additionally, for $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$ the finite energy geodesic $t\to u_{t}$ joining $u_{0},u_{1}$ (given by Proposition 3.18) is a $d_{p}$-geodesic.*

Recall that a pseudo–metric is just a metric that may not satisfy the non-degeneracy condition. Also, given a pseudo–metric space $(M,d)$, we say that a curve $[0,1]\ni t\to\gamma_{t}\in M$ is a $d$–geodesic if

[TABLE]

A geodesic pseudo–metric space $(M,d)$ is pseudo–metric space in which any two points can be joining by a $d$–geodesic.

In the next section we will show that in fact $d_{p}$ is in fact a bona fide metric, but this will require additional machinery. Finally, as the last major theorem of this chapter, we will prove that the resulting metric space $(\mathcal{E}_{p}(X,\omega),d_{p})$ is the completion of $(\mathcal{H}_{\omega},d_{p})$.

The proof of Theorem 3.19 will be split into a sequence of lemmas and propositions. Our first one is an estimate for “comparable” potentials:

Proposition 3.20**.**

Suppose $u,v\in\mathcal{H}_{\omega}$ with $u\leq v$. Then we have:

[TABLE]

Proof.

Suppose $[0,1]\ni t\to w_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ is the $C^{1,\bar{1}}$-geodesic segment joining $w_{0}=u$ and $w_{1}=v$. By (3.23) we have

[TABLE]

Since $u\leq v$, we have that $u\leq w_{t}$, as follows from (3.36) (or Theorem 3.2). Since $(t,x)\to w_{t}(x)$ is convex in the $t$ variable, we get $0\leq\dot{w}_{0}\leq v-u\leq\dot{w}_{1}$, and together with the above identity we obtain part of (3.41):

[TABLE]

Now we prove the rest of (3.41). Using $\omega_{u}^{n}\leq 2^{n}\omega_{(u+v)/2}^{n}$ we obtain that

[TABLE]

Since $u\leq(u+v)/2$, the first estimate of (3.42) allows us to continue and write:

[TABLE]

The lemma below implies that $d_{p}((u+v)/2,u)\leq d_{p}(v,u)$, giving the remaining estimate in (3.41). ∎

Lemma 3.21**.**

Suppose $u,v,w\in\mathcal{H}_{\omega}$ and $u\geq v\geq w$. Then $d_{p}(v,w)\leq d_{p}(u,w)$ and $d_{p}(u,v)\leq d_{p}(u,w)$.

Proof.

We introduce the $C^{1,\bar{1}}$ geodesics $[0,1]\ni t\to\alpha_{t},\beta_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ connecting $\alpha_{0}:=w,\alpha_{1}:=v$ and $\beta_{0}:=w,\beta_{1}:=u$ respectively. From (3.36) (or Theorem 3.2) it follows that both of these curves are increasing in $t$. Additionally, $\alpha\leq\beta$ by Theorem 3.2. As $\alpha_{0}=\beta_{0}$, it follows that $0\leq\dot{\alpha}_{0}\leq\dot{\beta}_{0}$. Using this and Theorem 3.6 we obtain that $d_{p}(w,v)\leq d_{p}(w,u)$. The estimate $d_{p}(u,v)\leq d_{p}(u,w)$ is proved similarly. ∎

Next we turn our attention to smooth approximants of finite energy potentials:

Lemma 3.22**.**

Suppose $u\in\mathcal{E}_{p}(X,\omega)$ and $\{u_{k}\}_{k}\subset\mathcal{H}_{\omega}$ is a sequence decreasing to $u$. Then $d_{p}(u_{l},u_{k})\to 0$ as $l,k\to\infty$.

Proof.

We can suppose that $l\leq k$. Then $u_{k}\leq u_{l}$, hence by Proposition 3.20 we have:

[TABLE]

We clearly have $u-u_{l},u_{k}-u_{l}\in\mathcal{E}_{p}(X,\omega_{u_{l}})$ and $u-u_{l}\leq u_{k}-u_{l}\leq 0$. Hence, applying Proposition 2.7 for the class $\mathcal{E}_{p}(X,\omega_{u_{l}})$ we obtain that

[TABLE]

As $u_{l}$ decreases to $u\in\mathcal{E}_{p}(X,\omega)$, it follows from the dominated convergence theorem that $d_{p}(u_{l},u_{k})\to 0$ as $l,k\to\infty$. ∎

Our next lemma confirms that the way we proposed to extend the $d_{p}$ to $\mathcal{E}_{p}(X,\omega)$ (see (3.39)) does not have inconsistencies:

Lemma 3.23**.**

Given $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$, the limit in (3.39) is finite and independent of the approximating sequences $u^{k}_{0},u^{k}_{1}\in\mathcal{H}_{\omega}$.

In particular, this result implies that for $u_{0},u_{1}\in\mathcal{H}_{\omega}$ the distance $d_{p}(u_{0},u_{1})$ will be the same according to both (3.39) and our original definition in (3.4). Lastly, the triangle inequality will also hold, hence $d_{p}$ is a pseudo–metric on $\mathcal{E}_{p}(X,\omega)$, as claimed in Theorem 3.19.

Proof.

By the triangle inequality and Lemma 3.22 we can write:

[TABLE]

proving that $d_{p}(u^{k}_{0},u^{k}_{1})$ is indeed convergent.

Now we prove that the limit in (3.39) is independent of the choice of approximating sequences. Let $v^{l}_{0},v^{l}_{1}\in\mathcal{H}_{\omega}$ be different approximating sequences. By adding small constants if necessary, we can arrange that the sequences $u^{l}_{0},u^{l}_{1}$, respectively $v^{l}_{0},v^{l}_{1}$, are strictly decreasing to $u_{0},u_{1}$.

Fixing $k$ for the moment, the sequence $\{\max\{u^{k+1}_{0},v^{j}_{0}\}\}_{j\in\mathbb{N}}$ decreases pointwise to $u^{k+1}_{0}$. By Dini’s lemma the convergence is uniform, hence there exists $j_{k}\in\mathbb{N}$ such that for any $j\geq j_{k}$ we have $v^{j}_{0}<u^{k}_{0}$. By repeating the same argument we can also assume that $v^{j}_{1}<u^{k}_{1}$ for any $j\geq j_{k}$. By the triangle inequality again

[TABLE]

From (3.43) it follows that for $k$ big enough the quantities $d(u^{j}_{0},v^{k}_{0})$, $d(u^{j}_{1},v^{k}_{1}),\ j\geq j_{k}$ are arbitrarily small, hence $d_{p}(u_{0},u_{1})$ is independent of the choice of approximating sequences.

When $u_{0},u_{1}\in\mathcal{H}_{\omega}$, then it is possible to approximate with a constant sequence, hence our argument implies that the restriction of $d_{p}$ (as extended in (3.39)) to $\mathcal{H}_{\omega}$ coincides with the original definition in (3.4). ∎

By the previous result, the triangle inequality is inherited by the extension of $d_{p}$ to $\mathcal{E}_{p}(X,\omega)$, making $(\mathcal{E}_{p}(X,\omega),d_{p})$ a pseudo–metric space. In the last part of this section we turn our attention to showing that finite energy geodesic segments of Proposition 3.18 are in fact $d_{p}$–geodesics:

Lemma 3.24**.**

Suppose $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$ and $[0,1]\ni t\to u_{t}\in\mathcal{E}_{p}(X,\omega)$ is the finite energy geodesic segment connecting $u_{0},u_{1}$. Then $t\to u_{t}$ is a $d_{p}$-geodesic.

Proof.

First we prove that

[TABLE]

By Theorem 2.1, suppose $u^{k}_{0},u^{k}_{1}\in\mathcal{H}_{\omega}$ are strictly decreasing approximating sequences of $u_{0},u_{1}$ and let $[0,1]\ni t\to u^{k}_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ be the decreasing sequence of $C^{1,\bar{1}}$ geodesics connecting $u_{0}^{k},u_{1}^{k}$. By the definition of (3.39) and Theorem 3.6 we can write:

[TABLE]

By Proposition 3.15(i), the geodesic segments $[0,1]\ni t\to u^{k}_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ are decreasing pointwise to $[0,1]\ni t\to u_{t}\in\mathcal{E}_{p}(X,\omega)$. In particular, this implies that $u^{k}_{l}\searrow u_{l}$. We want to find a decreasing sequence $\{w^{k}_{l}\}_{k}\subset\mathcal{H}_{\omega}$ such that $u^{k}_{l}\leq w^{k}_{l}$, $w^{k}_{l}\searrow u_{l}$ and

[TABLE]

where $t\to w^{k}_{t}$ is the $C^{1,\bar{1}}$–geodesic segment connecting $u^{k}_{0}$ and $w^{k}_{l}$. By the definition of $d_{p}$, letting $k\to\infty$ in (3.45) would give us (3.44).

Finding such sequence $w_{l}^{k}$ is always possible by an (iterated) application of Lemma 3.25 applied to $v_{0}:=u^{k}_{0}$ and $v_{1}:=u^{k}_{l}$, as we observe that the bounded geodesic segment connecting $u^{k}_{0}$ and $u^{k}_{l}$ is exactly $t\to u_{lt}^{k}$ (see Proposition 3.15(ii)).

Finally, to finish the proof, we argue that for $t_{1},t_{2}\in[0,1],\ t_{1}\leq t_{2}$ we have

[TABLE]

Let $h_{0}=u_{t_{2}}$ and $h_{1}=u_{0}$. From Proposition 3.15(ii) it follows that $[0,1]\ni t\to h_{t}:=u_{t_{2}(1-t)}\in\mathcal{E}_{p}(X,\omega)$ is the finite energy geodesic connecting $h_{0},h_{1}$. Applying 3.44 to $t\to h_{t}$ and $l=1-t_{1}/t_{2}$ we obtain

[TABLE]

Now applying (3.44) for $t\to u_{t}$ and $l=t_{2}$ we have

[TABLE]

Putting these last two formulas together we obtain (3.46), finishing the proof. ∎

Lemma 3.25**.**

Suppose $v_{0},v_{1}\in\textup{PSH}(X,\omega)\cap L^{\infty}$ and $\{v^{j}_{1}\}_{j\in\mathbb{N}}\subset\textup{PSH}(X,\omega)\cap L^{\infty}$ is sequence decreasing to $v_{1}$. By $[0,1]\ni t\to v_{t},v_{t}^{j}\in\textup{PSH}(X,\omega)\cap L^{\infty}$ we denote the bounded geodesics connecting $v_{0},v_{1}$ and $v_{0},v^{j}_{1}$ respectively. By convexity in the $t$ variable, we can introduce $\dot{v}_{0}=\lim_{t\to 0}(v_{t}-v_{0})/t$ and $\dot{v}^{j}_{0}=\lim_{t\to 0}(v^{j}_{t}-v_{0})/t$, and the following holds:

[TABLE]

Proof.

By (3.38) there exists $C>0$ such that $\|\dot{v}_{0}\|_{L^{\infty}(X)},\|\dot{v}^{j}_{0}\|_{L^{\infty}(X)}\leq C$. We also have $v\leq v^{j},\ j\in\mathbb{N}$ by (3.36) (or Theorem 3.2). As we have convexity in the $t$ variable and all our bounded geodesics share the same starting point, it also follows that $\dot{v}^{j}_{0}\searrow\dot{v}_{0}$ pointwise. Consequently, the lemma follows now from Lebesgue’s dominated convergence theorem. ∎

3.5 The Pythagorean formula and applications

In this section we explore the geometry of the operator $(u,v)\to P(u,v)$ restricted to the spaces $\mathcal{E}_{p}(X,\omega)$. The main focus will be on the following result, establishing a metric relationship between the vertices of the “triangle” $(u,v,P(u,v))$. This will help in proving that $d_{p}$ is non–degenerate, that $v\to P(u,v)$ is a $d_{p}$–contraction, and a number of other properties. Again, throughout this section we will assume the volume normalization condition (3.24) holds. The main result of this section is the Pythagorean formula of [48]:

Theorem 3.26** (Pythagorean formula, [48, Corollary 4.14]).**

Given $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$, we have $P(u_{0},u_{1})\in\mathcal{E}_{p}(X,\omega)$ and

[TABLE]

The name of the above formula comes from the particular case $p=2$, in which case it suggests that $u_{0},u_{1}$ and $P(u_{0},u_{1})$ form a right triangle with hypotenuse $t\to u_{t}$.

Before we give the proof of this result, we argue how it implies the non–degeneracy of $d_{p}$, giving that $(\mathcal{E}_{p}(X,\omega),d_{p})$ is a metric space:

Proposition 3.27**.**

Given $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$ if $d_{p}(u_{0},u_{1})=0$ then $u_{0}=u_{1}$.

Proof.

By Theorem 3.26 it follows that $d_{p}(u_{0},P(u_{0},u_{1}))=0$ and $d_{p}(u_{1},P(u_{0},u_{1}))=0$. By the first estimate of the next lemma, which generalizes Proposition 3.20, it follows that $u_{0}=P(u_{0},u_{1})$ a.e. with respect to $\omega_{P(u_{0},u_{1})}^{n}$, and similarly, $u_{1}=P(u_{0},u_{1})$ a.e. with respect to $\omega_{P(u_{0},u_{1})}^{n}$. By the domination principle (Proposition 2.21) it follows that $u_{0}\leq P(u_{0},u_{1})$ and $u_{1}\leq P(u_{0},u_{1})$. As the reverse inequalities are trivial, we obtain that $u_{0}=P(u_{0},u_{1})=u_{1}$. ∎

Lemma 3.28**.**

Suppose $u,v\in\mathcal{E}_{p}(X,\omega)$ with $u\leq v$. Then we have:

[TABLE]

Proof.

By Proposition 2.9, all the integrals in (3.47) are finite. By Theorem 2.1, we can choose $u_{k},v_{k}\in\mathcal{H}_{\omega}$ such that $v_{k}\searrow v$ and $u_{k}\searrow u$ and $u_{k}\leq v_{k}$. By Proposition 3.20, estimate (3.47) holds for $u_{k},v_{k}$. Using Proposition 2.20 and definition (3.39) we can take the limit $k\to\infty$ in the estimates for $u_{k},v_{k}$, and obtain (3.47). ∎

By the next corollary we will only need to prove Theorem 3.26 for smooth potentials $u_{0},u_{1}$:

Corollary 3.29**.**

If $\{w_{k}\}_{k\in\mathbb{N}}\subset\mathcal{E}_{p}(X,\omega)$ decreases (increases a.e.) to $w\in\mathcal{E}_{p}(X,\omega)$ then $d_{p}(w_{k},w)\to 0$.

Proof.

By the previous lemma, we have $d_{p}(w,w_{k})^{p}\leq\int_{X}|w-w_{k}|^{p}(\omega_{w_{k}}^{n}+\omega_{w}^{n})$. We can use Proposition 2.20 again to conclude that $d_{p}(w,w_{k})\to 0$. ∎

As $P(u_{0},u_{1})\in\mathcal{H}_{\omega}^{1,\bar{1}}$ for $u_{0},u_{1}\in\mathcal{H}_{\omega}$ (Corollary 2.14), we need to generalize Theorem 3.6 for endpoints in $\mathcal{H}_{\omega}^{1,\bar{1}}$, before we can prove Theorem 3.26. The first step is the next lemma:

Lemma 3.30**.**

Suppose $u_{0},u_{1}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ and $t\to u_{t}$ is the bounded geodesic connecting them. Then the following holds:

[TABLE]

Recall that it is not yet known if $t\to u_{t}$ is $C^{1}$ in the $t$-direction when $u_{0},u_{1}\in\mathcal{H}_{\omega}^{1,\bar{1}}$. However, since $t\to u_{t}$ is $t$-convex, it makes sense to define $\dot{u}_{0}$ as the right derivative at $t=0$ and $\dot{u}_{1}$ as the left derivative at $t=0$.

Proof.

To obtain (3.48) we prove:

[TABLE]

[TABLE]

Using Remark 2.16 and Lemma 3.17 multiple times we can write:

[TABLE]

where in the second line we used Lemma 3.17, in the third line we used Remark 2.16 and in the sixth line we used Lemma 3.17 again, but this time for the “reversed” geodesic $t\to u_{1-t}$. Formula (3.50) follows if we apply (3.49) to the bounded geodesic $t\to u_{1-t}$. ∎

Proposition 3.31**.**

Suppose $u_{0},u_{1}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ and $t\to u_{t}$ is the bounded geodesic connecting them. Then we have:

[TABLE]

Proof.

As usual, let $u^{k}_{0},u^{k}_{1}\in\mathcal{H}_{\omega}$ be a sequence of potentials decreasing to $u_{0},u_{1}$. Let $[0,1]\ni t\to u^{kl}_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ be the $C^{1,\bar{1}}$–geodesic joining $u_{0}^{k},u^{l}_{1}$. By Theorem 3.6 we have

[TABLE]

If we let $l\to\infty$, by Lemma 3.25 and (3.39) we obtain that that

[TABLE]

where $t\to u^{k}_{t}$ is bounded geodesic connecting $u^{k}_{0}$ with $u_{1}$. Using the previous lemma we can write:

[TABLE]

Letting $k\to\infty$, another application of Lemma 3.25 yields (3.51) for $t=1$, and the case $t=0$ follows by symmetry. ∎

Proof of Theorem 3.26.

By Corllary 3.29, it is enough to prove the Pythgaorean formula for $u_{0},u_{1}\in\mathcal{H}_{\omega}$. According to Corollary 2.14 we have $P(u_{0},u_{1})\in\mathcal{H}_{\omega}^{1,\bar{1}}$. Suppose $[0,1]\ni t\to u_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ is the $C^{1,\bar{1}}$–geodesic connecting $u_{0},u_{1}$. By Theorem 3.6 we have:

[TABLE]

To complete the argument we will prove the following:

[TABLE]

[TABLE]

We prove now (3.52). Using Lemma 3.17 we can write:

[TABLE]

Suppose $t\to\tilde{u}_{t}$ is the bounded geodesic connecting $P(u_{0},u_{1}),u_{1}$. Since $P(u_{0},u_{1})\leq\tilde{u}_{t},\ t\in[0,1]$, the correspondence $(t,x)\to\tilde{u}_{t}(x)$ is increasing in the $t$ variable, hence $\dot{\tilde{u}}_{0}\geq 0$. By (3.51), Lemma 3.17 and Proposition 2.15 we can write:

[TABLE]

where in the third line we have used Lemma 3.17, in the sixth line we have used Proposition 2.15 together with the fact that $\{P(u_{0},u_{1})=u_{1}\}\cap\{P(u_{0},u_{1}-\tau)=P(u_{0},u_{1})\}$ is empty for $\tau>0$.

Comparing our above calculations (3.52) follows. One can conclude (3.53) from (3.52) after reversing the roles of $u_{0},u_{1}$ and then using (3.49). ∎

As another application of Theorem 3.26 we will show that the $d_{p}$ metric is comparable to a concrete analytic expression:

Theorem 3.32** ([48]).**

For any $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$ we have

[TABLE]

Proof.

To obtain the first estimate we use the triangle inequality and Lemma 3.28:

[TABLE]

Now we deal with the second estimate in (3.54). By the next result result and Theorem 3.26 we can write

[TABLE]

By a similar reasoning as above, and the fact that $2^{n}\omega^{n}_{(u_{0}+u_{1})/2}\geq\omega^{n}_{u_{0}}$ we can write:

[TABLE]

Adding the last two estimates, and using the convexity of $t\to t^{p}$ we obtain:

[TABLE]

By symmetry we also have $2^{2n+3p+2}d_{p}(u_{0},u_{1})^{p}\geq\int_{X}|u_{0}-u_{1}|^{p}\omega_{u_{1}}^{n}$, and adding these last two estimates together the second inequality in (3.54) follows. ∎

Lemma 3.33**.**

Suppose $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$. Then we have

[TABLE]

Proof.

Using Theorem 3.26 and Lemma 3.21 we can start writing:

[TABLE]

where in the second line we have used Lemma 3.21 and the fact that $P(u_{0},u_{1})\leq P(u_{0},(u_{0}+u_{1})/2)$, in the third and fifth line Lemma 3.28, and in the sixth line we have used Theorem 3.26 again. ∎

3.6 The complete metric spaces $(\mathcal{E}_{p}(X,\omega),d_{p})$

To show completeness of $(\mathcal{E}_{p}(X,\omega),d_{p})$ we have to argue that limits of Cauchy sequences land in $\mathcal{E}_{p}(X,\omega)$. According to our first result, which extends Corollary 3.29, monotone $d_{p}$–bounded sequences are Cauchy, and their limit is in $\mathcal{E}_{p}(X,\omega)$:

Lemma 3.34**.**

Suppose $\{u_{k}\}_{k\in\mathbb{N}}\subset\mathcal{E}_{p}(X,\omega)$ is a decreasing/increasing $d_{p}$–bounded sequence. Then $u=\lim_{k\to\infty}u_{k}\in\mathcal{E}_{p}(X,\omega)$ and additionally $d_{p}(u,u_{k})\to 0$.

Proof.

First we assume that $u_{k}$ is decreasing. Without loss of generality, we can also suppose that $u_{k}<0$. By Lemma 3.28 it follows that

[TABLE]

As $\int_{X}|u_{j}|^{p}\omega^{n}$ is uniformly bounded, it follows that the limit function $u=\lim_{k}u_{k}$ exists and $u\in\textup{PSH}(X,\omega)$. Since $E_{p}(u_{j})=\int_{X}|u_{j}|^{p}\omega_{u_{j}}^{n}$ is uniformly bounded we can use Lemma 2.10, to obtain that $u\in\mathcal{E}_{p}(X,\omega)$ and Corollary 3.29 gives that $d_{p}(u_{k},u)\to 0$.

Now we turn to the case when $u_{k}$ is increasing. We know that $d_{p}(u_{k},0)$ is uniformly bounded, hence using Theorem 3.32 we obtain that $\int_{X}|u_{k}|^{p}\omega^{n}$ is uniformly bounded as well. By the $L^{1}$–compactness of subharmonic functions [60, Chapter I, Proposition 4.21] we obtain that $u_{k}\nearrow u\in\textup{PSH}(X,\omega)$ a.e. with respect to $\omega^{n}$. By the monotonicity property (Corollary 2.8) we obtain that $u\in\mathcal{E}_{p}(X,\omega)$ and another application of Corollary 3.29 gives that $d_{p}(u_{k},u)\to 0$. ∎

As another consequence of the Pythagorean identity, we note the contractivity of the operator $v\to P(u,v)$, when restricted to finite energy spaces. This result will be essential in proving that $(\mathcal{E}_{p}(X,\omega),d_{p})$ is a complete metric space.

Proposition 3.35**.**

Given $u,v,w\in\mathcal{E}_{p}(X,\omega)$ we have

[TABLE]

Proof.

By Corollary 3.29 we can assume that $u,v,w\in\mathcal{H}_{\omega}^{1,\bar{1}}$ and $P(u,v),P(u,w)\in\mathcal{H}_{\omega}^{1,\bar{1}}$ (see Corollary 2.14).

