The isometries of the space of K\"ahler metrics

TL;DR

This paper characterizes all global isometries of the space of K"ahler metrics on a compact K"ahler manifold, showing they are induced by biholomorphisms or anti-biholomorphisms, and explores the non-existence of symmetries in Mabuchi's metric.

Contribution

It proves that all global isometries are induced by biholomorphisms or anti-biholomorphisms, and demonstrates the absence of local symmetries in the Mabuchi metric completion.

Findings

All global isometries are induced by biholomorphisms or anti-biholomorphisms.

No global symmetries exist for Mabuchi's metric.

Constructs examples of geodesic segments that cannot be extended at one end.

Abstract

Given a compact K\"ahler manifold, we prove that all global isometries of the space of K\"ahler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symmetries. Closely related to these findings, we provide a large class of metric geodesic segments that can not be extended at one end, pointing out the first such examples in the literature.

Equations

$${\mathcal{H}}_{\omega }:=\{u\in C^{\infty }(X)\text{ s.t. }\omega +i\partial \bar{\partial }u>0\}.$$

$$⟨\xi ,\zeta {⟩}_v:=\frac{1}{V}{\int }_X\xi \zeta {\omega }_v^n,v\in {\mathcal{H}}_{\omega },\xi ,\zeta \in T_v{\mathcal{H}}_{\omega }\simeq C^{\infty }(X),$$

$${\mathcal{H}}_{\omega }\simeq \frac{{\text{Ham}}_{\omega }^{\mathbb{C}}}{{\text{Ham}}_{\omega }},$$

$${\int }_X|\xi {|}^2{\omega }_v^n={\int }_X|F_{\ast }\xi {|}^2{\omega }_{G(v)}^n,v\in {\mathcal{H}}_{\omega },\xi \in T_v{\mathcal{H}}_{\omega }.$$

$$I(u)=\frac{1}{V(n+1)}\sum _{j=0}^n{\int }_{X}u{\omega }^j\wedge {\omega }_u^{n-j}.$$

$$\|\xi\|_{p,v}=\bigg{(}\frac{1}{V}\int_{X}|\xi|^{p}\omega_{v}^{n}\bigg{)}^{\frac{1}{p}},\ \ v\in\mathcal{H}_{\omega},\ \xi\in T_{v}\mathcal{H}_{\omega}.$$

$$V={\int }_X{\omega }^n=1.$$

$$F_{\ast }(u)\xi =a\xi \circ G_u-b{\int }_X\xi {\omega }_u^n,\xi \in T_u{\mathcal{H}}_{\omega }\simeq C^{\infty }(X),$$

$$F(u)-F(v)={\int }_0^1\frac{d}{dt}F({\gamma }_t)dt={\int }_0^1-(u-v)\circ G_{{\gamma }_t}dt\ge 0.$$

$$\xi\circ G_{u+c}=F_{*}(u+c)\xi=\frac{d}{dt}\bigg{|}_{t=0}F(u+t\xi+c)=\frac{d}{dt}\bigg{|}_{t=0}F(u+t\xi)=F_{*}(u)\xi=\xi\circ G_{u}.$$

$$u_t:=\sup \{v_t|\text{where }t\to v_t\text{ is a subgeodesic}\},t\in (0,1).$$

$$\underset{t\to 0}{\lim }{u}_t(x)=u_0(x)\text{ and }\underset{t\to 1}{\lim }{u}_t(x)=u_1(x)$$

$$\int_{X}\Big{(}\frac{d}{dt}\mathcal{I}(\gamma_{t})\Big{)}^{2}\omega_{\gamma_{t}}^{n}=\int_{X}\Big{(}\dot{\gamma}_{t}-2\int_{X}\dot{\gamma}_{t}\omega_{\gamma_{t}}^{n}\Big{)}^{2}=\int_{X}\dot{\gamma}_{t}^{2}\omega_{\gamma_{t}}^{n},$$

$$\frac{d}{dt}F(\mathcal{I}(\gamma_{t}))=F_{*}(\mathcal{I}_{*}\dot{\gamma}_{t})=F_{*}\Big{(}\dot{\gamma}_{t}-2\int_{X}\dot{\gamma}_{t}\omega_{\gamma_{t}}^{n}\Big{)}.$$

$$L_f(u)=L_f(0)+u\circ F,u\in I^{-1}(0),$$

$$L_f(v)=L_f(v-I(v))+I(v),v\in {\mathcal{H}}_{\omega }.$$

$$N_g(u)=N_g(0)+u\circ g,u\in I^{-1}(0).$$

$$\omega +i\partial \bar{\partial }(N_g(0)+u\circ g)=-g^{\ast }\omega -g^{\ast }i\partial \bar{\partial }u=-g^{\ast }{\omega }_u={\omega }_{N_g(u)}=\omega +i\partial \bar{\partial }{N}_g(u).$$

$$I(N_g(0)+u\circ g)$$

$$=\frac{\pm 1}{n+1}\sum _{j=0}^n{\int }_X(u\circ g)g^{\ast }({\omega }_u^j\wedge {\omega }^{n-j})$$