First we assume that $v\leq w$. Let $t\to\phi_{t}$ be the $C^{1,\bar{1}}$ geodesic connecting $v,w$, and $t\to\psi_{t}$ be the bounded geodesic connecting $P(u,v),P(u,w)$. By Proposition 3.31 we have to argue that

[TABLE]

Proposition 2.15 implies that

[TABLE]

We argue that the first term in this sum is zero. As $P(u,v)\leq P(u,w)\leq u$, by (3.36) (or Theorem 3.2), it is clear that $P(u,v)\leq\psi_{t}\leq u,\ t\in[0,1]$. Hence, if $x\in\{P(u,v)=u\}$ then $\psi_{t}(x)=u(x),\ t\in[0,1]$, implying \dot{\psi}_{0}\big{|}_{\{P(u,v)=u\}}\equiv 0.

At the same time, using Theorem 3.2 again, it follows that $\psi_{t}\leq\phi_{t},\ t\in[0,1]$. This implies that 0\leq\dot{\psi}_{0}\big{|}_{\{P(u,v)=v\}}\leq\dot{\phi}_{0}\big{|}_{\{P(u,v)=v\}}, which in turn implies (3.56).

The general case follows now from an application of the Pythagorean formula (Theorem 3.26) and what we just proved above:

[TABLE]

∎

We arrive to the main result of this section. Using the previous proposition we will show that any Cauchy sequence of $(\mathcal{E}_{p},d_{p})$ is equivalent to a monotonic Cauchy sequence. By the Lemma 3.34, such sequences have limit in $\mathcal{E}_{p}(X,\omega)$, showing that this space is complete with respect to $d_{p}$:

Theorem 3.36** ([48]).**

$(\mathcal{E}_{p}(X,\omega),d_{p})$* is a geodesic metric space, which is the metric completion of $(\mathcal{H}_{\omega},d_{p})$. Additionally, the finite energy geodesic $t\to u_{t}$ connecting $u_{0},u_{1}\in\mathcal{E}_{p}(X,\omega)$ is a $d_{p}$-geodesic.*

Proof.

By Theorem 2.1 and Corollary 3.29 $\mathcal{H}_{\omega}$ is a $d_{p}$–dense subset of $\mathcal{E}_{p}(X,\omega)$. The statement about geodesics was addressed in Theorem 3.19, hence we only need to argue completeness.

Suppose $\{u_{k}\}_{k\in\mathbb{N}}\subset\mathcal{E}_{p}(X,\omega)$ is a $d_{p}$–Cauchy sequence. We will prove that there exists $v\in\mathcal{E}_{p}(X,\omega)$ such that $d_{p}(u_{k},v)\to 0.$ After passing to a subsequence we can assume that

[TABLE]

By Proposition 2.17 we can introduce $v^{k}_{l}=P(u_{k},u_{k+1},\ldots,u_{k+l})\in\mathcal{E}_{p}(X,\omega),\ l,k\in\mathbb{N}$. We argue first that each decreasing sequence $\{v^{k}_{l}\}_{l\in\mathbb{N}}$ is $d_{p}$–Cauchy. We observe that $v^{k}_{l+1}=P(v^{k}_{l},u_{k+l+1})$ and $v^{k}_{l}=P(v^{k}_{l},u_{k+l})$. Using this and Proposition 3.35 we can write:

[TABLE]

By Lemma 3.34, it follows now that each sequence $\{v^{k}_{l}\}_{l\in\mathbb{N}}$ is $d_{p}$–convergening to some $v^{k}\in\mathcal{E}_{p}(X,\omega)$. Using the same trick as above, we can write:

[TABLE]

[TABLE]

Consequently, $\{v^{k}\}_{k\in\mathbb{N}}$ is an increasing $d_{p}$–bounded $d_{p}$–Cauchy sequence that is equivalent to $\{u_{k}\}_{k\in\mathbb{N}}$. By Lemma (3.34) there exists $v\in\mathcal{E}_{p}(X,\omega)$ such that $d_{p}(v_{k},v)\to 0$, which in turn implies that $d_{p}(u_{k},v)\to 0$, finishing the proof. ∎

3.7 Special features of the $L^{1}$ Finsler geometry

When it comes to applications of measure theory, the most important $L^{p}$ spaces are the $L^{1}$ space, its dual $L^{\infty}$, and the Hilbert space $L^{2}$. A similar pattern can be observed with the $L^{p}$ Finsler geometries on $\mathcal{H}_{\omega}$. One can show that the $d_{\infty}$ metric is a multiple of the usual $L^{\infty}$ metric. As follows from the work of Calabi–Chen [31], the completion of $(\mathcal{H}_{\omega},d_{2})$ is non–postively curved and as such it provides fertile ground to the study of geometric gradient flows explored in [100, 101, 13].

In this section we will focus exclusively on the $L^{1}$ Finsler geometry of $\mathcal{H}_{\omega}$, whose path length metric structure has a number of interesting properties making it suitable in our study of canonical Kähler metrics, detailed in the next chapter.

The starting point is the Monge–Ampère energy $I:\textup{PSH}(X,\omega)\cap L^{\infty}\to\mathbb{R}$ (sometimes called Aubin–Yau, or Aubin–Mabuchi energy):

[TABLE]

The following lemma explains the choice of name for the $I$ energy, as it turns out that the first order variation of this functional is exactly the complex Monge–Ampère operator:

Lemma 3.37**.**

Suppose $[0,1]\ni t\to v_{t}\in\mathcal{H}_{\omega}$ is a smooth curve. Then $t\to I(v_{t})$ is differentiable and

[TABLE]

Proof.

Using the definition of $I$ we can calculate the differential of $t\to I(v_{t})$ directly:

[TABLE]

∎

Based on the last lemma, we deduce a number of properties of the Monge–Ampère energy:

Proposition 3.38**.**

*Given $u,v\in\textup{PSH}(X,\omega)\cap L^{\infty}(X)$, the following hold:

[TABLE]

[TABLE]

Proof.

First we show (3.58) for $u,v\in\mathcal{H}_{\omega}$. Let $[0,1]\ni t\to h_{t}:=(1-t)v+tu\in\mathcal{H}_{\omega}$ be the smooth affine curve joining $v$ and $u$. By the previous lemma and the binomial theorem we can write

[TABLE]

After an elementary calculation involving integration by parts we obtain that

[TABLE]

giving (3.58) for $u,v\in\mathcal{H}_{\omega}$. For the general case $u,v\in\mathcal{E}^{1}$, by Theorem 2.1 we can find $u_{k},v_{k}\in\mathcal{H}_{\omega}$ decreasing to $u$ and $v$ respectively. We can use Proposition 2.20 to conclude that $I(u_{k})\to I(u)$, $I(v_{k})\to I(v)$ and $\int_{X}(u_{k}-v_{k})\omega_{u_{k}}^{j}\wedge\omega_{v_{k}}^{n-j}\to\int_{X}(u-v)\omega_{u}^{j}\wedge\omega_{v}^{n-j}$, giving (3.57) for elements of $\textup{PSH}(X,\omega)\cap L^{\infty}(X)$.

Now we turn to the estimates in (3.59). First we prove the following estimate:

[TABLE]

Using this estimate inductively for the terms of (3.58) yields (3.59). ∎

As a consequence of (3.58) we note the monotonicity property of $I$:

Corollary 3.39**.**

Suppose $u,v\in\textup{PSH}(X,\omega)\cap L^{\infty}$ such that $u\geq v$. Then $I(u)\geq I(v)$.

This results allows to extend the definition of $I$ to $\textup{PSH}(X,\omega)$. Indeed, if $u\in\textup{PSH}(X,\omega)$ then using the canonical cutoffs $u_{k}=\max(u,-k)$ we can write:

[TABLE]

Though the limit in the above definition is well defined, it may equal $-\infty$ for certain potentials $u$. By our next result $I$ is finite exactly on $\mathcal{E}_{1}(X,\omega)$. What is more, $I$ is $d_{1}$–continuous on $\mathcal{E}_{1}(X,\omega)$, and Proposition 3.38 also extends to this bigger space:

Proposition 3.40**.**

Let $u\in\textup{PSH}(X,\omega)$. Then $I(u)>-\infty$ if and only if $u\in\mathcal{E}_{1}(X,\omega)$. In addition, the following hold for $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)$:

[TABLE]

[TABLE]

[TABLE]

Proof.

By definition, $I(u+c)=I(u)+c$ for all $c\in\mathbb{R}$, hence we can assume that $u\leq 0$. Consequently, by (3.59) for the canonical cutoffs $u_{k}=\max(u,-k)$ we have:

[TABLE]

If $u\in\mathcal{E}_{1}(X,\omega)$ then by the fundamental estimate (Proposition 2.7) we have that $E_{1}(u_{k})$ is uniformly bounded, hence $I(u)=\lim_{k}I(u_{k})$ is finite. On the other hand, if $I(u)$ is finite then $I(u_{k})$ is uniformly bounded, hence so is $E_{1}(u_{k})$. By Lemma 2.10 we obtain that $u\in\mathcal{E}_{1}(X,\omega)$.

Now we turn to (3.62). By Corollary 3.29 and (3.61) it follows that it is enough to prove (3.62) for bounded potentials. Furthermore, using Theorem 2.1 it is actually enough to prove (3.62) for potentials in $\mathcal{H}_{\omega}$. For $u_{0},u_{1}\in\mathcal{H}_{\omega}$ let $t\to u_{t}$ be the $C^{1,\bar{1}}$ geodesic connecting $u_{0},u_{1}$. By (3.65), proved in the next lemma, we can finish our argument:

[TABLE]

where in the last equality we have used Theorem 3.6.

Finally, (3.63) and (3.64) follow from Proposition 3.38 and Proposition 2.20 ∎

Lemma 3.41**.**

Suppose $t\to u_{t}$ is the $C^{1,\bar{1}}$ geodesic connecting $u_{0},u_{1}\in\mathcal{H}_{\omega}$. Then $t\to I(u_{t})$ is differentiable, moreover

[TABLE]

Proof.

As $t\to u_{t}$ is $C^{1,\bar{1}}$ it follows that $(u_{t+\delta}-u_{t})/\delta\to\dot{u}_{t}$ uniformly as $\delta\to 0$. Using this and (3.58) we can write:

[TABLE]

where we have used that $\omega_{u_{t+\delta}}^{j}\wedge\omega_{u_{t}}^{n-j}\to\omega_{u_{t}}^{n}$ weakly. ∎

Further linking the Monge–Ampère energy to the $L^{1}$ Finsler geometry is the fact that $I$ is affine along finite energy geodesics:

Proposition 3.42**.**

Suppose $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)$ and $t\to u_{t}$ is the finite energy geodesic connecting $u_{0},u_{1}$. Then $t\to I(u_{t})$ is affine: $I(u_{t})=(1-t)I(u_{0})+tI(u_{1}),\ t\in[0,1].$

Proof.

First we show that $t\to I(u_{t})$ is affine for the $C^{1,\bar{1}}$ geodesic connecting $u_{0},u_{1}\in\mathcal{H}_{\omega}$. Let $[0,1]\ni t\to u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ be the smooth $\varepsilon$–geodesic connecting $u_{0},u_{1}$ (see (3.12)). By Lemma 3.37 we have that

[TABLE]

As we take one more derivative of the above formula, we can use the $L^{2}$ Mabuchi Levi–Civita connection (see (3.7)) and (3.11) to obtain:

[TABLE]

For fixed $t\in[0,1]$ we have that $u^{\varepsilon}_{t}\nearrow u_{t}$ uniformly, hence (3.58) gives that $I(u^{\varepsilon}_{t})\to I(u_{t})$. By an elementary argument, (3.67) implies that $t\to I(u_{t})=\lim_{\varepsilon\to 0}I(u_{t}^{\varepsilon})$ is affine.

Next we show the result for $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)$. In this case let $u^{k}_{0},u^{k}_{1}\in\mathcal{H}_{\omega}$ be decreasing approximating sequences that exist by Theorem 2.1. Let $t\to u^{k}_{t}$ be the $C^{1,\bar{1}}$ geodesics joining $u^{k}_{0},u^{k}_{1}$, and $t\to u_{t}$ be the finite energy geodesic joining $u_{0},u_{1}$. By Proposition 3.15 it follows that for fixed $t$ we have $u^{k}_{t}\searrow u_{t}$. Corollary 3.29 now gives that $d_{1}(u^{k}_{t},u_{t})\to 0$. We can now use Proposition 3.40 to conclude that $I(u^{k}_{t})\to I(u_{t})$. Since $t\to I(u^{k}_{t})$ is affine, so is $t\to I(u_{t})$. ∎

By revisiting the Pythagorean formula of the $L^{1}$ geometry (Theorem 3.26) we obtain the following concrete formula for the $d_{1}$ metric in terms of the Monge–Ampère energy:

Proposition 3.43**.**

If $u,v\in\mathcal{E}_{1}(X,\omega)$ then

[TABLE]

In particular, if $u\geq v$ then $d_{1}(u,v)=I(u)-I(v)$.

Proof.

We show that $d_{1}(u,v)=I(u)-I(v)$ when $u\geq v$. Then the Pythagorean formula (Theorem 3.26) will imply (3.68). By Theorem 2.1 , Corollary 3.29 and Proposition 3.40 it is enough to show $d_{1}(u,v)=I(u)-I(v)$ for $u,v\in\mathcal{H}_{\omega}$. Let $[0,1]\ni t\to u_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ be the $C^{1,\bar{1}}$ geodesic joining $u_{0}:=v$ and $u_{1}:=u$. As $u_{1}\geq u_{0}$, by Theorem 3.2 we obtain that $u_{0}\leq u_{t}$. Using convexity in the $t$ variable, we obtain that $0\leq\dot{u}_{0}\leq\dot{u}_{t}.$ Since $\dot{u}_{t}\geq 0$, by Theorem 3.6 we can conclude that

[TABLE]

where in the second equality we have used (3.65). ∎

Recall that the correspondence $u\to\omega_{u}$ is one-to-one up to a constant. To make the correspondence between metrics and potentials one-to-one, we need to restrict the map $u\to\omega_{u}$ to a hyper–surface of $\mathcal{H}_{\omega}$. There are many normalizations that come to mind, but the most convenient choice is to use the following space and its completion:

[TABLE]

As $I:\mathcal{E}_{1}(X,\omega)\to\mathbb{R}$ is $d_{1}$-continuous, we see that $\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$ is indeed the $d_{1}$–completion of $\mathcal{H}_{\omega}\cap I^{-1}(0)$. We focus on the preimage of $I$ due to the conclusion of Proposition 3.42. Indeed, according to this result, if $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$ then $t\to u_{t}$, the finite energy geodesic connecting $u_{0},u_{1}$, satisfies $u_{t}\in\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$, hence the “hypersurface” $\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$ is totally geodesic.

The $J$ energy is closely related to the Monge–Ampère energy and is given by the following formula:

[TABLE]

By (3.64) it follows that $J(u)\geq 0$ for all $u\in\mathcal{E}_{1}(X,\omega)$.

For $u\in\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$, the $J$ energy is given by the especially simple formula $J(u)=\frac{1}{V}\int_{X}u\omega^{n}$. Additionally, on $\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$ the growth of the $d_{1}$ metric and the $J$ energy is closely related:

Proposition 3.44** ([48, 57]).**

There exists $C=C(X,\omega)>1$ such that

[TABLE]

Proof.

Let $u\in\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)$. By Theorem 3.32 we have

[TABLE]

implying the first estimate in (3.71). For the second estimate, since $I(u)=0$, we have that $\sup_{X}u\geq 0$ and

[TABLE]

Clearly, $u-\sup_{X}u\leq\min(0,u)$, so $u-\sup_{X}u\leq P(0,u)$. Thus, $-\sup_{X}u=I(u-\sup_{X}u)\leq I(P(0,u)).$ Combined with (3.72), we obtain that $d_{1}(0,u)=-2I(P(0,u))\leq 2\sup_{X}u.$ Finally, according to the next basic lemma, $\sup_{X}u\leq C^{\prime}\int_{X}u\omega^{n}+C^{\prime}$ for some $C^{\prime}(X,\omega)>1$, finishing the proof. ∎

Lemma 3.45**.**

There exists $C:=C(X,\omega)>1$ such that for any $u\in\textup{PSH}(X,\omega)$ we have

[TABLE]

This is a well known result in pluripotential theory. A proof using compactness can be found in [72, Proposition 1.7]. Below we give a constructive argument using only the sub-mean value property of psh functions.

Proof.

The first estimate is trivial. To argue the second estimate, it is enough to show that $\int_{X}u\omega^{n}$ is uniformly bounded from below, for all $u\in\textup{PSH}(X,\omega)$ with $\sup_{X}u=0$. We fix such potential $u$ for the rest of the proof.

We pick nested coordinate charts $U_{k}\subset W_{k}\subset X$, such that $\{U_{k}\}_{1\leq k\leq N}$ covers $X$, and there exist holomorphic diffeomorphisms $\varphi_{k}:B(0,4)\to W_{k}$ such that $\varphi_{k}(B(0,1))=U_{k}$ and there exists $\psi_{k}\in C^{\infty}(W_{k})$ such that $\omega|_{W_{k}}=i\partial\bar{\partial}\psi_{k}$. Notice that it is enough to show the existence of $C:=C(X,\omega)<0$ such that

[TABLE]

where $d\mu$ is the Euclidean measure on $\mathbb{C}^{n}$.

Using our setup, we obtain that $\psi_{k}+u\in\textup{PSH}(W_{k})$ for all $k\in\{1,\ldots,N\}$. As $u$ is usc, its supremum is realized at some $x_{1}\in X$, i.e. $u\leq u(x_{1})=0$. As $\{U_{k}\}_{k}$ covers $X$, $x_{1}\in U_{l}$ for some $l\in\{1,\ldots,N\}$. For simplicity, we can assume that $l=1$.

Let $z_{1}:=\varphi^{-1}_{1}(x_{1})\in B(0,1)$. As $B(z_{1},2)\subset B(0,4)$, by the sub-meanvalue property of psh functions we can write:

[TABLE]

Since $u\leq 0$ and $B(0,1)\subset B(z_{1},2)$, there exists $C_{1}<0$, independent of $u$, such that

[TABLE]

As $\{U_{k}\}_{k}$ is a covering of $X$, $U_{1}$ needs to intersect at least another member of the covering. We can assume that $U_{1}\cap U_{2}$ is non-empty.

Since $u\leq 0$ and (3.75) holds, there exists $x_{2}\in U_{2}\cap U_{1},r_{2}\in(0,1)$ and $\tilde{C}_{1}<0$, independent of $u$, such that for $z_{2}=\varphi_{2}^{-1}(x_{2})$ we have $\varphi_{2}(B(z_{2},r_{2}))\subset U_{1}\cap U_{2}$ and

[TABLE]

Since $u\circ\varphi_{2}+\psi_{2}\circ\varphi_{2}$ is psh in $B(0,4)$, we can further write that:

[TABLE]

Since $B(0,1)\subset B(z_{2},2)$ and $u\leq 0$, from here we conclude the existence of $C_{2}<0$, independent of $u$, such that

[TABLE]

Continuing the above process, after $N-2$ more steps we eventually arrive at (3.73). ∎

3.8 Relation to classical notions of convergence

Theorem 3.32 gave a characterization of $d_{p}$–convergence using concrete analytic expressions. When dealing with $d_{1}$, it turns out that an even more convenient characterization can be given with the help of the Monge–Ampère energy. This is the content of the next theorem, which also shows that $d_{1}$–convergence implies classical $L^{1}$ convergence of potentials and also the weak convergence of the associated complex Monge–Ampère measures:

Theorem 3.46** ([48]).**

*Suppose $u_{k},u\in\mathcal{E}_{1}(X,\omega)$. Then the following hold:

(i) $d_{1}(u_{k},u)\to 0$ if and only if $\int_{X}|u_{k}-u|\omega^{n}\to 0$ and $I(u_{k})\to I(u)$.

(ii) If $d_{1}(u_{k},u)\to 0$ then $\omega_{u_{k}}^{n}\to\omega_{u}^{n}$ weakly, and $\int_{X}|u_{k}-u|\omega_{v}^{n}\to 0$ for any $v\in\mathcal{E}_{1}(X,\omega)$.*

The proof of this result is given in a number of propositions and lemmas below. The new analytic input will be given by the bi–functional $\mathcal{I}(\cdot,\cdot):\mathcal{E}_{1}(X,\omega)\times\mathcal{E}_{1}(X,\omega)\to\mathbb{R}$ and its properties:

[TABLE]

Observe that by Lemma 2.9 the above expression is indeed finite. Moreover, if $d_{1}(u_{j},u)\to 0$ then by Theorem 3.32 we get $\mathcal{I}(u_{j},u)\to 0$. As it turns out, the relationship between $d_{1}$ and $\mathcal{I}$ is much deeper then this simple observation might suggest.

When $u_{0},u_{1}$ are bounded, then Bedford–Taylor theory allows to integrate by parts and obtain:

[TABLE]

In particular, $\mathcal{I}(u_{0},u_{1})\geq 0$ for bounded potentials. Applying the next lemma to canonical cutoffs, we deduce that this also holds in general:

Lemma 3.47**.**

*Suppose $u_{0},u_{0}^{j},u_{1},u_{1}^{j}\in\mathcal{E}_{1}(X,\omega)$ and $u^{j}_{0}\searrow u_{0}$, $u^{j}_{1}\searrow u_{1}$. Then the following hold:

(i) $\mathcal{I}(u_{0},u_{1})=\mathcal{I}(u_{0},\max(u_{0},u_{1}))+\mathcal{I}(\max(u_{0},u_{1}),u_{1}).$

(ii) $\lim_{j\to\infty}\mathcal{I}(u^{j}_{0},u^{j}_{1})=\mathcal{I}(u_{0},u_{1}).$*

Proof.

(i) follows from the locality of the complex Monge–Ampère measure on plurifine open sets (see (2.1)). Indeed, $\mathbbm{1}_{\{u_{1}>u_{0}\}}(\omega_{\max(u_{0},u_{1})}^{n}-\omega_{u_{0}}^{n})=\mathbbm{1}_{\{u_{1}>u_{0}\}}(\omega_{u_{1}}^{n}-\omega_{u_{0}}^{n})$, consequently

[TABLE]

and for similar reasons, $\mathcal{I}(\max(u_{0},u_{1}),u_{1})=\int_{\{u_{0}>u_{1}\}}(u_{0}-u_{1})(\omega_{u_{1}}^{n}-\omega_{u_{0}}^{n})$. Adding these last two identities yields (i).

To address (ii), we start with $\mathcal{I}(u^{j}_{0},u^{j}_{1})=\mathcal{I}(u^{j}_{0},\max(u^{j}_{0},u^{j}_{1}))+\mathcal{I}(\max(u^{j}_{0},u^{j}_{1}),u^{j}_{1})$. Trivially, $u^{j}_{0},u^{j}_{1}\leq\max(u^{j}_{0},u^{j}_{1})$, hence we can apply Proposition 2.20 to each term on the right hand side of (3.77) to conclude that $\mathcal{I}(u^{j}_{0},\max(u^{j}_{0},u^{j}_{1}))\to\mathcal{I}(u_{0},\max(u_{0},u_{1}))$ and $\mathcal{I}(\max(u^{j}_{0},u^{j}_{1}),u^{j}_{1})\to\mathcal{I}(\max(u_{0},u_{1}),u_{1})$. Another application of (i) yields (ii). ∎

The next proposition contains the main analytic properties of the $\mathcal{I}$ functional:

Proposition 3.48**.**

[11, Lemma 3.13, Lemma 5.8]* Suppose $C>0$ and $\phi,\psi,u,v\in\mathcal{E}_{1}(X,\omega)$ satisfies*

[TABLE]

Then there exists $f_{C}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ (only dependent on $C$) continuous with $f_{C}(0)=0$ such that

[TABLE]

[TABLE]

Before we get into the argument, we note that by the lemma following this proposition, the condition (3.78) is seen to be equivalent with

[TABLE]

Proof.

By repeated application of the dominated convergence theorem, (2.3) and Lemma 3.47(ii), one can see that it is enough to show (3.79) and (3.80) for bounded potentials.

We focus now on (3.79) and introduce the quantities $a_{k}:=\int_{X}\phi\omega_{u}^{k}\wedge\omega_{v}^{n-k}$. We note that (3.79) follows if we are able to prove

[TABLE]

As our potentials are bounded, we can use integration by parts and start writing:

[TABLE]

Consequently, by the Cauchy–Schwarz inequality we can write:

[TABLE]

By the above, (3.82) would follow if we can show that $\int_{X}i\partial\phi\wedge\bar{\partial}\phi\wedge\omega_{u}^{k}\wedge\omega_{v}^{n-k-1}$ is uniformly bounded, and this exactly what we argue:

[TABLE]

where in the last estimate we have used Theorem 3.32. To finish, by the triangle inequality we have to argue that $d_{1}(0,u/8+v/8)$ and $d_{1}(0,\phi/4+u/8+v/8)$ are bounded. This can be deduced from (3.81) by repeated application of Lemma 3.33 and the triangle inequality for $d_{1}$.

Now we turn to (3.80). The proof has similar philosophy, but it is slightly more intricate. We introduce $\alpha:=u-v$, and the quantities $b_{k}:=\int_{X}\alpha\omega_{u}^{k}\wedge\omega_{\phi}^{n-k}$. We will show that

[TABLE]

This will imply that \Big{|}\int_{X}(u-v)(\omega_{u}^{n}-\omega_{\phi}^{n})\Big{|}\leq f_{C}(\mathcal{I}(u,v)), and using the symmetry in $u,v$, and basic properties of the absolute value, we obtain (3.80). Integration by parts yields the following:

[TABLE]

Introducing $\beta:=(u+v)/2$ and $c_{k}:=\int_{X}i\partial\alpha\wedge\bar{\partial}\alpha\wedge\omega_{\beta}^{k}\wedge\omega_{\phi}^{n-k-1}$, by the Cauchy–Schwarz inequality we deduce that

[TABLE]

where in the last inequality we have used that $\mathcal{I}(u,\phi)$ is bounded. Indeed, this follows from (3.81), the triangle inequality for $d_{1}$, and Theorem 3.32. Consequently, to prove (3.83) it suffices to show that

[TABLE]

To obtain this, we integrate by parts again:

[TABLE]

Using that $\omega_{u}\leq 2\omega_{\beta}$ and the Cauchy–Schwarz inequality we can estimate the first term from the right hand side in the following manner:

[TABLE]

where in the last inequality we used that $\mathcal{I}({(u+v)}/{2},\phi)$ is bounded. This follows from (3.81), as repeated application of Lemma 3.33 and the triangle inequality for $d_{1}$ yields that $d_{1}(\phi,(u+v)/2)$ is bounded.

We can similarly estimate the other term on the right hand side of (3.8) and putting everything together we obtain that $c_{k}\leq c_{k+1}+2D^{\prime\prime}c_{k+1}^{1/2}$. After a successive application of this inequality, we find that $c_{k}\leq f_{C}(c_{n-1})$, which is equivalent to (3.85). ∎

Lemma 3.49**.**

Suppose $u\in\mathcal{E}_{1}(X,\omega)$ and $-C\leq I(u)\leq\sup_{X}u\leq C$ for some $C>0$. Then $d_{1}(0,u)\leq 3C$.

Proof.

By the triangle inequality and Proposition 3.43 we can write:

[TABLE]

∎

We note the following important corollary of Proposition 3.48:

Corollary 3.50**.**

For any $C>0$ there exists $\tilde{f}_{C}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ continuous with $\tilde{f}_{C}(0)=0$ such that

[TABLE]

for $u,v,\psi\in\mathcal{E}_{1}(X,\omega)$ satisfying $d_{1}(0,u),d_{1}(0,v),d_{1}(0,\psi)\leq C$.

Proof.

By Lemma 3.40 and Theorem 3.32 we note that $d_{1}(0,u),d_{1}(0,v),d_{1}(0,\psi)\leq C$ implies that $I(u),I(v),I(\psi)$ and $\sup_{X}u,\sup_{X}v,\sup_{X}\psi$ are uniformly bounded. Consequently (3.80) gives that

[TABLE]

Next, by Theorem 3.32, $d_{1}(u,\max(u,v))$ is uniformly bounded. By the triangle inequality, so is $d_{1}(0,\max(u,v))$. Consequently, since $\mathcal{I}(\max(u,v),v)\leq\mathcal{I}(u,v)$ (see Lemma (3.47)) we also have:

[TABLE]

Using $|u-v|=2(\max(u,v)-v)-(u-v)$, we can add these last two inequalities to obtain (3.87).∎

As another corollary of Proposition 3.48 we obtain the second part of Theorem 3.46:

Corollary 3.51**.**

*Suppose $u_{k},u,v\in\mathcal{E}_{1}(X,\omega)$ with $d_{1}(u_{k},u)\to 0$. The following hold:

(i) $\omega_{u_{k}}^{n}\to\omega_{u}^{n}$ weakly.

(ii) $\int_{X}|u_{k}-u|\omega_{v}^{n}\to 0.$*

Proof.

Given $\phi\in C^{\infty}(X)$, after possibly multiplying with a small positive constant, we can assume that $\phi\in\mathcal{H}_{\omega}$. As $d_{1}(u_{k},u)\to 0$ implies that $\mathcal{I}(u_{k},u)\to 0$ (Theorem 3.32), we see that (3.79) gives (i). The convergence statement of (ii) follows immediately from the previous corollary. ∎

Now we prove the rest of Theorem 3.46:

Proposition 3.52**.**

Suppose $u_{k},u\in\mathcal{E}_{1}(X,\omega)$. Then $d_{1}(u,u_{k})\to 0$ if and only if $\int_{X}|u-u_{k}|\omega^{n}\to 0$ and $I(u_{k})\to I(u)$.

Proof.

The fact that $d_{1}(u_{k},u)\to 0$ implies $\int_{X}|u-u_{k}|\omega^{n}\to 0$, follows from the previous corollary. $I(u_{k})\to I(u)$ follows from the $d_{1}$–continuity of $I$.

We focus on the reverse direction. The first step is to show that

[TABLE]

By Theorem 2.1, pick $v_{k}\in\mathcal{H}_{\omega}$ decreasing to $u$. Consequently $\sup_{X}v_{k}$ and $I(v_{k})$ is uniformly bounded, hence by (3.79) we can write:

[TABLE]

As $v_{j}$ is smooth and $\int_{X}|u_{k}-u|\omega^{n}\to 0$, we can write

[TABLE]

As $v_{j}\searrow u$, we can use Proposition 2.20 to get $\limsup_{k}\int_{X}u_{k}\omega_{u}^{n}=\int_{X}u\omega_{u}^{n}$. The analogous statement also holds for $\liminf$, and we obtain (3.88).

Next we use the following estimate, which is a consequence of (3.63) and (3.7):

[TABLE]

By (3.88) we have $\mathcal{I}(u,u_{k})\to 0$. Using (3.88) again and (3.80) we can write

[TABLE]

Since, $\mathcal{I}(u,u_{k})\to 0$, Lemma 3.47(i) gives that $\mathcal{I}(\max(u_{k},u),u)\to 0$. Also, by our assumptions, $\int_{X}(\max(u_{k},u)-u)\omega^{n}\to 0$, hence (3.80) implies that

[TABLE]

where in the last limit we also used the locality of the complex Monge–Ampère operator with respect to the plurifine topology (see (2.1)). Using the fact that $|u_{k}-u|=2(\max(u_{k},u)-u)-(u_{k}-u)$, together with (3.88) and (3.89) we get that

[TABLE]

Theorem 3.32 now gives $d_{1}(u_{k},u)\to 0$. ∎

For the last result of this section we return to the general $d_{p}$ metric topologies. First observe that for any $u\in\mathcal{H}_{\omega}$ and $\xi\in T_{u}\mathcal{H}_{\omega}$ the Hölder inequality gives $\|\xi\|_{u,p^{\prime}}\leq\|\xi\|_{u,p}$ for any $p^{\prime}\leq p$. This in turn implies that the $L^{p^{\prime}}$ length of smooth curves in $\mathcal{H}_{\omega}$ is shorter then their $L^{p}$ length, ultimately giving that the $d_{p}$ metric dominates $d_{p}^{\prime}$. Consequently, all $d_{p}$ metrics dominate the $d_{1}$ metric, hence Theorem (3.46)(ii) holds for $d_{p}$–convergence as well. We record this (in a slightly stronger form) in our last result:

Proposition 3.53**.**

Suppose $v,u,u_{k}\in\mathcal{E}_{p}(X,\omega)$ and $d_{p}(u_{k},u)\to 0$. Then $\omega_{u_{k}}^{n}\to\omega_{u}^{n}$ weakly and $\int_{X}|u-u_{k}|^{p}\omega_{v}^{n}\to 0.$

Proof.

By the previous result, we only need to argue that $\int_{X}|u-u_{k}|^{p}\omega_{v}^{n}\to 0.$ In fact, it is enough to prove this convergence for a subsequence of $\{u_{k}\}_{k}$.

Given an arbitrary subsequence of $u_{k}$, there exists a sub-subsequence, again denoted by $u_{k}$, satisfying the sparsity condition:

[TABLE]

Using this sparsity condition and Proposition 3.35 we can write

[TABLE]

Hence, the decreasing sequence $h_{k}=P(u,u_{0},u_{1},\ldots,u_{k}),\ k\geq 1$ is $d_{p}$-bounded and Lemma 3.34 (or completeness) implies that the limit satisfies $h:=\lim_{k}h_{k}\in\mathcal{E}_{p}(X,\omega).$ As $d_{p}(u_{k},0),d_{p}(u,0)$ are bounded, by Theorem 3.32 and Lemma 3.45 there exists $M>0$ such that $h\leq u_{k},u\leq M$. Putting everything together we get

[TABLE]

By Theorem 3.46(ii) we have $\int_{X}|u-u_{k}|\omega_{v}^{n}\to 0$. Hence, after passing to a further subsequence, $u_{k}\to u$ a.e. with respect to $\omega_{v}^{n}$. Finally, using (3.91), the dominated convergence theorem gives $\int_{X}|u_{k}-u|^{p}\omega_{v}^{n}\to 0.$ ∎

Brief historical remarks.

In this chapter we put extensive focus on the particular case of the $L^{1}$ geometry. Historically however, the study of the $L^{2}$ structure was the one developed first. In particular, Calabi-Chen showed that $C^{1,1}$ geodesics of $\mathcal{H}_{\omega}$ satisfy the CAT(0) inequality [31]. As shown in [47], this implies that the completion $(\mathcal{E}_{2}(X,\omega),d_{2})$ is a CAT(0) geodesic metric space. The possibility of identifying $\mathcal{E}_{2}(X,\omega)$ with the metric completion of $\mathcal{H}_{\omega}$ was conjectured by Guedj [71], who worked out the case of toric Kähler manifolds. As detailed in this chapter, this conjecture was confirmed in [47], and later generalized to the case of $L^{p}$ Finsler structures [48].

As noticed by J. Streets [100, 101], the CAT(0) property allows to study of the Calabi flow [40] in the context of the metric completion, leading to precise convergence results for this flow in [13].

Endowing the space of Kähler metrics with natural geometries goes back to the work of Calabi in the 50’s [30]. Calabi’s Riemannian metric is defined in terms of the Laplacian of the potentials, and the resulting geometry differs from that of Mabuchi. The study of this structure was taken up by Calamai [32] and Clarke-Rubinstein [45]. In the latter work the completion of the Calabi path lengh metric was identified, and was compared to the Mabuchi geometry in [50].

It is also possible to introduce a Dirichlet type Riemannian metric on $\mathcal{H}_{\omega}$ in terms of the gradient of the potentials [32, 33]. Not much is known about the metric theory of this structure. However properties of this space seem to be closely immeresed with the study of the Kähler-Ricci flow [16].

Chapter 4 Applications to Kähler–Einstein metrics

Given a Kähler manifold $(X,\omega)$, we will be interested in picking a metric with special curvature properties from $\mathcal{H}$, the space of all Kähler metrics $\omega^{\prime}$ whose de Rham cohomology class equals that of $\omega$:

[TABLE]

By the $\partial\bar{\partial}$–lemma of Hodge theory, there exists $u\in\mathcal{H}_{\omega}$ (2.4), unique up to a constant, s.t. $\omega^{\prime}=\omega_{u}$ (see [23, Theorem 3]). As follows from Stokes’ theorem (see more generally Lemma 2.2), the total volume of each metric in $\mathcal{H}$ is the same, and we introduce the constant

[TABLE]

As a result of the above observations, we will focus on the space of potentials $\mathcal{H}_{\omega}$ instead of $\mathcal{H}$, and our goal will be to find $u\in\mathcal{H}_{\omega}$ whose Ricci curvature is a multiple of the metric $\omega_{u}$:

[TABLE]

Such metrics are called Kähler–Einstein (KE) metrics. Recall from Appendix 6.1 that $\textup{Ric }\omega_{u}$ is always closed. Moreover, by the well known formula (A.9) for the change of Ricci curvature

[TABLE]

we deduce that the de Rham class $[\textup{Ric }\omega_{u}]$ does not depend on the choice of $u\in\mathcal{H}_{\omega}$. What is more, $[\textup{Ric }\omega_{u}]$ agrees with $c_{1}(X)$, the first Chern class of $X$ (see [113, Section III.3] for more details). Consequently, if (4.2) holds then $X$ needs to have special cohomological properties depending on the sign of $\lambda\in\mathbb{R}$:

(i) if $\lambda=0$ then $c_{1}(X)=0$, i.e., $X$ is Calabi–Yau.

(ii) if $\lambda<0$ then $K_{X}$ is an ample line bundle, i.e., $X$ is of general type.

(iii) if $\lambda>0$ then $-K_{X}$ is an ample line bundle, i.e., $X$ is Fano.

After close inspection it turns out that (4.2) is a fourth order PDE in terms of the derivatives of $u$. As we show now, with the help of (4.3) one can write down a scalar equation equivalent to (4.2), that is merely a second order PDE, greatly simplifying our subsequent treatment. Indeed, from $\lambda[\omega]=c_{1}(X)$ it follows that for each $u\in\mathcal{H}_{\omega}$ there exists a unique $f_{u}\in C^{\infty}(X)$ such that $\int_{X}e^{f_{u}}\omega_{u}^{n}=V$ and

[TABLE]

Fittingly, the potential $f_{u}$ is called the Ricci potential of $\omega_{u}$, and $\omega_{u}$ is KE if and only if $f_{u}=0$. Consequently, by (4.3), (4.2) is equivalent to

[TABLE]

And now the (magical!) drop of order takes place. As there are only constants in the kernel of $i\partial\bar{\partial}$, there exists $c\in\mathbb{R}$ such that

[TABLE]

When $\lambda\neq 0$, the constant $c$ can be “contracted” into $u$. When $\lambda=0$, then the normalization condition on $f_{0}$ implies that $c=0$. Summarizing, we arrive at the scalar KE equation:

[TABLE]

When $\lambda<0$, the existence of unique solutions was proved by Aubin and Yau [1, 112]. In the case $\lambda=0$, existence and uniqueness was obtained by Yau, as a particular case of the solution of the Calabi conjecture [112].

When $X$ is Fano ($\lambda>0$) the situation is more involved. Uniqueness up to holomorphic automorpshims was shown by Bando–Mabuchi ([2], see also Section 4.5 below). As it turns out, on a general Fano manifold $(X,\omega)$, there are numerous obstructions to existence of KE metrics (see [88, 70]). Recently the algebro–geometric notion of K–stability has been found to be equivalent with existence of KE metrics ([39, 108]), with this verifying an important particular case of the Yau–Tian–Donaldson conjecture. Our goal in this chapter is to give equivalent characterizations, more in line with the variational study of partial differential equations, eventually verifying related conjectures of Tian.

As mentioned above, in the rest of this chapter we will assume that $(X,\omega)$ is Fano. Also, after possibly rescaling $\omega$, we can also assume that $\lambda=1$, i.e., $[\omega]=c_{1}(X)$. Next we introduce Ding’s $\mathcal{F}$ functional [63]:

[TABLE]

where $I$ is the Monge–Ampère energy (see (3.57)). By definition, $\mathcal{F}$ is invariant under adding constants, i.e., $\mathcal{F}(u+c)=\mathcal{F}(u)$, and as a consequence of Lemma 3.37 it follows that the critical points of $\mathcal{F}$ are exactly the KE metrics:

Lemma 4.1**.**

Suppose $[0,1]\ni t\to v_{t}\in\mathcal{H}_{\omega}$ is a smooth curve. Then

[TABLE]

Comparing with (4.5), we deduce that existence of KE potentials in $\mathcal{H}_{\omega}$ is equivalent with existence of critical points of $\mathcal{F}$. As we will see, the natural domain of definiton of $\mathcal{F}$ is not $\mathcal{H}_{\omega}$, but rather its $d_{1}$ metric completion $\mathcal{E}_{1}(X,\omega)$. In addition to this, $\mathcal{F}$ will be shown to be $d_{1}$–continuous and it will be convex along the finite energy geodesics of $\mathcal{E}_{1}(X,\omega)$. Using convexity we will deduce that critical points of $\mathcal{F}$ are exactly the minimizers of $\mathcal{F}$, and we can use properness properties of $\mathcal{F}$ to characterize existence of these minimizers. All this will be done in the forthcoming sections.

4.1 The action of the automorphism group

We continue to assume that $(X,\omega)$ is Fano with $[\omega]=c_{1}(X)$. Let $\textup{Aut}_{0}(X,J)$ denote the connected component of the complex Lie group of biholomorphisms of $(X,J)$, and denote by $\textup{aut}(M,J)$ its complex Lie algebra of holomorphic vector fields. $\textup{Aut}_{0}(X,J)$ acts on $\mathcal{H}$ by pullback of metrics. Indeed, $f^{\star}\eta\in\mathcal{H}$ for any $f\in\textup{Aut}_{0}(X,J)$ and $\eta\in\mathcal{H}.$

Given the one-to-one correspondence between $\mathcal{H}$ and $\mathcal{H}_{\omega}\cap I^{-1}(0)$ (recall (3.69)), the group $\textup{Aut}_{0}(X,J)$ also acts on $\mathcal{H}_{\omega}\cap I^{-1}(0)$ and we describe this action more precisely in the next lemma:

Lemma 4.2**.**

[57, Lemma 5.8]* For $\varphi\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ and $f\in\textup{Aut}_{0}(X,J)$ let $f.\varphi\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ be the unique element such that $f^{*}\omega_{\varphi}=\omega_{f.\varphi}$. Then,*

[TABLE]

Proof.

First we note that the right hand side of (4.7) is a Kähler potential for $f^{*}\omega_{\varphi}$. Indeed, since $f\in\textup{Aut}_{0}(X,J)$ we have $f^{\star}i\partial\bar{\partial}\varphi=i\partial\bar{\partial}\varphi\circ f$. The identity $I(f.0+\varphi\circ f)=0$ follows from (3.58) as we have:

[TABLE]

∎

With the formula of the above lemma, we show that $\textup{Aut}_{0}(X,J)$ acts on $\mathcal{H}_{\omega}\cap I^{-1}(0)$ by $d_{p}$–isometries:

Lemma 4.3**.**

The action of $\textup{Aut}_{0}(X,J)$ on $\mathcal{H}_{\omega}\cap I^{-1}(0)$ is by $d_{p}$–isometries, for any $p\geq 1$.

Proof.

From (4.7) it follows that ${d}(f.\varphi_{t})/dt=\dot{\varphi}_{t}\circ f,$ for any smooth curve $[0,1]\ni t\to\varphi_{t}\in\mathcal{H}_{\omega}\cap I^{-1}(0)$. Thus, the $d_{p}$–length of $t\to f.\varphi_{t}$ satisfies:

[TABLE]

Since this last quantity is exactly the the $d_{p}$-length of $t\to\varphi_{t}$, it follows that $\textup{Aut}_{0}(X,J)$ acts by $d_{p}$–isometries. ∎

As a consequence of this last result (see also Lemma 4.6 below), the action of $\textup{Aut}_{0}(X,J)$ on $\mathcal{H}_{\omega}\cap I^{-1}(0)$ has a unique $d_{p}$–isometric extension to the $d_{p}$–metric completion $\mathcal{E}_{p}(X,\omega)\cap I^{-1}(0)$. In what follows we will focus on the case $p=1$.

Recall the definition of the $J$ functional (3.70) and let $H\leq\textup{Aut}_{0}(X,J)$ be a subgroup. The “$H$–dampened” functional J_{H}:\big{(}\mathcal{E}_{1}\cap I^{-1}(0)\big{)}/H\to\mathbb{R} is introduced by the formula

[TABLE]

As a direct consequence of Proposition 3.44 we have the following estimates for this functional:

Lemma 4.4**.**

There exists $C:=C(X,\omega)>1$ such that for any $u\in\mathcal{E}_{1}\cap I^{-1}(0)$ we have

[TABLE]

where $d_{1,H}$ is the pseudo–metric of the quotient \big{(}\mathcal{E}_{1}\cap I^{-1}(0)\big{)}/H given by the formula $d_{1,H}(Hu,Hv):=\inf_{f\in H}d_{1}(u,f.v)$.

Lastly, we note that $\mathcal{F}$ is affine along one parameter subgroups of $\textup{Aut}_{0}(X,J)$, and that the map $(u,v)\to\mathcal{F}(u)-\mathcal{F}(v)$ is $\textup{Aut}_{0}(X,J)$–invariant:

Lemma 4.5**.**

*Suppose $\mathbb{R}\ni t\to\rho_{t}\in\textup{Aut}_{0}(X,J)$ is a one parameter subgroup. Then:

(i) for any $u\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ the map $t\to\mathcal{F}(\rho_{t}.u)$ is affine.

(ii) if $u,v\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ and $g\in\textup{Aut}_{0}(X,J)$ then $\mathcal{F}(u)-\mathcal{F}(v)=\mathcal{F}(g.u)-\mathcal{F}(g.v)$.*

Proof.

First we show that for any $u\in\mathcal{H}_{\omega}$ we have that

[TABLE]

where $f_{u}$ is the Ricci potential of $u$ (see (4.4)). By definition, we have $\textup{Ric }\omega_{u}=\omega_{u}+i\partial\bar{\partial}f_{u}$ and $\textup{Ric }\omega=\omega+i\partial\bar{\partial}f_{0}$. Substituting this into (4.3) we see that (4.10) holds.

Now we argue (i). Let $h\in C^{\infty}(X)$ be the unique function such that $\int_{X}h\omega_{u}^{n}=0$ and \frac{d}{dt}\big{|}_{t=0}\rho_{t}^{*}\omega_{u}=i\partial\bar{\partial}h. We claim that

[TABLE]

Indeed, this follows from a simple calculation: $i\partial\bar{\partial}\frac{d}{dt}\rho_{t}.u=\frac{d}{dt}\omega_{\rho_{t}.u}=\frac{d}{dt}\rho_{t}^{*}\omega_{u}=\rho^{*}_{t}i\partial\bar{\partial}h=i\partial\bar{\partial}h\circ\rho_{t}$.

Finally, using Lemma 4.1, (4.10), (4.11), and the fact that $f_{u}\circ\rho_{t}=f_{\rho_{t}.u}$ we can conclude that $t\to\mathcal{F}(\rho_{t}.u)$ is indeed affine:

[TABLE]

Now we focus on (ii). Let $[0,1]\ni t\to\gamma_{t}\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ be any smooth segment connecting $u,v$. For example, one can take $\gamma_{t}:=(1-t)u+tv-I((1-t)u+tv)$. Consequently, $t\to g.\gamma_{t}$ connects $g.u,g.v$. Using Lemma 4.1, (4.7), and (4.10) we can finish the proof:

[TABLE]

∎

4.2 The existence/properness principle and relation to Tian’s conjectures

Staying with a Fano manifold $(X,\omega)$, our main goal is to show that existence of KE metrics in $\mathcal{H}$ is equivalent with properness of the Ding energy (and later the K-energy). As it turns out, our proof will rest on a very general existence/properness principle for abstract metric spaces, and we describe this now as it provides the skeleton for our later arguments concerning Kähler geometry.

To begin, we introduce $(\mathcal{R},d,F,G)$, a metric space structure with additional data satisfying the following axioms:

1. (A1)

$(\mathcal{R},d)$ is a metric space with a distinguished element $0\in\mathcal{R}$, whose metric completion is denoted by $(\overline{\mathcal{R}},d)$. 2. (A2)

$F:\mathcal{R}\to\mathbb{R}$ is $d$–lsc. Let $F:\overline{\mathcal{R}}\to\mathbb{R}\cup\{+\infty\}$ be the largest $d$–lsc extension of $F:\mathcal{R}\to\mathbb{R}$:

[TABLE] 3. (A3)

By $\mathcal{M}$ we denote the set of minimizers of $F$ on $\overline{\mathcal{R}}$:

[TABLE] 4. (A4)

$G$ is a group acting on ${\mathcal{R}}$ by $G\times{\mathcal{R}}\ni(g,u)\to g.u\in{\mathcal{R}}$. Denote by ${\mathcal{R}}/G$ the orbit space, by $Gu\in{\mathcal{R}}/G$ the orbit of $u\in{\mathcal{R}}$, and define $d_{G}:{\mathcal{R}}/G\times{\mathcal{R}}/G\to\mathbb{R}_{+}$ by

[TABLE]

In addition to the above, our data $(\mathcal{R},d,F,G)$ also enjoys the following properties:

1. (P1)

For any $u_{0},u_{1}\in\mathcal{R}$ there exists a $d$–geodesic segment $[0,1]\ni t\mapsto u_{t}\in\overline{\mathcal{R}}$ connecting $u_{0},u_{1}$ (see (3.40)) for which $t\mapsto F(u_{t})\textup{ is continuous and convex on }[0,1].$ 2. (P2)

If $\{u_{j}\}_{j}\subset\overline{\mathcal{R}}$ satisfies $\lim_{j\to\infty}F(u_{j})=\inf_{\overline{\mathcal{R}}}F$, and for some $C>0$, $d(0,u_{j})\leq C$ for all $j$, then there exists $u\in\mathcal{M}$ and a subsequence $\{u_{j_{k}}\}_{k}$ s.t. $d(u_{j_{k}},u)\to 0$. 3. (P2)

∗ If $\{u_{j}\}_{j}\subset\overline{\mathcal{R}}$ satisfies $F(u_{j})\leq C$, and $d(0,u_{j})\leq C,\ j\geq 0$ for some $C>0$, then there exists $u\in\mathcal{R}$ and a subsequence $\{u_{j_{k}}\}_{k}$ s.t. $d(u_{j_{k}},u)\to 0$. 4. (P3)

$\mathcal{M}\subset\mathcal{R}.$ 5. (P4)

$G$ acts on ${\mathcal{R}}$ by $d$-isometries. 6. (P5)

$G$ acts on $\mathcal{M}$ transitively. 7. (P6)

For all $u,v\in\mathcal{R}$ and $g\in G$, $F(u)-F(v)=F(g.u)-F(g.v)$.

We make three remarks. First, by (A2),

[TABLE]

Second, condition $\textup{\ref{p2*}}^{*}$ is stronger than (P2) and we will require that only one of these conditions holds.

Third, thanks to (P4) and the next lemma, the action of $G$, originally defined on $\mathcal{R}$ (A4), extends to an action by $d$–isometries on the completion $\overline{\mathcal{R}}$.

Lemma 4.6**.**

Let $(X,\rho)$ and $(Y,\delta)$ be two complete metric spaces, $W$ a dense subset of $X$ and $f:W\to Y$ a $C$-Lipschitz function, i.e.,

[TABLE]

Then $f$ has a unique $C$–Lipschitz continuous extension to a map $\bar{f}:X\to Y$.

Proof.

Let $w_{k}\in W$ be a sequence converging to some $w\in X$. Lipschitz continuity gives

[TABLE]

hence $\bar{f}(w):=\lim_{k}f(w_{k})\in Y$ exists and independent of the choice of approximating sequence $w_{k}$. Choose now another sequence $z_{k}\in W$ with limit $z\in X$, plugging in $w_{k},z_{k}$ in (4.13) and taking the limit gives that $\bar{f}:X\to Y$ is $C$-Lipschitz continuous. ∎

A $d$-geodesic ray $[0,\infty)\ni t\to u_{t}\in\overline{\mathcal{R}}$ is $G$-calibrated if the curve $t\to Gu_{t}$ is a $d_{G}$-geodesic with the same speed as $t\to u_{t}$, i.e.,

[TABLE]

The next result will provide the framework that relates existence of canonical Kähler metrics to energy properness and uniform geodesic stability.

Theorem 4.7**.**

[57, Theorem 3.4]*

Suppose $(\mathcal{R},d,F,G)$ satisfies (A1)–(A4) and (P1)–(P6). The following are equivalent:

(i)(existence of minimizers) $\mathcal{M}$ is nonempty.*

(ii)(energy properness) $F:{\mathcal{R}}\to\mathbb{R}$ is $G$-invariant, and for some $C,D>0$,

[TABLE]

If $\textup{\ref{p2*}}^{*}$ holds instead of (P2), then the above are additionally equivalent to:

(iii)(uniform geodesic stability) Fix $u_{0}\in\mathcal{R}$ with $F(u_{0})<+\infty$. Then $F:{\mathcal{R}}\to\mathbb{R}$ is $G$-invariant, and there exists $C>0$ such that for all geodesic rays $[0,\infty)\ni t\to u_{t}\in\overline{\mathcal{R}}$ we have that

[TABLE]

(iv)(uniform geodesic stability) Fix $u_{0}\in\mathcal{R}$ with $F(u_{0})<+\infty$. Then $F:{\mathcal{R}}\to\mathbb{R}$ is $G$-invariant, and there exists $C>0$ such that for all $G$-calibrated geodesic rays $[0,\infty)\ni t\to u_{t}\in\overline{\mathcal{R}}$ we have that

[TABLE]

Before arguing the above theorem we recall standard facts from metric geometry in the form of the following two lemmas:

Lemma 4.8**.**

If (P4) holds, then $(\overline{\mathcal{R}}/G,d_{G})$ and $(\mathcal{R}/G,d_{G})$ are pseudo–metric spaces.

Proof.

It is enough to show that $(\overline{\mathcal{R}}/G,d_{G})$ is a pseudo–metric space. Since $d$ is symmetric,

[TABLE]

Hence $d_{G}$ is also symmetric. Given $u,v,w\in\overline{\mathcal{R}}$ and $\varepsilon>0$, there exist $f,g\in G$ such that $d_{G}(Gu,Gw)>d(f.u,w)-\varepsilon$ and $d_{G}(Gv,Gw)>d(g.v,w)-\varepsilon$. The triangle inequality for $d$ and (P4) give

[TABLE]

Letting $\varepsilon\to 0$ shows $d_{G}$ satisfies the triangle inequality. Thus $d_{G}$ is a pseudo–metric. ∎

Lemma 4.9**.**

Suppose (P4) holds, $u_{0},u_{1}\in\mathcal{R}$ and $[0,1]\ni t\to u_{t}\in\overline{\mathcal{R}}$ is a $d$–geodesic connecting $u_{0},u_{1}$. If $d(u_{0},u_{1})-\varepsilon\leq d_{G}(Gu_{0},Gu_{1})\leq d(u_{0},u_{1})$ for some $\varepsilon>0$ then

[TABLE]

Proof.

The proof is by contradiction. Suppose $d_{G}(Gu_{a},Gu_{b})<d(u_{a},u_{b})-\varepsilon$. Since $d(u_{0},u_{a})+d(u_{a},u_{b})+d(u_{b},u_{1})=d(u_{0},u_{1})$ we can write

[TABLE]

This is a contradiction, finishing the proof. ∎

Proof of Theorem 4.7.

First we show (ii)$\Rightarrow$(i). If condition (ii) holds, then $F$ is bounded from below. By (4.12), (4.14), the $G$–invariance of $F$, and the definition of $d_{G}$ there exists $u_{j}\in\mathcal{R}$ such that $\lim_{j}F(u_{j})=\inf_{\overline{\mathcal{R}}}F$ and $d(0,u_{j})\leq d_{G}(G0,Gu_{j})+1<C$ for $C$ independent of $j$. By (P2), $\mathcal{M}$ is non-empty.

We now show that (i)$\Rightarrow$(ii). First we argue that $F:\mathcal{R}\to\mathbb{R}$ is $G$–invariant. Let $v\in\mathcal{M}$. By (P3), $v\in\mathcal{R}$. By (P4), $f.v\in\mathcal{M}$ for any $f\in G$. Thus, $F(v)=F(f.v)$. Consequently, $F(u)-F(v)=F(u)-F(f.v)$. By (P6), we get $F(u)-F(v)=F(f^{-1}.u)-F(v)$, so $F(u)=F(f^{-1}.u)$ for every $f\in G$, i.e., $F$ is $G$–invariant.

For $v\in\mathcal{M}\subset\mathcal{R}$ we define

[TABLE]

If $C>0$, then we are done. Suppose $C=0$. Then there exists $\{u^{k}\}_{k}\subset\mathcal{R}$ such that

[TABLE]

and $d_{G}(Gv,Gu^{k})\geq 2$. By $G$–invariance of $F$ we can also assume that $d(v,u^{k})-1/k\leq d_{G}(Gv,Gu^{k})\leq d(v,u^{k})$. Thus,

[TABLE]

Using (P1), let $[0,d(v,u^{k})]\ni t\mapsto u^{k}_{t}\in\overline{\mathcal{R}}$ be a unit speed $d$–geodesic connecting $u^{k}_{0}:=v$ and $u^{k}_{d(v,u^{k})}:=u^{k}$ such that $t\mapsto F(u^{k}_{t})$ is convex. As $v$ is a minimizer of $F$, by convexity we obtain

[TABLE]

Trivially, $d(v,u^{k}_{1})=1$, hence (P2) and (4.20) imply that $d(u^{k}_{1},\tilde{v})\to 0$ for some $\tilde{v}\in\mathcal{M}$ (after perhaps passing to a subsequence of $u^{k}_{1}$). By (P5), $\tilde{v}=f.v$ for some $f\in G$, hence $d_{G}(Gv,G\tilde{v})=0$.

From Lemma 4.9 we obtain $1-1/k\leq d_{G}(Gv,Gu^{k}_{1})\leq 1$. Since $d(u^{k}_{1},\tilde{v})\to 0$, we also have $d_{G}(Gu^{k}_{1},G\tilde{v})\to 0$, which gives $d_{G}(Gv,G\tilde{v})=1$, a contradiction with $d_{G}(Gv,G\tilde{v})=0$. This implies that $C>0$, finishing the proof of the implication (i)$\Rightarrow$(ii).

The implications (ii)$\Rightarrow$(iii)$\Rightarrow$(iv) is trivial and we finish the proof by showing that (iv)$\Rightarrow$(i). Suppose (i) does not hold but (iv) does. We will derive a contradiction. Since $F$ is $G$-invariant there exists $\{u^{k}\}\subset\mathcal{R}$ such that $d(u_{0},u^{k})-1/k\leq d_{G}(Gu_{0},Gu^{k})$ and $F(u^{k})$ decreases to $\inf_{u\in\mathcal{R}}F(u)$. By $\textup{\ref{p2}}^{*}$ we must have $d(u_{0},u^{k})\to\infty$, otherwise there would exists $u\in\overline{\mathcal{R}}$ such that $F(u)=-\infty$, a contradiction.

Let $[0,d(u_{0},u^{k})]\ni t\to u^{k}_{t}\in\overline{\mathcal{R}}$ be the $d$-geodesic joining $u_{0},u^{k}$ from (P1). We note that by Lemma 4.9 it follows that

[TABLE]

Fix $l\in\mathbb{Q}_{+}$. For big enough $k$ using convexity of $F$ we can write:

[TABLE]

As $d(u^{k}_{l},u_{0})=l$, we can use $\textup{\ref{p2}}^{*}$ and a Cantor diagonal process, to conclude the existence of a sequence $k_{j}\to\infty$ and $u_{l}\in\overline{\mathcal{R}}$ for all $l\in\mathbb{Q}_{+}$ such that $d(u^{k_{j}}_{l},u_{l})\to 0$. As each curve $t\to u^{k}_{t}$ is $d$-Lipschitz, it follows that in fact we can extend the curve $\mathbb{Q}_{+}\ni l\to u_{l}\in\overline{\mathcal{R}}$ to a curve $[0,\infty)\ni t\to u_{t}\in\mathcal{\overline{}}{\mathcal{R}}$ such that $d(u^{k_{t}}_{l},u_{t})\to 0$.

For elementary reasons $t\to u_{t}$ is a $d$-geodesic. By (A2) and (4.22) we get that

[TABLE]

Finally, we argue that $t\to u_{t}$ is a $G$-calibrated geodesic ray, yielding a contradiction with (4.16). Let $g\in G$ be arbitrary, from (4.21) it follows that $d(u_{0},u^{k_{j}}_{t})-1/{k_{j}}\leq d(g.u_{0},u^{k_{j}}_{t})$. Letting $k_{j}\to\infty$, we obtain $d(u_{0},u_{t})\leq d(g.u_{0},u_{t})$ for $t\in\mathbb{Q}_{+}$ and by density for all $t\geq 0$. Consequently, $t\to u_{t}$ is $G$-calibrated. ∎

Later, when dealing with the K-energy functional, we will make use of the following observation:

Remark 4.10**.**

The direction (ii)$\Rightarrow$(i) in the above argument only uses the compactness condition (P2).

The next result, together with Theorem 4.36, represents the main application of Theorem 4.7:

Theorem 4.11**.**

[57, Theorem 7.1]*

Suppose $(X,\omega)$ is Fano and set $G:=\textup{Aut}_{0}(X,J)$. The following are equivalent:

(i) There exists a KE metric in $\mathcal{H}$.*

(ii) For some $C,D>0$ the following holds:

[TABLE]

(iii) For some $C,D>0$ the following holds:

[TABLE]

Proof.

The equivalence between (ii) and (iii) is the content of Lemma 4.4.

For the equivalence between (i) and (ii) we wish to apply Theorem 4.7 to the data

[TABLE]

We first have to show that (4.24) implies a bit of extra information: that $\mathcal{F}$ is invariant under the action of $\textup{Aut}_{0}(X,J)$. Indeed, (4.24) implies that $\mathcal{F}$ is bounded from below. This and Lemma 4.5(i) implies that $\mathbb{R}\ni t\to\mathcal{F}(\rho_{t}.u)\in\mathbb{R}$ is affine and bounded for all one parameter subgroups $\mathbb{R}\ni t\to\rho_{t}\in\textup{Aut}_{0}(X,J)$. Consequently $t\to\mathcal{F}(\rho_{t}.u)$ has to be constant equal to $\mathcal{F}(u)$. As $\textup{Aut}_{0}(X,J)$ is connected, we obtain that $\mathcal{F}$ is invariant under the action of this group, as claimed.

To finish the proof, we go over the axioms and properties of $(\mathcal{R},d,F,G)$, as imposed in the statement of Theorem 4.7:

1. (A1)

By Theorem 3.36 and the fact that $I$ is $d_{1}$–continuous (Proposition 3.40) we obtain that $\overline{(\mathcal{H}_{\omega}\cap I^{-1}(0),d_{1})}=(\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0),d_{1}).$ 2. (A2)

That $\mathcal{F}$ is $d_{1}$–continuous on $\mathcal{H}_{\omega}$ will be proved in Theorem 4.17. As we will see, the $d_{1}$–continuous extension $\mathcal{F}:\mathcal{E}_{1}(X,\omega)\to\mathbb{R}$ is given by the original formula for smooth potentials (see (4.6)). 3. (A3)

We choose $\mathcal{M}$ as the minimizer set of the extended functional $\mathcal{F}:\mathcal{E}_{1}(X,\omega)\cap I^{-1}(0)\to\mathbb{R}$.

1. (P1)

This fact is due to Berndtsson [19, Theorem 1.1], and we present this result in Theorem 4.22 below. 2. (P2)

This property will be verfied in Theorem 4.18. 3. (P3)

That elements of $\mathcal{M}$ are in fact smooth KE potentials follows after combination of results due to Berman, Tosatti–Székelyhidi and Berman–Boucksom–Guedj–Zeriahi. We present this in Theorem 4.30 below. 4. (P4)

This is Lemma 4.3 for $p=1$. 5. (P5)

This follows from (P3) and the Bando–Mabuchi uniqueness theorem that we will prove in Theorem 4.23 below. 6. (P6)

This is exactly the content of Lemma 4.5(ii) above.

∎

Remark 4.12**.**

The equivalence between (i) and (iii) in the above theorem verifies the analogue of a conjecture of Tian for the Ding functional $\mathcal{F}$ (see [107, p. 127],[106, Conjecture 7.12]).

In Theorem 4.36 below we will verify Tian’s original conjecture for the K–energy as well, giving another characterization for existence of KE metrics on Fano manifolds. For other results of the same spirit, as well as relation to the literature, we refer to [57, 14, 37, 38, 54].

4.3 Continuity and compactness properties of the Ding functional

Given a Fano manifold $(X,\omega)$, in this section we will show that the $\mathcal{F}$ functional is $d_{1}$–continuous on $\mathcal{H}_{\omega}$, hence naturally extends to the $d_{1}$–completion $\mathcal{E}_{1}(X,\omega)$ (Theorem 4.17). In addition to this, we will show that $d_{1}$–bounded sequences that are $\mathcal{F}$ minimizing are $d_{1}$–subconvergent, with this establishing an important compactness property of $\mathcal{F}$ (Theorem 4.18).

To start, we need to prove the following preliminary compactness lemma:

Lemma 4.13**.**

For $B,D\in\mathbb{R}$ we consider the following subset of $\mathcal{E}_{1}(X,\omega)$:

[TABLE]

Then $\mathcal{C}$ is compact with respect to the weak $L^{1}(\omega^{n})$ topology of $\textup{PSH}(X,\omega)$.

Proof.

Since $\sup_{X}u$ is bounded for any $u\in\mathcal{C}$, by Lemma 3.45 it follows that $\int_{X}|u|\omega^{n}$ is bounded as well. By the Montel property of psh functions ([60, Proposition I.4.21]) it follows that $\mathcal{C}$ is precompact with respect to the weak $L^{1}(\omega^{n})$ topology.

Now we argue that $\mathcal{C}$ is $L^{1}$–closed. Let $u_{k}\in\mathcal{C}$ and $u\in\textup{PSH}(X,\omega)$ such that $\int_{X}|u-u_{k}|\omega^{n}\to 0$. After possibly taking a subsequence, we can suppose that $u_{k}\to u$ a.e.. As $u_{k}\leq D$ and $u_{k}\to u$ a.e., it follows that $\sup_{X}u\leq D$.

Since $u_{k}\to u$ a.e., it follows that $v_{k}\searrow u$, where $v_{k}:=(\sup_{j\geq k}u_{j})^{*}\in\textup{PSH}(X,\omega)$. By the monotonicity property of $I$ we have that $B\leq I(u_{k})\leq I(v_{k})\leq\sup_{X}v_{k}\leq D,\ k\geq 1$. By Lemma 3.49 and Lemma 3.34 it follows that $d_{1}(v_{k},u)\to 0$. Consequently $I(v_{k})\to I(u)$ (Proposition 3.40), hence $u\in\mathcal{C}$. ∎

In contrast with Proposition 3.40, as a consequence of the above argument, we also obtain that $I$ is $L^{1}(\omega^{n})$–usc:

Corollary 4.14**.**

Suppose that $u_{k},u\in\mathcal{E}_{1}(X,\omega)$ satisfies $u_{k}\to_{L^{1}}u$. Then

[TABLE]

Before we proceed, we recall Zeriahi’s uniform version of the famous Skoda integrability theorem [114]:

Theorem 4.15**.**

Suppose $\mathcal{S}\subset\textup{PSH}(X,\omega)$ is an $L^{1}(\omega^{n})$–compact family whose elements have zero Lelong numbers. Then for any $p\geq 1$ there exists $C:=C(p,\mathcal{S},\omega)>1$ such that

[TABLE]

For a full account of this result we refer to [74, Theorem 2.50]. Since full mass potentials have zero Lelong numbers (Propostion 2.11), we obtain the following corollary:

Corollary 4.16**.**

For $D,p\geq 1$ there exists $C:=C(p,D,\omega)>0$ such that for any $u\in\mathcal{E}_{1}(X,\omega)$ with $d_{1}(0,u)\leq D$ we have

[TABLE]

Proof.

From Theorem 3.32 it follows that $\int_{X}u\omega^{n}$ is uniformly bounded. By Lemma 3.45 so is $\sup_{X}u$. By Proposition 3.40 it follows that $I(u)$ is bounded as well. This allows to apply Lemma 4.13, and together with Proposition 2.11 the conditions of of Theorem 4.15 are satisfied to conclude the result. ∎

In particular, this last corollary implies that the original definition of the $\mathcal{F}$ functional for smooth potentials (see (4.6)) extends to $\mathcal{E}_{1}(X,\omega)$ as well. Additionally, our next theorem shows that this extension is in fact $d_{1}$–continuous:

Theorem 4.17**.**

The map

[TABLE]

is $d_{1}$–continuous.

Proof.

We know that $u\to I(u)$ is $d_{1}$–continuous and finite on $\mathcal{E}_{1}(X,\omega)$, hence we only have to argue that so is $u\to\int_{X}e^{-u+f_{0}}\omega^{n}$. For $u,v\in\mathcal{E}_{1}(X,\omega)$, using the inequality $|e^{x}-e^{y}|\leq|x-y|(e^{x}+e^{y}),\ x,y\in\mathbb{R}$, we have the following estimates:

[TABLE]

Using the fact that $f_{0}$ is bounded, the Hölder inequality gives

[TABLE]

Suppose $u_{k}\in\mathcal{E}_{1}(X,\omega)$ such that $d_{1}(u_{k},u)\to 0$. Then Proposition 3.53 implies that $\int_{X}|u_{k}-u|\omega^{n}\to 0$. As all $L^{p}$ topologies on $\textup{PSH}(X,\omega)$ are equivalent (see [80, Theorem 4.1.8]) we obtain that $\int_{X}|u_{k}-u|^{2}\omega^{n}\to 0$. Finally, (4.27) and Corollary 4.16 implies that $\int_{X}e^{-u_{k}+f_{0}}\omega^{n}\to\int_{X}e^{-u+f_{0}}\omega^{n}$. ∎

Lastly, we argue the compactness property of $\mathcal{F}$ that is a vital ingredient in the existence/properness principle of the previous section:

Theorem 4.18**.**

Suppose $u_{k}\in\mathcal{E}_{1}(X,\omega)$ such that $\mathcal{F}(u_{k})\to\inf_{\mathcal{E}_{1}(X,\omega)}\mathcal{F}$ and $d_{1}(u_{k},0)\leq C$ for some $C>0$. After possibly taking a subsequence, there exists $u\in\mathcal{E}_{1}(X,\omega)$ such that $d_{1}(u_{k},u)\to 0$. In particular, $\mathcal{F}(u)=\inf_{\mathcal{E}_{1}(X,\omega)}\mathcal{F}$.

Proof.

First we construct a candidate for the minimizer $u\in\mathcal{E}_{1}(X,\omega)$. From Theorem 3.32 it follows that $\int_{X}u_{k}\omega^{n}$ is uniformly bounded. By Lemma 3.45 so is $\sup_{X}u_{k}$. By Proposition 3.40 it follows that $I(u_{k})$ is bounded as well. Now Lemma 4.13 implies that after possibly taking a subsequence we can find $u\in\mathcal{E}_{1}(X,\omega)$ such that $\int_{X}|u_{k}-u|\omega^{n}\to 0$.

We now show that $u$ is actually a minimizer of $\mathcal{F}$. Using (4.27) in the same way as in the proof of the previous result, we obtain that $\int_{X}e^{-u_{k}+f_{0}}\omega^{n}\to\int_{X}e^{-u+f_{0}}\omega^{n}$. By Corollary 4.14 $I$ is usc with respect to the weak $L^{1}(\omega^{n})$ topology, hence we can write

[TABLE]

As $\{u_{k}\}_{k}$ minimizes $\mathcal{F}$, it follows that all the inequalities above are equalities. Thus, $u$ minimizes $\mathcal{F}$.

Lastly, we show that there is a subsequence of $u_{k}$ that $d_{1}$-converges to $u$. As $\limsup_{k}I(u_{k})=I(u)$, after possibly passing to a subsequence, $\lim_{k}I(u_{k})=I(u)$. This together with $|u_{k}-u|_{L^{1}(\omega^{n})}\to 0$ and Theorem 3.46 gives that $d_{1}(u_{k},u)\to 0$. ∎

4.4 Convexity of the Ding functional

We stay with a Fano manifold $(X,\omega)$ for this section as well. Previously we extended Ding’s functional to $\mathcal{E}_{1}(X,\omega)$. In this short section we show that this extension is convex along the finite energy geodesics of $\mathcal{E}_{1}(X,\omega)$ (Theorem 4.22). We start by computing the Hessian of the Ding functional with respect to the $L^{2}$ Mabuchi metric of $\mathcal{H}_{\omega}$:

Proposition 4.19**.**

Suppose $u\in\mathcal{H}_{\omega}$ and $\phi,\psi\in C^{\infty}(X)\simeq T_{u}\mathcal{H}_{\omega}$. We have the following formula for the Hessian of $\mathcal{F}$ with respect to the $L^{2}$ Mabuchi metric:

[TABLE]

Proof.

First we compute the Hessian of the Aubin–Yau energy $I$. From Lemma 3.37 it follows that for small $t>0$ we have:

[TABLE]

Using the formulas $\nabla^{2}I(u+t\psi)(\phi,\psi)=\frac{d}{dt}dI(u+t\psi)(\phi)-dI(u+t\psi)(\nabla_{\frac{d}{dt}}\phi)$ and $\nabla_{\frac{d}{dt}}\phi=-\frac{1}{2}\langle\nabla^{\omega_{u}}\phi,\nabla^{\omega_{u}}\psi\rangle_{\omega_{u}}$, we can write:

[TABLE]

where in the last line we have used (A.4) in the appendix and the formula below it. Next we introduce $\mathcal{B}:\mathcal{H}_{\omega}\to\mathbb{R}$ by the formula

[TABLE]

Using (4.10) and $d\mathcal{B}(u+t\psi)(\phi)={\int_{X}\phi e^{-u-t\psi+f_{0}}\omega^{n}}/{\int_{X}e^{-u-t\psi+f_{0}}\omega^{n}},$ we obtain that $d\mathcal{B}(u+t\psi)(\phi)=\frac{1}{V}\int_{X}\phi e^{f_{u}}\omega_{u}^{n}$. Another differentiation gives

[TABLE]

where in the last line we have used (4.10) again. Reorganizing terms in this identity and using $\nabla^{2}\mathcal{B}(u+t\psi)(\phi,\psi)=\frac{d}{dt}d\mathcal{B}(u+t\psi)(\phi)-d\mathcal{B}(u+t\psi)(\nabla_{\frac{d}{dt}}\phi)$ we conclude that

[TABLE]

The proof is finished after we subtract (4.4) from this last formula. ∎

Next we will show that $\nabla^{2}\mathcal{F}(u)(\cdot,\cdot)$ is positive semi–definite for all $u\in\mathcal{H}_{\omega}$. Before this we introduce and study the following complex valued weighted complex Laplacian

[TABLE]

To clarify, by $\langle\partial h,\partial f_{u}\rangle_{\omega_{u}}$ we mean the quantity $g_{u}^{j\bar{k}}h_{j}{f_{u}}_{\bar{k}}$ (expressed in local coordinates). Also $\partial^{*}$ is the Hermitian $L^{2}$ adjoint of $\partial$ with respect to $\omega_{u}$. For $g,h\in C^{\infty}(X,\mathbb{C})$ integration by parts gives the following:

[TABLE]

Consequently $L^{f_{u}}$ is a self–adjoint elliptic operator with respect to the Hermitian inner product

[TABLE]

We conclude that $L^{2}(e^{f_{u}}\omega_{u}^{n})$ has an orthonormal base composed of eigenfunctions corresponding to the eigenvalues $\lambda_{0}<\lambda_{1}<\ldots$ of $L^{f_{u}}$. As another application of (4.33) we see that $\lambda_{0}=0$ and the eigenspace of this eigenvalue is composed by the constant functions. Moreover, we have the following general result about the eigenfunctions of $L^{f_{u}}$ due to Futaki:

Proposition 4.20** ([70]).**

Suppose $v\in C^{\infty}(X,\mathbb{R})$ and $u\in\mathcal{H}_{\omega}$ such that $\int_{X}ve^{f_{u}}\omega_{u}^{n}=0$. In addition, let $h$ be an eigenfunction $L^{f_{u}}$, i.e., $L^{f_{u}}h=\lambda h$. Then the following hold:

[TABLE]

[TABLE]

where $\mathcal{L}$ is the Lichnerowitz operator (see (A.12) in the appendix). Also, one has equality in (4.35) if and only if $\mathcal{L}v=0$, or equivalently, $\nabla^{\omega_{u}}_{0,1}v\in T^{1,0}_{\mathbb{C}}X$ is a holomorphic vector field.

Recall from the discussion following (A.12) in the appendix that the condition $\mathcal{L}v=0$ is indeed equivalent with $\nabla^{\omega_{u}}_{0,1}v\in T^{1,0}_{\mathbb{C}}X$ being holomorphic.

Proof.

By the the discussion preceding the proposition, the condition $\int_{X}ve^{f_{u}}\omega_{u}^{n}=0$ simply means that $v$ is orthogonal to the eigenspace of $\lambda_{0}$, hence to prove (4.35) we only need to argue (4.34). Indeed, the rightmost term in (4.34) is nonnegative hence either $\lambda=\lambda_{0}=0$ (in which case $h$ is a constant) or $\lambda\geq\lambda_{1}\geq 1$. Consequently the inequality between the first and last term of (4.35) follows after expressing $v$ using the orthonormal base of $L^{2}(e^{f_{u}}\omega_{u}^{n})$ composed of eigenfunctions of $L^{f_{u}}$.

Lastly, we note that the middle identities of (4.35) are simply a consequence of (4.33) and the formula following (A.4), since $v$ is real valued.

We now argue (4.34). To ease notation, we will drop the subscript of $f_{u}$ and $\omega_{u}$ in the rest of the proof. Also, recall that by a choice of normal coordinates identifying a neighborhood of $x\in X$ with that of $0\in\mathbb{C}^{n}$, we can assure that locally $\omega=i\partial\bar{\partial}g$, with $g_{j\bar{k}}(0)=\delta_{jk}$ and $g_{j\bar{k}l}(0)=g_{j\bar{k}\bar{l}}(0)=0$ (see Proposition A.2 below). With such a choice of coordinates we also have $\textup{Ric}\ \omega_{j\bar{k}}|_{x}=-g_{j\bar{k}a\bar{a}}(0)$ (see (A.11)). Making use of this and integrating by parts multiple times we get

[TABLE]

where in the last line we have used the identity $\textup{Ric }\omega_{u}-\omega_{u}=i\partial\bar{\partial}f_{u}$, and the expression of $\mathcal{L}$ in normal coordinates (see (A.12)). ∎

From (4.35) and Proposition 4.19 it follows that $\nabla^{2}\mathcal{F}(u)(\cdot,\cdot)$ is indeed positive semi–definite. As an additional consequence we obtain that $\mathcal{B}$ is convex along $\varepsilon$–geodesics:

Lemma 4.21**.**

Suppose $u_{0},u_{1}\in\mathcal{H}_{\omega}$ and $\varepsilon>0$. Let $[0,1]\ni t\to u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ be the smooth $\varepsilon$–geodesic joining $u_{0},u_{1}$ (see (3.11)). Then t\to\mathcal{B}(u^{\varepsilon}_{t}):=-\log\big{(}\int_{X}e^{-u_{t}^{\varepsilon}+f_{0}}\omega^{n}\big{)} is convex.

Proof.

As it will not cause confusion, we will drop the reference to $\varepsilon$ in our argument.

Using the $L^{2}$ Mabuchi structure of $\mathcal{H}_{\omega}$ we have the following formula:

[TABLE]

From Proposition 4.20 it follows that $\nabla^{2}\mathcal{B}(u_{t})(\dot{u}_{t},\dot{u}_{t})\geq 0$. In addition to this, the equation of $\varepsilon$–geodesics (see (3.11)) gives $\nabla_{\dot{u}_{t}}\dot{u}_{t}>0$, hence we obtain $d\mathcal{B}(u_{t})(\nabla_{\dot{u}_{t}}\dot{u}_{t})\geq 0$. Putting the last two facts together we get that $\frac{d^{2}}{dt^{2}}\mathcal{B}(u_{t})\geq 0$. ∎

Finally, we argue that $\mathcal{F}$ is convex along the finite energy geodesics of $\mathcal{E}_{1}(X,\omega)$.

Theorem 4.22**.**

Suppose $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)$ and let $t\to u_{t}$ be the finite energy geodesic connecting $u_{0},u_{1}$. Then $t\to\mathcal{F}(u_{t})$ is convex and continuous on $[0,1]$.

Proof.

As $t\to I(u_{t})$ is known to be affine (Proposition 3.42), we only need to argue that t\to\mathcal{B}(u_{t}):=-\log\big{(}\int_{X}e^{-u_{t}+f_{0}}\omega^{n}\big{)} is also affine. As $u\to\mathcal{F}(u)$ and $u\to I(u)$ is $d_{1}$-continuous (see Proposition 3.40 and Theorem 4.17), we obtain that so is $u\to\mathcal{B}(u)$.

Let us assume first that $u_{0},u_{1}\in\mathcal{H}_{\omega}$. In this case $u^{\varepsilon}_{t}\nearrow u_{t}$ as $\varepsilon\to 0$, where $t\to u^{\varepsilon}_{t}$ is the $\varepsilon$–geodesic joining $u_{0},u_{1}$ (see (3.19)). Consequently $d_{1}(u^{\varepsilon}_{t},u_{t})\to 0,\ t\in[0,1]$. Hence $\mathcal{B}(u^{\varepsilon}_{t})\to\mathcal{B}(u_{t}),\ t\in[0,1]$. By Lemma 4.21 it follows that $t\to\mathcal{B}(u_{t})$ is convex.

In the general case $u_{0},u_{1}\in\mathcal{E}_{1}(X,\omega)$, let $u_{0}^{j},u_{1}^{j}\in\mathcal{H}_{\omega}$ be smooth decreasing approximants that exist by Theorem 2.1. Let $t\to u^{j}_{t}$ be the $C^{1,\bar{1}}$–geodesics joining $u^{j}_{0},u^{j}_{1}$. By Proposition 3.15(i) it follows that $d_{1}(u^{j}_{t},u_{t})\to 0,\ t\in[0,1]$ and consequently $\mathcal{B}(u^{j}_{t})\to\mathcal{B}(u_{t}),\ t\in[0,1]$, implying that $t\to\mathcal{B}(u_{t})$ is convex, finishing the proof. ∎

For weak geodesics joining bounded potentials the above theorem was proved by Berndtsson in much more general context [17, Theorem 1.2]. The argument that we presented in this section follows more closely the simplified treatment in [77].

4.5 Uniqueness of KE metrics and reductivity of the automorphism group

In this section we will give the proof of an important theorem of Bando–Mabuchi according to which on a Fano manifold $(X,\omega)$ Kähler–Einstein metrics are unique up to pullback by an automorphism:

Theorem 4.23** ([2]).**

Suppose $u,v\in\mathcal{H}_{\omega}$ both solve (4.5), i.e., they are both KE potentials. Then there exists $g\in\textup{Aut}_{0}(X,J)$ such that $g^{*}\omega_{u}=\omega_{v}$.

As we plan to use some of the machinery that we developed in previous parts, we will not follow the original proof of Bando–Mabuchi. Instead our proof will be a combination of the arguments of Berndtsson [19] and Berman–Berndtsson [8, Section 4], and bears similarities with the treatment in [85].

The proof will need a sequence of preliminary results about the automorphism group of Kähler–Einstein manifolds and will also use the classical theory of self adjoint elliptic differential operators on compact manifolds (see [113, Chapter 4]).

Lemma 4.24**.**

Suppose $X$ is a Fano manifold. Then $H^{0,q}(X,\mathbb{C})$ is trivial for $q\in\{1,\ldots,n\}$.

Proof.

We start with the observation $H^{0,q}(X,\mathbb{C})\simeq H^{n,q}(X,-K_{X})$. As $-K_{X}>0$ by the Fano condition, the triviality of $H^{n,q}(X,-K_{X})$ follows from the Kodaira vanishing theorem ([60, Theorem VII.3.3]). ∎

Note that by the Hodge decomposition we also have $H^{1}(X,\mathbb{C})\simeq H^{1,0}(X,\mathbb{C})\oplus H^{0,1}(X,\mathbb{C})$, hence this group is trivial as well. Though we will not use this, we mention that by an argument involving the Bonnet–Myers theorem and the Euler characteristic, we can further deduce that $X$ is in fact simply connected.

Now we focus on the Lie algebra $\mathfrak{g}$ of $G:=\textup{Aut}_{0}(X,J)$. Pick $U\in\mathfrak{g}$. As $U=U^{1,0}+\overline{U^{1,0}}$ is real holomorphic it follows that $\bar{\partial}(U^{1,0}\lrcorner\omega_{u})=0$ for all $u\in\mathcal{H}_{\omega}$. Indeed, this is immediate after one computes $\partial_{\bar{l}}(U^{1,0}_{j}{g^{u}}_{j\bar{k}})=0,\ l\in\{1,\ldots,n\}$ in normal coordinates (here and below $g^{u}$ is a local potential of $\omega_{u}$).

Using the previous lemma it follows that there exists a unique $v_{\omega_{u}}^{U}\in C^{\infty}(X,\mathbb{C})$ with $\int_{X}v_{\omega_{u}}^{U}\omega_{u}^{n}=0$ such that

[TABLE]

Equivalently, using Hamiltonian formalism this can be written as $X^{\omega_{u}}_{1,0}({v_{\omega_{u}}^{U}})=U^{1,0},$ and we have the following identification for $\mathfrak{g}$ using $\omega_{u}$:

[TABLE]

Recall that the “complex” gradient $\nabla^{\omega_{u}}_{1,0}v=JX^{\omega_{u}}_{1,0}v$ (see (A.5)) is holomorphic precisely when $\mathcal{L}v=0$, where $\mathcal{L}$ is the Lichnerowitz operator of the metric $\omega_{u}$ (see (A.12)).

When $\mathcal{H}_{\omega}$ contains a KE potential $u$ then $S_{\omega_{u}}$ is trivially constant, hence by Proposition A.3 we have that

[TABLE]

is a real differential operator. As a result, $v\in\ker\mathcal{L}^{*}\mathcal{L}=\ker\mathcal{L}$ if and only if $\textup{Re }v,\textup{Im }v\in\ker\mathcal{L}$. Consequently, for KE Fano manifolds the above description of $\mathfrak{g}$ can be sharpened, to imply that $\textup{Aut}_{0}(X,J)$ is reductive, which was one of the first known obstructions to existence of KE metrics:

Proposition 4.25** ([88]).**

Suppose $(X,\omega_{u})$ is a Fano KE manifold. Introducing $\mathfrak{k}_{\omega_{u}}=\mathfrak{g}_{\omega_{u}}\cap C^{\infty}(X,\mathbb{R})$ we can write

[TABLE]

In particular, $\textup{Aut}_{0}(X,J)$ is the complexification of the compact connected Lie group $\textup{Isom}_{0}(X,\omega_{u},J)$, with Lie algebra $\mathfrak{k}_{\omega_{u}}$.

Here $\textup{Isom}_{0}(X,\omega_{u},J)$ is the identity component of the group of holomorphic isometries of the KE metric $\omega_{u}$. As $X$ is compact, the group of smooth isometries of $(X,\omega_{u})$ is compact as well, hence so is its subgroup $\textup{Isom}_{0}(X,\omega_{u},J)$.

Proof.

The decomposition $\mathfrak{g}_{\omega_{u}}=\mathfrak{k}_{\omega_{u}}\oplus i\mathfrak{k}_{\omega_{u}}$ follows from the discussion preceding the proposition. We have to argue that the Lie algebra of $\textup{Isom}_{0}(X,\omega_{u},J)$ is exactly $\mathfrak{k}_{\omega_{u}}$.

Suppose $U\in\mathfrak{k}_{\omega_{u}}$. Trivially $v:=v^{U}_{\omega_{u}}\in C^{\infty}(X,\mathbb{R})$, and since $U^{1,0}=X^{\omega_{u}}_{1,0}v$, by conjugation we obtain that in fact $U=X^{\omega_{u}}_{v}$. Consequently,

[TABLE]

hence $U$ represents an infinitesimal symplectomorphism. Since $U$ is holomorphic, by (A.2) $U$ represents an infinitesimal isometry as well, i.e., $U\in\textup{Lie}(\textup{Isom}_{0}(X,\omega_{u},J))$ as claimed.

Conversely, if $U\in\textup{Lie}(\textup{Isom}_{0}(X,\omega_{u},J))$ then $0=d(U\lrcorner\omega_{u})=d(U^{1,0}+U^{0,1})\lrcorner\omega_{u}=d(\bar{\partial}v^{U}_{\omega_{u}}+\partial\overline{v^{U}_{\omega_{u}}})=2i\partial\bar{\partial}\textup{Im }v^{U}_{\omega_{u}}$. Consequently $v^{U}_{\omega_{u}}\in\mathfrak{g}_{\omega_{u}}\cap C^{\infty}(X,\mathbb{R})=\mathfrak{k}_{\omega_{u}}$. ∎

According to the next result the action of a one parameter subgroup of automorphisms in the direction of $i\mathfrak{k}$ gives $d_{p}$–geodesic rays inside $\mathcal{H}_{\omega}$:

Lemma 4.26**.**

Suppose $u\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ is a KE potential. Let $U\in i\mathfrak{k}_{\omega_{u}}$ and $\mathbb{R}\ni t\to\rho_{t}\in\textup{Aut}_{0}(X,J)$ be the associated one parameter subgroup. Then $t\to u_{t}:=\rho_{t}.u$ is a smooth $d_{p}$–geodesic ray for any $p\geq 1$.

Proof.

As $U\in i\mathfrak{k}_{\omega_{u}}$, it follows that $v^{U}_{\omega_{u}}=ih$ for some $h\in C^{\infty}(X,\mathbb{R})$ with $\int_{X}h\omega_{u}^{n}=0$. Differentiating $\rho_{t}^{*}\omega_{u}=\omega_{u_{t}}$ we find that $i\partial\bar{\partial}\dot{u}_{t}=\rho_{t}^{*}d(U\lrcorner\omega_{u})=\rho_{t}^{*}d(\bar{\partial}v^{U}_{\omega_{u}}+\partial\overline{v^{U}_{\omega_{u}}})=2i\rho_{t}^{*}\partial\bar{\partial}(\textup{Im }v^{U}_{\omega_{u}})=2i\partial\bar{\partial}h\circ\rho_{t}$.

Since $I(u_{t})=0,\ t\geq 0$, Lemma 3.37 gives that $\int_{X}\dot{u}_{t}\omega_{u_{t}}^{n}=0$. Also, $\int_{X}h\circ\rho_{t}\omega_{u_{t}}^{n}=\int_{X}h\circ\rho_{t}\rho_{t}^{*}\omega_{u}^{n}=\int_{X}h\omega_{u}^{n}=0$, so we conclude that

[TABLE]

Differentiating this identity we obtain

[TABLE]

where we used (A.5) in the second to last equality, and (4.37) again in the last equality. The above arguments show that $t\to u_{t}$ satisfies (3.8), hence by Theorem 3.36 we obtain that $t\to u_{t}$ is a geodesic ray for any $p\geq 1$. ∎

The following result will play an important role in the proof of Theorem 4.23:

Proposition 4.27**.**

Let $(X,\omega)$ be Fano. Suppose $u\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ is a KE potential. Then the map $\textup{Aut}_{0}(X,J)\ni h\to J_{\omega}(h.u)\in\mathbb{R}$ admits a minimizer $g\in\textup{Aut}_{0}(X,J)$ that satisfies

[TABLE]

Recall the definition of the $J$ functional from (3.70). To avoid the possibility of confusion with the complex structure, we denoted this functional with $J_{\omega}$ in the above proposition, and will continue to do so in the rest of this section.

Proof.

By Proposition 3.44 the functional $J_{\omega}$ has the same growth as the metric $d_{1}$. We turn to the group $\textup{Aut}_{0}(X,J)$ which is reductive by Proposition 4.25, hence we can apply Proposition A.14 in the appendix to deduce that the map $C:\textup{Isom}_{0}(X,\omega_{u},J)\oplus\mathfrak{k}_{\omega_{u}}\to\textup{Aut}_{0}(X,J)$ given by $C(k,U)=k\textup{exp}_{I}(JU)$ is surjective.

For any $k\in\textup{Isom}_{0}(X,\omega_{u},J)$ and any $U\in\mathfrak{k}_{\omega_{u}}$ the previous proposition gives that $t\to k\textup{exp}_{I}(tJU).u=\textup{exp}_{I}(tJU).u=:u_{t}\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ is a smooth $d_{1}$–geodesic. As the growth of $J_{\omega}$ is equivalent with the growth of the $d_{1}$ metric, it follows that the map

[TABLE]

is proper (meaning that $\Theta(k_{j},U_{j})=\Theta(I,U_{j})\to\infty$ if $|U_{j}|\to\infty$), hence it admits a minimizer. As $C$ is surjective, it follows that $\textup{Aut}_{0}(X,J)\ni h\to J_{\omega}(h.u)\in\mathbb{R}$ admits a minimizer $g\in\textup{Aut}_{0}(X,J)$ as well.

Fix $v\in\mathfrak{k}_{\omega_{g.u}}$. We introduce the vector field $W=X^{\omega_{g.u}}v$, and by the previous proposition $[0,\infty)\ni t\to h_{t}:=\textup{exp}(tJW).(g.u)\in\mathcal{H}_{0}\cap I^{-1}(0)$ is a geodesic ray.

As the identity element minimizes $\textup{Aut}_{0}(X,J)\ni h\to J_{\omega}(h.(g.u))\in\mathbb{R}$, we can write

[TABLE]

Using (4.37) we conclude that $\dot{h}_{0}=2v$, hence $\int_{X}v\omega^{n}=0$. Finally, the decomposition formula of Proposition 4.25 implies that (4.38) in fact holds for any $v\in\mathfrak{g}_{\omega_{g.u}}$. ∎

For $u\in\mathcal{H}_{\omega}$, recall the self adjoint differential operator $L^{f_{u}}$ from (4.32). Motivated by the explicit formula for the Hessian of $\mathcal{F}$ (see 4.29) we introduce the differential operator $\mathcal{D}^{u}:C^{\infty}(X,\mathbb{C})\to\mathbb{R}$:

[TABLE]

Integrating by parts in (4.29) we get the following formula, relating $\nabla^{2}\mathcal{F}(u)$ and $\mathcal{D}^{u}$:

[TABLE]

This implies that $\mathcal{D}^{u}$ is a self–adjoint differential operator as well. Additionally, the kernel of $\mathcal{D}^{u}$ is exactly equal to the eigenspace of $\mathcal{L}^{f_{u}}$ corresponding to the eigenvalue $\lambda=1$. In Proposition 4.20 (see especially the identity (4.34)) we gave an exact description of this space that we now recall:

[TABLE]

In case $u\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ is a KE potential we trivially have $f_{u}=0$, and comparing with (4.36) we get the following identification:

[TABLE]

As $\mathcal{D}^{u}$ is self–adjoint and elliptic, for any $h\in C^{\infty}(X,\mathbb{C})$ such that $h\perp\textup{Ker }\mathcal{D}^{u}$ there exists $v\in C^{\infty}(X,\mathbb{C})$ such that $\mathcal{D}^{u}(v)=h$ (see [113, Theorem IV.4.11]). We note this fact in slightly more precise form in the following lemma:

Lemma 4.28**.**

Let $(X,\omega_{u})$ is a KE manifold. Suppose that $\int_{X}v\omega^{n}=0$ for all $v\in\mathfrak{g}_{\omega_{u}}$. Then there exists $g\in C^{\infty}(X,\mathbb{R})$ such that

[TABLE]

Proof.

Denote $h:=\omega^{n}/\omega_{u}^{n}\in C^{\infty}(X,\mathbb{R})$. By our assumption we have that $h\perp\mathfrak{g}_{\omega_{u}}$ with respect to the Hermitian product $\langle\alpha,\beta\rangle=\frac{1}{V}\int_{X}\alpha\overline{\beta}\omega_{u}^{n}$, hence by our above remarks there exists $g\in C^{\infty}(X,\mathbb{C})$ such that $\mathcal{D}^{u}(g)=h$. As $u$ is a KE potential, we have $f_{u}=0$, and as a result $\mathcal{D}^{u}$ is a real differential operator. Consequently, we can make sure that $g\in C^{\infty}(X,\mathbb{R})$, finishing the argument. ∎

Proof of Theorem 4.23.

Without loss of generality we can assume that our KE potentials $u,v$ satisfy $u,v\in\mathcal{H}_{\omega}\cap I^{-1}(0)$. Also, by Proposition 4.27, after possibly pulling back $u$ and $v$ by an element of $\textup{Aut}_{0}(X,J)$ we can assume that

[TABLE]

Using this, by the previous lemma we can find $g_{u},g_{v}\in C^{\infty}(X,\mathbb{R})$ such that

[TABLE]

For the rest of the proof we will be working with the twisted Ding functional $\mathcal{F}_{s}:\mathcal{H}_{\omega}\to\mathbb{R},\ s\geq 0$ given by the formula:

[TABLE]

For small enough $s>0$ we can suppose that the potentials $u_{0}^{s}:=u+sg_{u}$ and $u_{1}^{s}:=v+sg_{u}$ satisfy $u^{s}_{0},u^{s}_{1}\in\mathcal{H}_{\omega}$.

The differential of $\mathcal{F}_{s}$ is equal to $d\mathcal{F}+sdJ$. Choosing $w\in C^{\infty}(X,\mathbb{R})$ with $\int_{X}w\omega_{u}^{n}=0$ a simple differentiation gives

[TABLE]

Since $u$ minimizes $\mathcal{F}$ we have that $d\mathcal{F}(u)(\nabla_{\frac{d}{ds}}w)=0$. Since $\int_{X}w\omega_{u}^{n}=0$ we also have $dJ_{\omega}(u)(w)=\frac{1}{V}\int_{X}w\omega^{n}$, so we can continue the above identity and write:

[TABLE]

where in the last identity we have used (4.41). Consequently $s\to d\mathcal{F}_{s}(u^{s}_{0})(w)=O(s^{2})$. Since $d\mathcal{F}_{s}(u^{s}_{0})(w)=\frac{1}{V}\int_{X}wf_{s}\omega^{n}$ for some smooth curve $s\to f_{s}$, we can conclude that $f_{s}=O(s^{2})$ and we have

[TABLE]

A similar estimate holds for \big{|}d\mathcal{F}_{s}(u^{s}_{1})(w)\big{|} as well.

Let $[0,1]\ni t\to u^{s}_{t}\in\mathcal{H}_{\omega}^{1,\bar{1}}$ be the $C^{1,\bar{1}}$–geodesic connecting $u_{0}^{s}$ and $u_{1}^{s}$. By convexity of $t\to\mathcal{F}(u^{s}_{t})$ and $t\to J_{\omega}(u^{s}_{t})$ it follows that

[TABLE]

where the last inequality is a consequence of (4.42). Taking the limit $s\searrow 0$ in the above estimate, (by convexity) we obtain that $t\to J_{\omega}(u^{0}_{t})$ is affine. By the lemma below, this implies that $u=u^{0}_{0}=u^{0}_{1}=v$, finishing the proof. ∎

Lemma 4.29**.**

Suppose $u_{0},u_{1}\in\mathcal{H}_{\omega}\cap I^{-1}(0)$ and $t\to u_{t}$ is the $C^{1,\bar{1}}$–geodesic connecting $u_{0},u_{1}$. If $t\to J_{\omega}(u_{t})$ is affine then $u_{0}=u_{1}$.

Proof.

We know that $t\to I(u_{t})$ is affine (Proposition 3.42). Using this, our assumption implies that $t\to\int_{X}u_{t}\omega^{n}$ is linear as well, hence \frac{d}{d_{t}}\big{|}_{t=0}\int_{X}u_{t}\omega^{n}=\frac{d}{d_{t}}\big{|}_{t=1}\int_{X}u_{t}\omega^{n}. This implies that $\int_{X}(\dot{u}_{1}-\dot{u}_{0})\omega^{n}=0$. By convexity in the $t$ variable we have $\dot{u}_{1}\geq\dot{u}_{0}$, so we conclude that $\dot{u}_{0}=\dot{u}_{1}$, hence $t\to u_{t}$ is affine, i.e.,

[TABLE]

Since $t\to I(u_{t})$ is affine too, we have \frac{d}{dt}\big{|}_{t=0}I(u_{t})=\frac{d}{dt}\big{|}_{t=1}I(u_{t}), and by (3.65) and (4.43) we get that $\int_{X}(u_{1}-u_{0})\omega_{u_{0}}^{n}=\int_{X}(u_{1}-u_{0})\omega_{u_{0}}^{n}$. Subtracting the right hand side from the left and integrating by parts we obtain

[TABLE]

As all terms in the above sum are nonnegative, we get that $\int_{X}i\partial(u_{1}-u_{0})\wedge\bar{\partial}(u_{1}-u_{0})\wedge\omega_{u_{0}}^{n-1}=0$. Consequently, $\langle\nabla^{\omega_{u_{0}}}(u_{1}-u_{0}),\nabla^{\omega_{u_{0}}}(u_{1}-u_{0})\rangle_{\omega_{u_{0}}}=0$, hence $u_{1}=u_{0}+c$. As $I(u_{0})=I(u_{1})=0$, we have in fact $u_{0}=u_{1}$. ∎

4.6 Regularity of weak minimizers of the Ding functional

In this short section we will show that minimizers of the extended $\mathcal{F}$ functional are actually smooth, with this verifying another important condition in the existence/properness principle described earlier (see Theorem 4.7):

Theorem 4.30**.**

If $u\in\mathcal{E}_{1}(X,\omega)$ minimizes $\mathcal{F}:\mathcal{E}_{1}(X,\omega)\to\mathbb{R}$ then $u$ is a smooth Kähler–Einstein potential.

In case $u\in\mathcal{H}_{\omega}$ minimizes $\mathcal{F}$, then after computing the first order variation of $t\to\mathcal{F}(u+tv)$ for all $v\in C^{\infty}(X)$, Lemma 4.1 allows to conclude that:

[TABLE]

hence $u$ is indeed a KE potential. In case $u\in\mathcal{E}_{1}(X,\omega)$ we can’t even guarantee that $t\to u+tv\in\mathcal{E}_{1}(X,\omega)$ for small $t$. Getting around this obstacle will represent one of main technical ingredients in the proof of Theorem 4.30.

Eventually we will be able to show that (4.44) holds for minimizers from $\mathcal{E}_{1}(X,\omega)$ as well. Notice that by Zeriahi’s version of Skoda’s theorem (Corollary 4.16) the right hand side of this equation does indeed makes sense for potentials of $\mathcal{E}_{1}(X,\omega)$. The proof is completed by appealing to work of Kolodziej and Tosatti–Székelyhidi on the apriori regularity theory of such equations:

Theorem 4.31**.**

If $u\in\mathcal{E}_{1}(X,\omega)$ solves (4.44) then $u\in\mathcal{H}_{\omega}$, i.e., $u$ is a smooth KE potential.

Sketch of proof..

As $u\in\mathcal{E}_{1}(X,\omega)$ Corollary 4.16 implies that $e^{-u+f_{0}}\in L^{p}(\omega^{n})$ for all $p>1$. By Kolodziej’s theorem [81], since $u$ solves (4.44), we obtain that $u$ is bounded (for a full proof of this fact see [74, Theorem 14.1]). Using a result of Tosatti–Székelyhidi [103, Theorem 1.1] we obtain that $u$ is actually a smooth KE potential (see also [74, Theorem 14.1]). ∎

Proof of Theorem 4.30.

Let $v\in C^{\infty}(X)$. Recall from (2.15) that $P(u+tv)\in\textup{PSH}(X,\omega),$ $t\in\mathbb{R}$ is defined as follows:

[TABLE]

Since $u-t\sup_{X}|v|\leq P(u+tv)$, Corollary 2.8 implies that $P(u+tv)\in\mathcal{E}_{1}(X,\omega)$, and this allows to introduce the function

[TABLE]

We claim that $g(t)$ is differentiable at $t=0$ and the following formula holds:

[TABLE]

From the proposition below it follows that \frac{d}{dt}\big{|}_{t=0}I(P(u+tv))=\frac{1}{V}\int_{X}v\omega_{u}^{n}. Consequently, to prove (4.45) it suffices to show that

[TABLE]

Using the elementary inequality $|e^{x}-e^{y}|\leq|x-y|(e^{x}+e^{y})$ we can write that

[TABLE]

Corollary 4.16 implies that the right most quantity in this inequality is integrable, hence we can conclude (4.46) using the dominated convergence theorem.

Since $P(u+tv)\leq u+tv$, we notice that $g(0)=\mathcal{F}(u)\leq\mathcal{F}(P(u+tv))\leq g(t),\ t\in\mathbb{R}.$ This implies that \frac{d}{dt}\big{|}_{t=0}g(t)=0, and by (4.45) we can conclude that (4.44) holds for $u\in\mathcal{E}_{1}(X,\omega)$. Now using Theorem 4.31 we conclude that $u$ is smooth, finishing the argument. ∎

Proposition 4.32**.**

[9, Lemma 3.10]* Suppose $u\in\mathcal{E}_{1}(X,\omega)$ and $v\in C^{\infty}(X)$. Then $P(u+tv)\in\mathcal{E}_{1}(X,\omega)$ for all $t\in\mathbb{R}$, and $t\to I(P(u+tv))$ is differentiable at $t=0$. More precisely,*

[TABLE]

In our approach we will follow closely the simplified argument proposed by Lu and Nguyen [86].

Proof.

We want to show that

[TABLE]

After changing $v$ to $-v$, it suffices to consider $t>0$ in the above limit. Using (3.58) we can write

[TABLE]

By the same inequality we also have that

[TABLE]

By the lemma below $\omega_{P(u+tv)}^{n}$ is concentrated on the coincidence set $\{P(u+tv)=u+tv\}$, thus we have

[TABLE]

We also have $|P(u+tv)-u|\leq t\sup_{X}|v|$, and since the complex Monge–Ampère operator is continuous under uniform convergence we conclude that

[TABLE]

finishing the argument. ∎

Finally, we provide the lemma promised in the proof of the above proposition:

Lemma 4.33**.**

Suppose $u\in\mathcal{E}_{1}(X,\omega)$ and $v\in C^{\infty}(X)$. Then $\int_{\{u+tv>P(u+tv)\}}\omega_{P(u+tv)}^{n}=0.$

Proof.

Using Theorem 2.1 we choose $u_{k}\in\mathcal{H}_{\omega}$ such that $u_{k}\searrow u$. By a classical Perron type argument it follows that $\int_{\{u_{k}+tv>P(u_{k}+tv)\}}\omega_{P(u_{k}+tv)}^{n}=0$ (see [3, Corollary 9.2]). This is equivalent to $\int_{X}(u_{k}+tv-P(u_{k}+tv))\omega_{P(u_{k}+tv)}^{n}=0.$ As $u_{k}+t\sup_{X}|v|\geq P(u_{k}+tv)$, Proposition 2.20 allows to take the limit $k\to\infty$ and conclude that

[TABLE]

which is equivalent to $\int_{\{u+tv>P(u+tv)\}}\omega_{P(u+tv)}^{n}=0.$ ∎

4.7 Properness of the K–energy and existence of KE metrics

Given $u\in\mathcal{H}_{\omega}$, the average of the scalar curvature of the metric $\omega_{u}$ is independent of $u$, as by (A.9) and integration by parts yields

[TABLE]

Next we introduce Mabuchi’s K–energy functional $\mathcal{K}:\mathcal{H}_{\omega}\to\mathbb{R}$ [87]:

[TABLE]

The reason behind this specific definition is the following variational formula, which shows that critical points of the K–energy are exactly the constant scalar curvature Kähler (csck) metrics:

Proposition 4.34**.**

For a smooth curve $(0,1)\ni t\to u_{t}\in\mathcal{H}_{\omega}$ we have

[TABLE]

Proof.

By straightforward calculations we arrive at the identities:

[TABLE]

[TABLE]

Consequently, integration by parts and (A.9) gives:

[TABLE]

[TABLE]

The desired formula now follows after differentiating $t\to\mathcal{K}(u_{t})$, and using the last two identities together with Lemma 3.37. ∎

Trivially, KE metrics are csck. By the next result, in case $c_{1}(X)$ is a multiple of $[\omega]$, the reverse is also true:

Lemma 4.35**.**

Suppose $c_{1}(X)=\lambda[\omega]$. Then $u\in\mathcal{H}_{\omega}$ is a csck potential if and only if it is a KE potential.

Proof.

Suppose $u$ is a csck potential, i.e., $S_{\omega_{u}}=\bar{S}$. Since $c_{1}(X)=\lambda[\omega]$, by definition of the Ricci potential (see (4.4)) and invariance of $\bar{S}$ (see (4.47)) we have

[TABLE]

hence $\Delta^{\omega_{u}}f_{u}=0$. The condition $\int_{X}e^{f_{u}}\omega^{n}=V$ gives $f_{u}=0$, i.e., $\omega_{u}$ is a KE metric. ∎

By this last lemma, in case $(X,\omega)$ is Fano, the critical points of $\mathcal{K}$ are exactly the KE potentials, hence the K–energy plays a role similar to the $\mathcal{F}$ functional. Developing this analogy further, our main result in this section parallels Theorem 4.11, giving another characterization of existence of KE metrics, confirming a related conjecture of Tian (see [106, Conjecture 7.12],[107, p. 127]):

Theorem 4.36**.**

([57, Theorem 2.4])*

Suppose $(X,\omega)$ is Fano with $c_{1}(X)=[\omega]$, and set $G:=\textup{Aut}(X,J)_{0}$. The following are equivalent:

(i) there exists a KE metric in $\mathcal{H}$.*

(ii) For some $C,D>0$ the following holds:

[TABLE]

(iii) For some $C,D>0$ the following holds:

[TABLE]

For the resolution of other closely related conjectures we refer to [57]. It is possible to give a proof for this theorem by verifying the conditions of the existence/properness principle (Theorem 4.7) directly, the same way as we did with Theorem 4.11. Instead of doing this, we will rely on the special relationship between the K–energy and the $\mathcal{F}$ functional (see Proposition 4.41 below).

First we have to show that $\mathcal{K}$ is $d_{1}$–lsc, and (4.48) gives the $d_{1}$–lsc extension of this functional to $\mathcal{E}_{1}(X,\omega)$. This will be done in a series of lemmas and propositions:

Lemma 4.37**.**

[57, Lemma 5.23]*

Suppose $\alpha$ is a smooth closed $(1,1)$-form on $X$. The functional $I_{\alpha}:\mathcal{H}_{\omega}\to\mathbb{R}$ given by*

[TABLE]

is $d_{1}$–continuous, and extends uniquely to a $d_{1}$–continuous functional $I_{\alpha}:\mathcal{E}_{1}(X,\omega)\to\mathbb{R}$. Additionally, $I_{\alpha}$ is bounded on $d_{1}$–bounded subsets of $\mathcal{E}_{1}(X,\omega)$.

Proof.

Let $u_{k}\in\mathcal{H}_{\omega}$ and $u\in\mathcal{E}_{1}(X,\omega)$ be such that $d_{1}(u_{k},u)\to 0$. An argument similar to that yielding (3.58) shows that

[TABLE]

For some $D>0$ we have $-D\omega\leq\alpha\leq D\omega$. Thus, $\omega_{(u_{l}+u_{k})/{4}}=\omega/2+\omega_{u_{l}}/4+\omega_{u_{k}}/4$ and for some $C>0$ we can write

[TABLE]

Using Lemma 3.33 and the triangle inequality, $d_{1}(0,(u_{l}+u_{k})/4)$ is bounded. By Corollary 3.50, we obtain that $\{I_{\alpha}(u_{k})\}_{k}$ is a Cauchy sequence, showing that $I_{\alpha}$ is $d_{1}$–continuous and it that extends $d_{1}$–continuously to $\mathcal{E}_{1}(X,\omega)$.

To argue $d_{1}$–boundedness of $I_{\alpha}$, we turn again to (4.51) (with $u_{k}=0$). By this estimate and Corollary 3.50, it is enough to show that if $d_{1}(0,u)$ is bounded then so is $d_{1}(0,u/4)$. This is a consequence of Lemma 3.33. ∎

As we will see shortly, the entropy functional u\to\int_{X}\log\big{(}\frac{\omega_{u}^{n}}{\omega^{n}}\big{)}\omega_{u}^{n} is only $d_{1}$–lsc. In fact, this functional is already lsc with respect to weak convergence of measures, as it follows from our discussion below and the next proposition.

Suppose $\nu,\mu$ are Borel probability measures on $X$. If $\nu$ is absolutely continuous with respect to $\mu$ then the entropy of $\nu$ with respect to $\mu$ is \textup{Ent}(\mu,\nu)=\int_{X}\log\big{(}\frac{\nu}{\mu}\big{)}\frac{\nu}{\mu}\mu, otherwise $\textup{Ent}(\mu,\nu)=\infty.$

The next well known result follows from the classical Jensen inequality:

Lemma 4.38**.**

Suppose $\nu,\mu$ are Borel probability measures on $X$. Then $\textup{Ent}(\mu,\nu)\geq 0$ and equality holds if and only if $\nu=\mu$.

Proof.

$\textup{Ent}(\mu,\nu)\geq 0$ follows from an application of Jensen’s inequality to the convex weight $\rho(x):=x\log x$. As $\rho$ is strictly convex on $\mathbb{R}^{+}$, the proof of Jensen’s inequality implies that $\textup{Ent}(\mu,\nu)=0$ if and only if $\frac{\mu}{\nu}=1$ (see [96, Chapter 3, Theorem 3.3]). ∎

Let us recall the following classical formula for the entropy of two measures:

Proposition 4.39**.**

Suppose $\mu,\nu$ are probability Borel measures on $X$. Then the following holds:

[TABLE]

where $B(X)$ is the set of bounded Borel measurable functions on $X$.

Proof.

In case $\nu$ is not absolutely continuous with respect to $\mu$ then there exists a Borel set $M\subset X$ with $\mu(M)=0$ but $\nu(M)>0$. Then we trivially have that $\int_{X}c\mathbbm{1}_{M}\nu-\log\int_{X}e^{c\mathbbm{1}_{M}}\mu=\int_{X}c\mathbbm{1}_{M}\nu-\log\int_{X}e^{0}\mu=c\nu(M)$ for any $c>0$ and consequently,

[TABLE]

which is equal to $\textup{Ent}(\mu,\nu)$ by definition.

We assume now that $\nu$ is absolutely continuous with respect to $\mu$, i.e. $\nu=g\mu$ for some non–negative Borel measurable function $g$. To conclude (4.52) we need to show that

[TABLE]

By choosing $f_{k}=\log g_{k}:=\log(\min(\max(g,1/k),k)),\ k\in\mathbb{N}$. We get that the right hand side of (4.53) is greater then $\int_{X}g\log g_{k}\mu-\log\int_{X}g_{k}\mu$. Letting $k\to\infty$, we conclude that the right hand side of (4.53) is greater then the left hand side.

For the other direction, we need to argue that $\int_{X}g\log g\mu\geq\int_{X}fg\mu-\log\int_{X}e^{f}\mu$ for any $f\in B(X)$. For this it is enough to invoke Jensen’s inequality:

[TABLE]

∎

As the supremum of continuous functionals is lsc, it follows that $\nu\to\textup{Ent}(\mu,\nu)$ is lsc with respect to weak convergence of Borel measures.

Theorem 3.46(ii) implies that for any $u_{k},u\in\mathcal{E}_{1}(X,\omega)$ we have that $d_{1}(u_{k},u)\to 0$ implies $\omega_{u_{k}}^{n}\to\omega_{u}^{n}$ weakly. We arrive at the following important corollary:

Corollary 4.40**.**

The functional \mathcal{E}_{1}(X,\omega)\ni u\to\textup{Ent}\big{(}\frac{1}{V}\omega^{n},\frac{1}{V}\omega_{u}^{n}\big{)}\in\mathbb{R} is $d_{1}$–lsc.

Comparing with (4.48), we observe that for $u\in\mathcal{H}_{\omega}$ we actually have

[TABLE]

This observation together with Proposition 3.40, Lemma 4.37 and Corollary 4.40 allows to conclude that $\mathcal{K}$ is $d_{1}$–lsc on $\mathcal{H}_{\omega}$ and it extends to $\mathcal{E}_{1}(X,\omega)$ in a $d_{1}$–lsc manner, using the formula of (4.54).

Lastly, before we prove Theorem 4.36, we provide a precise inequality between the K–energy and $\mathcal{F}$ functional, and we point out the relationship between the minimizers of these functionals:

Proposition 4.41** ([5]).**

Suppose $(X,\omega)$ is a Fano manifold with $c_{1}(X)=[\omega]$. For any $u\in\mathcal{E}_{1}(X,\omega)$ we have

[TABLE]

*Moreover, for $u\in\mathcal{E}_{1}(X,\omega)$ the following are equivalent:

(i) $\mathcal{F}(u)=\mathcal{K}(u)-\frac{1}{V}\int_{X}f_{0}\omega^{n}$.

(ii) $u$ minimizes $\mathcal{F}$.

(iii) $u$ minimizes $\mathcal{K}$.

(iv) $u$ is a smooth KE potential.*

Consequently, the minimizers of $\mathcal{K}$ and $\mathcal{F}$ on $\mathcal{E}_{1}(X,\omega)$ are the same and coincide with the set of smooth KE potentials.

Proof.

Let $u\in\mathcal{H}_{\omega}$. As both $\mathcal{K}$ and $\mathcal{F}$ are constant invariant, we can assume that $\int_{X}e^{-u+f_{0}}\omega^{n}=V$ and note the following identity:

[TABLE]

By the formula of the next lemma, we can write that

[TABLE]

By Lemma 4.38 we have \textup{Ent}\big{(}\frac{1}{V}e^{-u+f_{0}}\omega^{n},\frac{1}{V}\omega_{u}\big{)}\geq 0, hence (4.55) follows.

Moreover, by this same lemma, equality holds in (4.55) if and only if $\omega_{u}=e^{-u+f_{0}}\omega^{n}$, which is equivalent with $u$ being a smooth KE potential (see Theorem 4.31). By Theorem 4.30 it immediately follows that (i), (ii) and (iv) are equivalent.

If (iv) holds, then $u$ minimizes $\mathcal{F}$, and $\mathcal{F}(u)=\mathcal{K}(u)-\frac{1}{V}\int_{X}f_{0}\omega^{n}$, by (i). Consequently $u$ minimizes $\mathcal{K}$ as well, hence (iii) holds.

Suppose (iii) holds. By Lemma 4.43(ii) proved below, there exists $v\in\textup{PSH}(X,\omega)\cap L^{\infty}$ such that

[TABLE]

As $u$ minimizes $\mathcal{K}$, it follows that the inequalities above are actually equalities, hence (i) holds. ∎

As pointed out in the above argument, we need a special expression for the K-energy:

Lemma 4.42**.**

Suppose $(X,\omega)$ is a Fano manifold with $c_{1}(X)=[\omega]$. For $u\in\mathcal{H}_{\omega}$ we have

[TABLE]

Proof.

We start by deriving an alternative formula for $I_{\operatorname{Ric}\omega}$:

[TABLE]

Since $c_{1}(X)=[\omega]$, we have $\bar{S}=n$ (see (4.47)), so by the above we conclude that

[TABLE]

We note that \textup{Ent}\big{(}\frac{1}{V}\omega^{n},\frac{1}{V}\omega_{u}^{n}\big{)}=\textup{Ent}\big{(}\frac{1}{V}e^{f_{0}}\omega^{n},\frac{1}{V}\omega_{u}^{n}\big{)}+\frac{1}{V}\int_{X}f_{0}\omega_{u}^{n}. Adding this identity to the above, and comparing with (4.54) finishes the proof. ∎

In the next lemma, we will make us of the inverse Ricci operator, introduced in [93] in connection with the so-called Ricci iteration. Given a potential $u\in\mathcal{E}_{1}(X,\omega)$, Corollary 4.16 and Kolodziej’s estimate [81, 82] give another potential $\operatorname{Ric}^{-1}(u)\in\textup{PSH}(X,\omega)\cap L^{\infty}$, unique up to a constant, such that

[TABLE]

In case $u\in\mathcal{H}_{\omega}$, we notice that $\operatorname{Ric}\omega_{\operatorname{Ric}^{-1}u}=\omega_{u},$ motivating the terminology.

By the next lemma, the inverse Ricci operator decreases the $\mathcal{F}$ functional and sheds further light on the intimate relationship between $\mathcal{K}$ and $\mathcal{F}$:

Lemma 4.43** ([93]).**

*Suppose $u\in\mathcal{E}_{1}(X,\omega)$. Then the following hold:

(i) $\mathcal{F}(\operatorname{Ric}^{-1}(u))\leq\mathcal{F}(u).$

(ii)$\ \mathcal{K}(\operatorname{Ric}^{-1}(u))-\frac{1}{V}\int_{X}f_{0}\omega^{n}\leq\mathcal{F}(u).$

(iii)$\ \mathcal{K}(\operatorname{Ric}^{-1}(u))\leq\mathcal{K}(u).$*

Proof.

First we argue (ii). We introduce $v:=\textup{Ric}^{-1}(u)$. As both $\mathcal{F}$ and $\mathcal{K}$ are constant invariant, we can assume that $\int_{X}e^{-u+f_{0}}\omega^{n}=V$. By the previous lemma, notice that we have

[TABLE]

where in the penultimate estimate we have used (3.64).

Lastly, (i) and (iii) follow from (ii) and (4.55). ∎

We now prove the main compactness theorem of the space $(\mathcal{E}_{1}(X,\omega),d_{1})$:

Theorem 4.44**.**

Let $u_{j}\in\mathcal{E}_{1}(X,\omega)$ be a $d_{1}$–bounded sequence for which \textup{Ent}\big{(}\frac{1}{V}\omega^{n},\frac{1}{V}\omega_{u_{j}}^{n}\big{)} is bounded. Then $\{u_{j}\}_{j}$ contains a $d_{1}$–convergent subsequence.

Proof.

As a consequence of Corollary 4.16, for any $p>0$ there exists $C(p)>0$ such that that $\int_{X}e^{-pu_{j}}\omega^{n}\leq C$. Since $|\sup_{X}u_{j}|$ is bounded, we get that

[TABLE]

Consider $\phi,\psi:\mathbb{R}\to\mathbb{R}^{+}$ given by

[TABLE]

An elementary calculation verifies that both of these functions are normalized Young weights and $\phi^{*}=\psi$ (in the sense of (1.1)).

Since \textup{Ent}\big{(}\frac{1}{V}\omega^{n},\frac{1}{V}\omega_{u_{j}}^{n}\big{)} is bounded, so is \int_{X}\phi\big{(}\omega_{u_{j}}^{n}/\omega^{n}\big{)}\omega^{n}. As $\phi$ is convex and $\phi(0)=0$, for some $D\in(0,1)$ we get that

[TABLE]

From $d_{1}$-boundedness we have that $|\sup_{X}u_{j}|$ and $I(u_{j})$ are bounded. By Lemma 4.13, after possibly taking a subsequence, we can find $u\in\mathcal{E}_{1}(X,\omega)$ such that $\int_{X}|u_{j}-u|\omega^{n}\to 0$.

Recalling the definition of Orlicz norms from (1.3), we can use the Hölder inequality (1.6) to deduce that

[TABLE]

From (4.57) it follows that $\|D{\omega_{u_{j}}^{n}}/{\omega^{n}}\|_{\phi,\frac{1}{V}\omega^{n}}\leq 1$. Since $\psi(t)\leq t^{2}e^{|t|}$, for any $r>0$ we can write

[TABLE]

The last two terms on right hand side are bounded by (4.56). As $\int_{X}|u_{j}-u|\omega^{n}\to 0$ it follows that $\int_{X}|u_{j}-u|^{6}\omega^{n}\to 0$, hence $\lim_{j}\|D^{-1}(u_{j}-u)\|_{\psi,\frac{1}{V}\omega^{n}}\leq\varepsilon$ for any $\varepsilon>0$.

Using (4.58), we conclude that $\int_{X}|u_{j}-u|\omega_{u_{j}}^{n}\to 0$. As a result, (3.64) gives that $\liminf_{j}I(u_{j})\geq I(u)$. Together with Corollary 4.14 we obtain that $\lim_{j}I(u_{j})=I(u)$. Since additionally $\int_{X}|u_{j}-u|\omega^{n}\to 0$, Theorem 3.46(i) implies that $d_{1}(u_{j},u)\to 0$, finishing the proof. ∎

Lastly, we prove our main theorem:

Proof of Theorem 4.36.

The equivalence between (ii) and (iii) follows from Lemma 4.4.

Suppose (i) holds. Then Proposition 4.41 and Theorem 4.11 implies properness of $\mathcal{K}$, giving (ii).

Now assume that (ii) holds. Then $\mathcal{K}$ is bounded from below and we can find $u_{j}\in\mathcal{E}_{1}(X,\omega)$ that is $d_{1}$–bounded and $\lim_{j}\mathcal{K}(u_{j})=\inf_{v\in\mathcal{E}_{1}(X,\omega)}\mathcal{K}(v)$.

In particular, $\{\mathcal{K}(u_{j})\}_{j}$ is bounded. Recalling (4.54), and the fact that both $I$ and $I_{\operatorname{Ric}\omega}$ are bounded on $d_{1}$–bounded sets (see Proposition 3.40 and Lemma 4.37), it follows that \textup{Ent}\big{(}\frac{1}{V}\omega^{n},\frac{1}{V}\omega_{u_{j}}^{n}\big{)} is bounded as well.

By the previous compactness theorem, after possibly passing to a subsequence, we have $d_{1}(u_{j},u)\to 0$ for some $u\in\mathcal{E}_{1}(X,\omega)$. Since $\mathcal{K}$ is $d_{1}$–lsc, we immediately obtain that $\mathcal{K}(u)=\inf_{v\in\mathcal{E}_{1}(X,\omega)}\mathcal{K}(v)$, i.e., $u$ minimizes $\mathcal{K}$.

By the equivalence between (iii) and (iv) in Theorem 4.41, $u$ is in fact a smooth KE potential. ∎

Brief historical remarks.

Motivated by results in conformal geometry, the relationship between energy properness and existence of canonical metrics in Kähler geometry goes back to the work of Tian and collaborators in the nineties [104, 106]. Numerous conjectures were proposed during this time, a number of which where adressed in the case of Fano manifolds without vectorfields [64, 105, 110]. For general Fano manifolds, all the remaining conjectures where addressed in [57], and we refer to this work for more details.

The sharpest form of the energy properness condition was identified in [89] and was later adopted in the literature, including in the present work.

Regarding general Kähler manifolds, in [57] the equivalence between energy properness and existence of consant scalar curvature (csck) metrics is linked to a regularity problem for fourth order PDE’s. In case a csck metric exists, this regularity conjecture was confirmed in [14], showing that on such manifolds the K-energy is indeed proper, partially generalizing Theorem 4.36.

In this chapter we tackled problems related to energy properness directly via the existence/properness principle of [57] (Theorem 4.7). The use of geodesic convexity in this context was initially proposed by X.X. Chen, however he advocated for the use of the $L^{2}$ Mabuchi geometry instead [36].

One of the advantages of our method (that uses pluripotential theory predominantly) over previous approaches in the literature is its adaptability to Kähler structures with mild singularities [49, 62]. For generalizations in other directions, as well as a more thourough overview of the vast related literature we refer to [57, 95].

Appendix A

A.1 Basic formulas of Kähler geometry

In this section we recall the most important facts about Kähler manifolds. Our minimalist approach follows [23] closely, and we refer to [60],[113] and [102] for more exhaustive treatments.

Suppose $(X,J)$ is compact connected complex manifold with holomorphic structure $J$. A Hermitian structure $(X,h)$ is the complex analog of a Riemannian structure, i.e., a smooth choice of $J-$compatible Hermitian metrics on each fiber of $TX.$ In local complex coordinates $h$ can be expressed as

[TABLE]

where we have used the Einstein summation. The real part of $h$ induces a Riemannian structure:

[TABLE]

This metric is compatible with $J$ in the sense that $\langle\cdot,\cdot\rangle=\langle J(\cdot),J(\cdot)\rangle$.

By $T_{\mathbb{C}}X=TX\otimes\mathbb{C}$ we denote the complexification of $TX$ and extend $J$ and $\langle\cdot,\cdot\rangle$ to $T_{\mathbb{C}}X$ in a $\mathbb{C}$–linear way. In local coordinates $z_{j}=x_{j}+iy_{j}$ the vector fields $\partial/\partial x_{j}$ , $\partial/\partial y_{j}$ span $TX$ over $\mathbb{R}$. We also have $J\partial/\partial x_{j}=\partial/\partial y_{j},\ J\partial/\partial y_{j}=-\partial/\partial x_{j}$ and the eigenbase of $J$ composed of the vector fields

[TABLE]

that span $T^{(1,0)}_{\mathbb{C}}X$ and $T^{(0,1)}_{\mathbb{C}}X$ respectively over $\mathbb{C}$. We also have the identities

[TABLE]

Compared to other works, we chose to multiply by a factor of two in the definition of $\langle\cdot,\cdot\rangle$ so that the following formula holds:

[TABLE]

Consequently, for any real vector field $Y\in C^{\infty}(X,TX)$ we have that $Y=Y^{j}\partial_{j}+\overline{Y^{j}}\partial_{\bar{j}}$ and $\|Y\|^{2}=\langle Y,Y\rangle=2Y^{j}\overline{Y^{k}}g_{j\bar{k}}$.

We will be also interested in the imaginary part of $h$:

[TABLE]

It is straightforward to see that

[TABLE]

We say that $\omega$ is a Kähler form if $d\omega=0$. In this case $(X,\omega)$ is called a Kähler manifold and we fix such a manifold for the remainder of this section.

By the Poincaré lemma [113, Lemma II.2.15], the Kähler condition implies that for any $x\in X$ there exists an open neighborhood $U\ni x$ and a local Kähler potential $g\in C^{\infty}(\overline{U})$ satisfying

[TABLE]

From this and (A.1) it follows that $g_{j\bar{k}}=\partial^{2}g/\partial z_{j}\partial\bar{z}_{k}$, hence in Kähler geometry, when doing calculations in local coordinates, one can interchange indices of the metric with its partial derivatives. We will do this quite frequently in our study.

The Christoffel symbols.

Given a hermitian structure $(X,J,h)$, it is possible to derive the following identity relating $\omega$, the Riemannian metric $\langle\cdot,\cdot\rangle$ and its Levi–Civita connection:

[TABLE]

where $W,Y,Z$ are smooth vector fields on $X$. Using the Kähler condition, this identity gives that

[TABLE]

As a consequence of this we obtain the following formulas for Christoffel symbols of $\nabla_{(\cdot)}(\cdot)$ in holomorphic local coordinates:

[TABLE]

More concretely, to derive the above formulas we used that :

(i) the Levi–Civita connection is torsion free, i.e. $\Gamma^{j}_{l\bar{k}}=\Gamma^{j}_{\bar{k}l},\ \Gamma^{j}_{lk}=\Gamma^{j}_{kl}$, etc.

(ii) the Levi–Civita connection is real, i.e., $\overline{\Gamma^{j}_{l\bar{k}}}=\Gamma^{\bar{j}}_{\bar{l}k},\ \overline{\Gamma^{j}_{lk}}=\Gamma^{\bar{j}}_{\bar{l}\bar{k}},$ etc.

(iii) $\nabla_{(\cdot)}J=0$, i.e., $i\nabla_{\partial_{j}}\partial_{k}=\nabla_{\partial_{j}}J\partial_{k}=J\nabla_{\partial_{j}}\partial_{k},\ -i\nabla_{\partial_{j}}\partial_{\bar{k}}=\nabla_{\partial_{j}}J\partial_{\bar{k}}=J\nabla_{\partial_{j}}\partial_{\bar{k}}$.

(iv) the product rule, $g_{lj\bar{k}}=\partial_{l}\langle\partial_{j},\partial_{\bar{k}}\rangle=\langle\nabla_{\partial_{l}}\partial_{j},\partial_{\bar{k}}\rangle+\langle\partial_{j},\nabla_{\partial_{l}}\partial_{\bar{k}}\rangle=\langle\nabla_{\partial_{l}}\partial_{j},\partial_{\bar{k}}\rangle=\Gamma_{lj}^{h}g_{h\bar{k}}.$

The volume form, gradient, Laplacian and Hamiltonian.

For Kähler geometers the volume form is equal to $\omega^{n}$. This is a non–degenerate top form that is a scalar multiple of the usual volume form of Riemannian geometry.

For a smooth function $v\in C^{\infty}(X)$ the Kähler gradient and Laplacian are given as follows:

[TABLE]

The length squared of the gradient is $\langle\nabla^{\omega}v,\nabla^{\omega}v\rangle=2ni\partial v\wedge\bar{\partial}v\wedge\omega^{n-1}/\omega^{n}=2g^{j\bar{k}}v_{j}v_{\bar{k}}.$ Also, given $u,v\in C^{\infty}(X)$, integration by parts gives the well known identity

[TABLE]

Lastly, from (A.2) it follows that $dv=\langle\nabla^{\omega}v,\cdot\rangle=-\omega(J\nabla^{\omega}v,\cdot)$, hence we have the following formula between the gradient and Hamiltonian of $v$:

[TABLE]

The Ricci and scalar curvatures.

We recall the definition of the curvature tensor. Suppose $T,Y,Z,W$ are smooth sections of the complexified tangent bundle $TX^{\mathbb{C}}$. The Riemannian curvature tensor is introduced by the formulas:

[TABLE]

[TABLE]

The classical curvature identities are:

[TABLE]

From $\nabla_{(\cdot)}J=0$ it follows that $R(T,Y)J=JR(T,Y)$, hence also

[TABLE]

From the curvature identity $R(T,Y,Z,W)=R(Z,W,T,Y)$ we also obtain

[TABLE]

hence $R(JT,JY)Z=R(T,Y)Z.$ We use this to compute the curvature in local coordinates:

[TABLE]

The Ricci curvature is the trace of the curvature tensor:

[TABLE]

From the curvature identities (A.1) it follows that Ric is symmetric. Moreover, from (A.1) it also follows that $\textup{Ric}(JT,JY)=\textup{Ric}(T,Y)$ and in local coordinates we have:

[TABLE]

Similar to the contruction of the Kähler form (A.1) we consider the Ricci form:

[TABLE]

This last formula tells us that the Ricci form is closed, and given two Kähler metrics $\omega,\omega^{\prime}$ on $X$ we have

[TABLE]

The scalar curvature $S_{\omega}$ is the trace of the Ricci curvature. In local coordinates it can be expressed as

[TABLE]

We compute the variation of the Ricci and scalar curvatures in the next proposition:

Proposition A.1**.**

Suppose $[0,1]\ni t\to\omega_{t}$ is a smooth curve of Kähler metrics on $X$. Then the following holds for the variation of the Ricci and scalar curvatures:

[TABLE]

[TABLE]

Proof.

The proof of the formulas follow after we differentiate (A.8) and (A.10). ∎

Normal coordinates.

One of the advantages of dealing with Kähler metrics is the fact that for local calculations one can choose coordinates in a convenient way:

Proposition A.2**.**

Suppose $(X,\omega)$ is a Kähler manifold. Given $x\in X$ there exist holomorphic local coordinates $z=(z_{1},\ldots,z_{n})$ mapping a neighborhood of $x$ to a neighborhood $U$ of $0\in\mathbb{C}^{n}$ such that in these coordinates $\omega$ can be expressed as:

[TABLE]

Proof.

First choose an arbitrary coordinate patch that maps $x$ and a neighborhood of $x$ to $0\subset\mathbb{C}^{n}$ and $V\subset\mathbb{C}^{n}$ respectively. It is possible to apply a linear change of coordinates in $\mathbb{C}^{n}$ so that $\omega|_{x}=i\partial\bar{\partial}g(0)=idz_{j}\wedge d\bar{z}_{j}$. Now we deal with the first order coefficients by making the following local change of coordinates:

[TABLE]

An elementary calculation for $F^{*}\omega=i\partial\bar{\partial}g\circ F=:i\partial\bar{\partial}\tilde{g}$ yields

[TABLE]

Choosing $c_{kjl}=-g_{j\bar{k}l}(0)/2$, we obtain that after composing our local coordinate map with $F$ the first order coefficients of $\omega$ vanish. Now we deal with the second order coefficients. For this we introduce a local diffeomorphism of the form

[TABLE]

Pulling back again by $H$, for $\tilde{g}=g\circ H$ we can write

[TABLE]

Choosing $b_{kjab}=-g_{j\bar{k}ab}(0)/6$ we get $\tilde{g}_{j\bar{k}ab}(0)=0$. By taking conjugates and applying the Leibniz rule for derivatives we also get $\tilde{g}_{j\bar{k}\bar{a}\bar{b}}(0)=0$, finishing the proof. ∎

It immediately follows that all Christoffel symbols of $\omega$ vanish at $x$ after a choice of normal coordinates around this point, as described in the above proposition. Also, from (A.1),(A.8) and (A.10) we obtain convenient formulas for geometric quantities related to curvature:

[TABLE]

The Lichnerowicz operator.

Given a complex valued function $u\in C^{\infty}(X,\mathbb{C})$, the Riemannian Hessian of $u$ is computed by the well known formula $\nabla^{2}u(X,Y)=Xdu(Y)-du(\nabla_{X}Y)$. We denote by $\mathcal{L}u$ the $(0,2)$ part of $\nabla^{2}u$. In local coordinates this can be expressed as

[TABLE]

After switching to normal coordinates, one can easily see that $\mathcal{L}u=0$ if and only if $(g^{j\bar{k}}u_{\bar{k}})_{\bar{l}}=0$ for all $\bar{l}\in\{1,\ldots,n\}$ which is equivalent to $\nabla^{\omega}_{1,0}u$ being a holomorphic vector field. $\mathcal{L}$ is the Lichnerowicz operator and the following formula for the self–adjoint complex operator $\mathcal{L}^{*}\mathcal{L}$ will be very useful for us:

Proposition A.3**.**

${\mathcal{L}}^{*}\mathcal{L}u=\frac{1}{4}\Delta^{\omega}(\Delta^{\omega}u)+\langle\textup{Ric}_{\omega},i\partial\bar{\partial}u\rangle_{\omega_{u}}+\langle\partial S_{\omega},\partial u\rangle,\ u\in C^{\infty}(X).$**

To clarify, in the above formula $\mathcal{L}^{*}$ is the Hermitian dual of $\mathcal{L}$, as will be detailed in the proof below. Also, $\langle\partial S_{\omega},\partial u\rangle$ denotes the Hermitian inner product of $\partial S_{\omega}$ and $\partial u$ (equaling $g^{j\bar{k}}{S_{\omega}}_{j}u_{\bar{k}}$ in local coordinates).

Proof.

Suppose $x\in X$. One has the following local expression for $\langle\mathcal{L}u,\mathcal{L}v\rangle$:

[TABLE]

We choose normal coordinates around $x$ as in Proposition A.2. Using integration by parts and (A.12) one can write

[TABLE]

As we are working in normal coordinates, all terms of type $g_{abc\bar{d}},g_{ab\bar{c}}$ vanish, hence the second term in the above sum is 0. By expanding the first term we arrive at:

[TABLE]

By (A.4) and (A.11), in normal coordinates we have $4\Delta^{\omega}(\Delta^{\omega}u)=g^{a\bar{b}}(g^{c\bar{d}}u_{c\bar{d}})_{a\bar{b}}=u_{a\bar{a}c\bar{c}}-u_{c\bar{d}}g_{a\bar{a}c\bar{d}}$, $\langle\partial S_{\omega},\partial u\rangle=g_{ljk\bar{j}\bar{k}}u_{\bar{l}}$ and $\langle\textup{Ric}_{\omega},i\partial\bar{\partial}u\rangle=-g_{lj\bar{j}\bar{k}}u_{k\bar{l}}$ hence the desired identity follows. ∎

A.2 Approximation of $\omega$–psh functions on Kähler manifolds

Given a compact Kähler manifold, we show that any $\omega$–psh function can be approximated by a decreasing sequence of smooth Kähler potentials. Much stronger results have been derived by Demailly [58, 59], using sophisticated techniques. Here we will follow closely the arguments of Blocki–Kolodziej [24]. Our main result is the following:

Theorem A.4**.**

Let $(X,\omega)$ be a compact Kähler manifold. Given $u\in\textup{PSH}(X,\omega)$, there exists a decreasing sequence $\{u_{k}\}_{k}\subset\mathcal{H}_{\omega}$ such that $u_{k}\searrow u$.

The main technical ingredient will be the following proposition, which provides an intermediate result:

Proposition A.5**.**

Let $(X,\omega)$ be a Kähler manifold (not necessarily compact). Let $X^{\prime}\subset X$ be a relatively compact open set. If the Lelong numbers of $\varphi$ are zero at any $x\in X$, then there exists $\varphi_{k}\in\textup{PSH}(X^{\prime},(1+\frac{1}{k})\omega)\cap C^{\infty}(X^{\prime})$ such that $\varphi_{k}\searrow\varphi$.

Based on this proposition we quickly give the proof of Theorem A.4:

Proof of Theorem A.4.

We can assume that $u\leq-1$, and consider the cutoffs $v_{k}:=\max(u,-k)\in\textup{PSH}(X,\omega)\cap L^{\infty}$. An application of Proposition A.5 to $\{v_{k}\}_{k}$ gives a sequence $w_{k}\in\mathcal{H}_{(1+\frac{1}{k})\omega}$ such that $w_{k}\searrow u$ and $v_{k}\leq w_{k}\leq 0$. Introducing $u_{k}:=\frac{1}{k}+(1-\frac{1}{k})w_{k}\in\mathcal{H}_{\omega}$, we see that $\{u_{k}\}_{k}$ is decreasing and $u_{k}\searrow u$. ∎

Returning to the local situation of a moment, let $v\in\textup{PSH}(U)$ for some open set $U\subset\mathbb{C}^{n}$. It is well known that the correspondence $r\to f(r)=\sup_{{B(z_{0},e^{r})}}v=\max_{\partial{B(z_{0},e^{r})}}v$ is convex for any $z_{0}\in U$ (this follows from the comparison principle for the complex Monge–Ampère operator and the fact that $i\partial\bar{\partial}(\log|z-z_{0}|)^{n}=0$ on $\mathbb{C}^{n}\setminus\{z_{0}\}$). Convexity of $r\to f(r)$ gives in particular that this map is continuous, hence so are the maps $v_{\delta}\in\textup{PSH}(U_{\delta})$, defined by the formula

[TABLE]

where $U_{\delta}=\{x\in U,\ B(x,\delta)\subset U\}.$ Lastly, as $v$ is usc, we also obtain that $v_{\delta}\searrow v$.

Given $u\in\textup{PSH}(X,\omega)$ and $x_{0}\in X$, recall the definition of the Lelong number $\mathcal{L}(u,x_{0})$ from (2.13):

[TABLE]

where $U_{r}\subset\mathbb{C}^{n}$ is some coordinate neighborhood of $x_{0}$ (dependent on $r$) that identifies $x_{0}$ with $0\in\mathbb{C}^{n}$. We now give an alternative description of $\mathcal{L}(u,x_{0})$ that will be of great use in the proof of Proposition A.5. Choose a coordinate neighborhood $U\subset\mathbb{C}^{n}$ of $x_{0}$, that identifies $x_{0}$ with $0\in\mathbb{C}^{n}$, and a potential $f\in C^{\infty}(X)$ such that $i\partial\bar{\partial}f=\omega$ on $U$. Clearly $u+f\in\textup{PSH}(U)$, and the following is well known:

[TABLE]

The limit on the right hand side is well defined as $\log r\to\sup_{B(x_{0},r)}(u+f)$ is convex, hence the quotients involved are decreasing as $r\to 0$. Also, this limit is clearly independent of the choice of potential $f$. For an extensive treatment of Lelong numbers we refer to [74, Section 2.3].

Finally, we arrive at the following result which will allow to compare approximation via (A.13) using different coordinate charts:

Lemma A.6**.**

Let $U,V\subset\mathbb{C}^{n}$ and $F:U\to V$ a biholomorphic map. Suppose $u\in\textup{PSH}(U)$ has zero Lelong numbers. For $u^{F}_{\delta}:=(u\circ F^{-1})_{\delta}\circ F$, the difference $u_{\delta}-u^{F}_{\delta}$ converges to zero uniformly on compact sets as $\delta\to 0$.

Proof.

Fix $a>1,\ r>0$ and $z\in U$. As $\log\delta\to u_{\delta}(z)$ is convex, for small enough $\delta>0$ we can write

[TABLE]

Consequently, as $\mathcal{L}(u,z)=0$, (A.14) implies that $u_{\delta}-u_{a\delta}\to 0$ uniformly on compact sets as $\delta\to 0$.

Next we notice that

[TABLE]

As $F$ is a biholomorphism, for a fixed compact set $K\subset U$ it is possible to find $a>1$ and $\delta_{0}>0$ such that for $\delta\in(0,\delta_{0})$ and $z\in K$ we have

[TABLE]

As a result of these containments, we get that $u^{F}_{\delta}\leq u_{a\delta}$ and $u_{\delta}\leq u^{F}_{a\delta}$ on $K$. Since $(u_{\delta}-u_{a\delta})\to 0$ uniformly on compacts, the statement of the lemma follows. ∎

Proof of Proposition A.5.

First we find $\tilde{\varphi}_{j}\in\textup{PSH}(X^{\prime},(1+\frac{1}{j})\omega)\cap C(X^{\prime})$ satisfying the requirements of the proposition.

Fix $\varepsilon>0$. We can find a finite number of nested charts $V_{\alpha}\subset U_{\alpha}$ such that $\{V_{\alpha}\}_{\alpha}$ covers $X^{\prime}$, $\overline{V_{\alpha}}\subset U_{\alpha}$, and $i\partial\bar{\partial}f_{\alpha}=\omega$ on $U_{\alpha}$, for some potentials $f_{\alpha}\in C^{\infty}(U_{\alpha})$.

Then $\varphi_{\alpha}:=\varphi+f_{\alpha}\in\textup{PSH}(U_{\alpha})$ and by the previous lemma we have

[TABLE]

locally uniformly on $U_{\alpha}\cap U_{\beta}$ as $\delta\to 0$, where $F$ is the change of coordinates on the overlap $U_{\alpha}\cap U_{\beta}$. Let $\eta_{\alpha}$ be smooth on $U_{\alpha}$ such that $\eta_{\alpha}=0$ on $V_{\alpha}$ and $\eta_{\alpha}=-1$ in a neighborhood of $\partial U_{\alpha}$. We have $i\partial\bar{\partial}\eta_{\alpha}>-C\omega$ some $C>0$.

We set

[TABLE]

By (A.15), for $\delta>0$ sufficiently small, the values of $\varphi_{\alpha,\delta}-f_{\alpha}+{\varepsilon\eta_{\alpha}}/{C}$ do not contribute to the maximum on the set $\{\eta_{\alpha}=-1\}$, thus $\tilde{\varphi}_{\delta}$ is continuous on $X^{\prime}$ and also $\tilde{\varphi}_{\delta}\in\textup{PSH}(X^{\prime},(1+\varepsilon)\omega)$. It is also clear that $\tilde{\varphi}_{\delta}$ decreases to $\varphi$ as $\delta\searrow 0$, allowing to construct $\tilde{\varphi}_{j}\in\textup{PSH}(X^{\prime},(1+\frac{1}{j})\omega)\cap C(X^{\prime})$ such that $\tilde{\varphi}_{j}\searrow\varphi$.

Next we argue that is also possible to approximate with smooth potentials. For this we only need to show that any $\psi$ that is continuous and $(1+\varepsilon)\omega$–psh on an open neighborhood $Y$ of $\overline{X^{\prime}}$ can be approximated uniformly on $X^{\prime}$ by smooth $(1+2\varepsilon\omega)$–psh potentials.

This can be done by using classical Richberg approximation. Indeed, let $V_{\alpha}\subset U_{\alpha}$, $f_{\alpha}$ and $\eta_{\alpha}$ be as in the first part of the proof. Additionally, let $\rho\in C^{\infty}_{c}(B(0,1))$ be a smooth, non–negative, compactly supported and spherically invariant bump function with $\int_{\mathbb{C}^{n}}\rho=1$.

We introduce the smooth mollifications $\psi_{\alpha,\delta}:=(\psi+(1+\varepsilon)f_{\alpha})*\rho_{\delta}\in\textup{PSH}(U_{\alpha,\delta})$ that converge uniformly to $\psi+(1+\varepsilon)f_{\alpha}$ on $V_{\alpha}$, since $\psi$ is continuous. As a result, we can apply the Richberg regularized maximum ([60, Corollary I.5.19 and Theorem II.5.21]) to get that

[TABLE]

is smooth on $\cup_{\alpha}V_{\alpha}$ (for small enough $\delta$) and also $\psi_{\delta}\in\textup{PSH}(\cup_{\alpha}V_{\alpha},(1+2\varepsilon)\omega)$. Consequently, $\psi_{\delta}\to\psi$ uniformly on $X^{\prime}\subset\cup_{\alpha}V_{\alpha}$, finishing the proof. ∎

A.3 Regularity of envelopes of $\omega$–psh functions

Suppose $(X,\omega)$ is a Kähler manifold and $f$ is an usc function on $X$. Recall the definition of the envelope $P(f)$ from (2.15):

[TABLE]

As it was argued in Section 3.4, $P(f)$ is $\omega$–psh and the purpose of this short section is to show the following regularity result:

Theorem A.7**.**

If $f\in C^{\infty}(X)$ then $\|P(f)\|_{C^{1,\bar{1}}}\leq C(X,\omega,\|f\|_{C^{1,\bar{1}}}).$

A bound on the $C^{1,\bar{1}}$ norm of $P(f)$ simply means a uniform bound on all mixed second order derivatives $\partial^{2}P(f)/\partial z_{j}\bar{\partial}z_{k}$. Since $P(f)$ is $\omega$-psh, this is equivalent to saying that $\Delta^{\omega}P(f)$ is bounded, and by the Calderon–Zygmund estimate [75, Chapter 9, Lemma 9.9], we automatically obtain that $P(f_{1},f_{2},\ldots,f_{k})\in C^{1,\alpha}(X)$.

Theorem A.7 was first proved by Berman–Demailly using methods from pluripotential theory[15]. Here we will follow an alternative path proposed by Berman, that uses more classical PDE techniques [6]. The point is to consider the following complex Monge–Ampère equation:

[TABLE]

As $f\in C^{\infty}(X)$, it follows from work of Aubin and Yau that this equation always has a unique smooth solution $u_{\beta}\in\mathcal{H}_{\omega}$ for any $\beta>0$ [1, 112] (for a survey see [74, Theorem 14.1]). Theorem A.7 will follow from the following regularity result:

Proposition A.8** ([6]).**

*The unique solutions $\{u_{\beta}\}_{\beta>0}$ of (A.17) satisfy the following:

(i) $\|u_{\beta}-P(f)\|_{C^{0}}\to 0$ as $\beta\to\infty$.

(ii) there exists $\beta_{0}>0$ and $C>0$ such that $\|\Delta^{\omega}u_{\beta}\|_{C^{0}}<C$ for all $\beta\geq\beta_{0}$.*

To argue Proposition A.8(i), first we prove the following comparison principle :

Lemma A.9**.**

Assume that $u,v\in\textup{PSH}(X,\omega)\cap L^{\infty}$ such that

[TABLE]

Then $v\leq u$.

Proof.

According to the comparison principle $\int_{\{u<v\}}\omega_{v}^{n}\leq\int_{\{u<v\}}\omega_{u}^{n}$ (Proposition 2.3). Consequently, we can write that $\int_{\{u<v\}}e^{\beta(v-f)}\omega^{n}\leq\int_{\{u<v\}}e^{\beta(u-f)}\omega^{n}$. As a result, $v\leq u$ a.e. with respect to the Lebesque measure. Since $u$ and $v$ are $\omega$–psh, we obtain that in fact $v\leq u$ globally on $X$. ∎

Now we argue Proposition A.8(i):

Lemma A.10**.**

There exists $C:=C(f)>0$ such that the unique solutions $\{u_{\beta}\}_{\beta>0}$ of (A.17) satisfy the following estimate:

[TABLE]

Proof.

Since $u_{\beta}$ is smooth, at the point $x_{0}\in X$ where the maximum of $u_{\beta}-f$ is attained we have $i\partial\bar{\partial}(u_{\beta}-f)\leq 0$. Consequently, (A.17) implies that at $x_{0}$ we have

[TABLE]

Denoting C:=\sup_{X}\log\big{(}\frac{\omega_{f}^{n}}{\omega^{n}}\big{)}, we obtain that $u_{\beta}-\frac{C}{\beta}\leq f$, hence

[TABLE]

Conversely, fix $v\in\mathcal{H}_{\omega}$ and $D>0$ such that $\omega_{v}\geq D\omega$, and $v\leq f$. Also, choose $\delta,\varepsilon\in(0,1)$ such that

[TABLE]

As a result, $u_{\delta,\varepsilon}:=(1-\delta)P(f)+\delta v-\varepsilon\leq f-\varepsilon$ and

[TABLE]

By Lemma A.9 we obtain that $u_{\delta,\varepsilon}\leq u_{\beta}$. Also, for $\beta:=\beta(D)>2$ big enough, the choice $\delta:=1/\beta$ and $\varepsilon:={2n\log\beta}/{\beta}$ satisfies (A.19), and we obtain that $(1-\frac{1}{\beta})P(f)+\frac{1}{\beta}v-\frac{2n\log\beta}{\beta}\leq u_{\beta}$. Putting this together with (A.18), we arive at

[TABLE]

In particular, since $P(f)$ is bounded, this implies that $u_{\beta}$ is uniformly bounded, and the estimate of the lemma follows. ∎

Lastly, we argue Proposition A.8(ii):

Lemma A.11**.**

There exists $C>0$ and $\beta_{0}>0$ such that the unique solutions $\{u_{\beta}\}_{\beta}$ of (A.17) satisfy the following estimate:

[TABLE]

where $C$ only depends on an upper bound for $\Delta^{\omega}f$.

Proof.

The lower bound on $\Delta^{\omega}u_{\beta}$ follows immediately from $\omega+i\partial\bar{\partial}u_{\beta}\geq 0$. To prove the upper bound, we recall a variant of a Laplacian estimate due to Aubin–Yau (provided by Siu in this context [99, page 99], for a survey we refer to [27, Proposition 4.1.2]): if $u\in\mathcal{H}_{\omega}$ satisfies $\omega_{u}^{n}=e^{g}\omega^{n}$, then

[TABLE]

where $B>0$ depends only on the magnitude of the holomorphic bisectional curvature of $\omega$. Applying this estimate to (A.17) we obtain

[TABLE]

Rearranging terms we conclude:

[TABLE]

Denoting $C:=\sup_{X}(2n+\Delta^{\omega}f)>0$, and multiplying this inequality with $\textup{Tr}_{\omega}\omega_{u_{\beta}}e^{-Bu_{\beta}}$, we arrive at:

[TABLE]

Let $s:=\sup_{X}\textup{Tr}_{\omega}\omega_{u_{\beta}}e^{-Bu_{\beta}}$ and suppose that this last supremum is realized at $x_{0}\in X$. By the previous estimate we can write:

[TABLE]

By the previous lemma, $u_{\beta}$ is uniformly bounded, hence for $\beta\geq\beta_{0}:=3nB$ we obtain an upper bound for $s$. Using again that $u_{\beta}$ is uniformly bounded, we conclude that $\Delta^{\omega}u_{\beta}\leq C$ for any $\beta\geq\beta_{0}$. ∎

Notice that $g:=\min(f_{1},f_{2},\ldots,f_{k})=-\max(-f_{1},-f_{2},\ldots,-f_{k})$. From this it follows that $\Delta^{\omega}g$ is uniformly bounded from above by a common upper bound for $\Delta^{\omega}f_{j},\ j\in\{1,\ldots,k\}$. Using the fact that the constant $C$ in the previous lemma only depends on an upper bound for the Laplacian of $f$, we can conclude the following more general result, proved in [56] using different methods:

Theorem A.12**.**

Given $f_{1},...,f_{k}\in C^{\infty}(X)$, then $P(f_{1},f_{2},...,f_{k})\in C^{1,\alpha}(X),\ \alpha\in(0,1)$. More precisely, the following estimate holds:

[TABLE]

A.4 Cartan type decompositions of Lie groups

In the proof of the Bando–Mabuchi uniqueness theorem, some basic facts about Lie groups are needed. This section is based on [57, Section 6.1]. For an extensive study of such groups we refer to [29, 79].

Recall the following form of the classical Cartan decomposition for complexifications of compact semisimple Lie groups (see [29, Proposition 32.1, Remark 31.1]):

Theorem A.13**.**

Let $K$ be a compact connected semisimple Lie group. Denote by $(K^{\mathbb{C}},J)$ the complexification of $K$, namely the unique connected complex Lie group whose Lie algebra is the complexification of $\mathfrak{k}$, the Lie algebra of $K$. Then the map $C:K\times\mathfrak{k}\to K^{\mathbb{C}}$ given by $C(k,X):=k\exp_{I}JX$ is a diffeomorphism.

The following result is a partial extension of the above classical theorem to compact but not necessary semisimple Lie groups. We state the result in a form that will be most useful for our applications in Kähler geometry, albeit it is likely not optimal.

Proposition A.14**.**

Let $K$ be a compact connected subgroup of a connected complex Lie group $(G,J)$ and denote by $\mathfrak{k}$ and $\mathfrak{g}$ their Lie algebras. If $\mathfrak{g}={\mathfrak{k}}\oplus J{\mathfrak{k}}$ then the map $C:K\times{\mathfrak{k}}\to G$ given by $C(k,X)=k\exp_{I}JX$ is surjective.

Proof.

First we note that

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where $\mathfrak{z}(\mathfrak{k})$ is the Lie algebra of $Z(K)$. This follows from [79, Proposition 6.6 (ii), p. 132], as $K$ is compact. Next we note the following identity for the Lie algebra of the center $Z(G)$:

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Since $\mathfrak{z}(\mathfrak{g})$ is complex, we immediately obtain that $\mathfrak{z}(\mathfrak{g})\supset\mathfrak{z}({\mathfrak{k}})\oplus J\mathfrak{z}({\mathfrak{k}})$. For the reverse inclusion, we use that $\mathfrak{g}=\mathfrak{k}\oplus J\mathfrak{k}$. Consequently, for any $X\in\mathfrak{z}(\mathfrak{g})$ we have $X=X_{1}+X_{2}$, with $X_{2}\in{\mathfrak{k}}\cap\mathfrak{z}(\mathfrak{g})=\mathfrak{z}({\mathfrak{k}})$, and $X_{2}\in J{\mathfrak{k}}\cap\mathfrak{z}(\mathfrak{g})=J({\mathfrak{k}}\cap\mathfrak{z}(\mathfrak{g}))=J\mathfrak{z}({\mathfrak{k}})$, since $\mathfrak{z}(\mathfrak{g})$ is complex. This finishes the proof of (A.21).

Next we claim that the map

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given by $(z,X)\mapsto z\exp_{I}JX$ is surjective. Indeed, (A.21) implies that $\dim Z(K)+\dim{\mathfrak{z}(\mathfrak{k})}=\dim Z(G)$ and the differential of $\Theta_{1}$ at $(I,0)$ is invertible (see [79, Proposition 1.6, p. 104]). As we are dealing with abelian groups it follows that $\Theta_{1}$ is a Lie group homomorphism, thus it must be surjective as its image is a connected subgroup of the same dimension as that of $Z(G)$.

Let $L$ denote the connected compact Lie subgroup of $K$ whose Lie algebra is $[{\mathfrak{k}},{\mathfrak{k}}]$ (since the Killing form is negative definite on $[{\mathfrak{k}},{\mathfrak{k}}]$, $L$ is indeed compact). By [79, Proposition 6.6 (i), p. 132], $L$ is semisimple. By Theorem A.13, the map

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given by $\Theta_{2}(l,X)=l\exp_{I}JX,$ is a diffeomorphism, where $L^{\mathbb{C}}$ is the complexification of $L$ inside $G$.

Next we note that the multiplication maps $Z(K)\times L\to K$, $Z(G)\times L^{\mathbb{C}}\to G$ are surjective. By (A.20) the multiplication map $Z(K)\times L\to K$ is a local isomorphism near $(I,I)$ by dimension count. The map is also a group homomorphism since elements of $Z(K)$ commute with elements of $L$. Thus, it is surjective. The same argument works for the multiplication map $Z(G)\times L^{\mathbb{C}}\to G$, with dimension count provided by (A.21).

Now we put all the above ingredients together. Given $k\in K$ and $X\in\mathfrak{k}$, observe that

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for some $z\in Z(K)$ and $l\in L$ such that $k=zl$ (these exist by the surjectivity of the multiplication map $Z(K)\times L\to K$). Now let $X_{1}$ and $X_{2}$ be the unique elements such that $X_{1}\in\mathfrak{z}({\mathfrak{k}})$, $X_{2}\in[{\mathfrak{k}},{\mathfrak{k}}]$, and $X=X_{1}+X_{2}$, given by A.20. Since $\exp_{I}JX_{1}\in Z(G)$ we can write

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Since both $\Theta_{2}$ and $\Theta_{1}$ are surjective, as well as the multiplication map $Z(G)\times L^{\mathbb{C}}\to G$, it follows that $C$ is surjective, concluding the proof. ∎