$$=\frac{\pm 1}{n+1}\sum _{j=0}^n{\int }_{X}u{\omega }_u^j\wedge {\omega }^{n-j}=\pm I(u)=0.$$

$$N_g(v)=N_g(v-I(v))+I(v),v\in {\mathcal{H}}_{\omega }.$$

$$\frac{d}{dt}{N}_g({\gamma }_t)=\frac{d}{dt}({\gamma }_t\circ g-I({\gamma }_t))+\frac{d}{dt}I(\gamma (t))={\dot{\gamma }}_t\circ g.$$

$$\underset{X}{\inf }{\dot{u}}_0=\underset{X}{\inf }(u_1-u_0),\underset{X}{\sup }{\dot{u}}_0=\underset{X}{\sup }(u_1-u_0).$$

$$\underset{X}{\sup }(F(u)-F(\varphi ))=-\underset{X}{\inf }(u-\varphi ).$$

$${\int }_X|{\dot{u}}_0{|}^p{\omega }_{\varphi }^n=d_p(\varphi ,u{)}^p=d_p(F(\varphi ),F(u){)}^p={\int }_X|{\dot{v}}_0{|}^p{\omega }_{F(\varphi )}^n,p\ge 1.$$

$$\underset{X}{\sup }{\dot{v}}_0=-\underset{X}{\inf }{\dot{u}}_0.$$

$${\dot{u}}_0\circ G_{u_0}=\dot{F(u_0)}.$$

$${\dot{u}}_0^{\epsilon }\circ G_{u_0}=\dot{F(u_0^{\epsilon })}\le \dot{F(u_0)}\le 0,\epsilon >0.$$

$${\int }_X({\dot{u}}_0\circ G_{u_0}{)}^2{\omega }_{F(u_0)}^n={\int }_X{\dot{u}}_0^2{\omega }_{u_0}^n=d_2(u_0,u_1{)}^2=d_2(F(u_0),F(u_1){)}^2={\int }_X\dot{F(u_0)}{}^2{\omega }_{F(u_0)}^n.$$

$$\underset{X}{\inf }(F(u)-F(v))=\underset{X}{\inf }(u-v).$$

Full text

1 The main results

Let $(X,\omega)$ be a compact connected Kähler manifold. Given a Kähler metric $\omega^{\prime}$ cohomologuos to $\omega$, by the $\partial\bar{\partial}$-lemma of Hodge theory there exists $u\in C^{\infty}(X)$ such that $\omega^{\prime}:=\omega+i\partial\bar{\partial}u.$ Such a metric $\omega^{\prime}$ is said to belong to the space of Kähler metrics $\mathcal{H}$. By the above, up to a constant, one can identify $\mathcal{H}$ with the space of Kähler potentials:

[TABLE]

This space can be endowed with a natural infinite dimensional $L^{2}$ type Riemannian metric [24, 26, 17]:

[TABLE]

where $V=\int_{X}\omega^{n}$. Additionally, Donaldson and Semmes pointed out that $(\mathcal{H}_{\omega},\langle\cdot,\cdot\rangle)$ can be thought of as a formal symmetric space [27, 17]:

[TABLE]

where $\textup{Ham}_{\omega}$ is the group of Hamiltonian symplectomorphisms of $\omega$, and $\textup{Ham}_{\omega}^{\mathbb{C}}$ is its formal complexification. Though not quite precise, the underlying heuristic of (2) led to many advances in the understanding of the geometry of $\mathcal{H}_{\omega}$, as well as the formulation of stability conditions aiming to characterize existence of canonical metrics (for an exposition see [28]).

Global $L^{2}$ isometries and symmetries of $\mathcal{H}_{\omega}$.

For finite dimensional Riemannian manifolds, the existence of a symmetric structure arising as a quotient of Lie groups, as in (2), is equivalent with existence of global symmetries at all points of the manifold [19]. Such maps are global involutive isometries reversing geodesics at a specific point. If such symmetries existed for $(\mathcal{H}_{\omega},\langle\cdot,\cdot\rangle)$ it would perhaps allow to make a precise sense of (2).

Recently a large class of local symmetries of $\mathcal{H}_{\omega}$ were constructed in [2], via complex Legendre transforms, that also found applications to interpolation of norms [3]. Moreover, it was shown in [21] that all local symmetries of $\mathcal{H}_{\omega}$ arise from the construction of [2]. Below we show that global symmetries actually do not exist, in particular these local symmetries can not be extended to $\mathcal{H}_{\omega}$. This will follow from our characterization of the isometry group of $(\mathcal{H}_{\omega},\langle\cdot,\cdot\rangle)$.

First we recall some terminology. Let $\mathcal{U},\mathcal{V}\subset\mathcal{H}_{\omega}$ be open sets. We say that a map $F:\mathcal{U}\to\mathcal{V}$ is $C^{1}$, or (with slight abuse of terminology) differentiable, if $(F,F_{*}):\mathcal{U}\times C^{\infty}(X)\to\mathcal{V}\times C^{\infty}(X)$ is continuous as a map of Fréchet spaces. Here $F_{*}$ is the differential of $F$ (see [22, p. 3] and references therein for more details). Moreover, $F:\mathcal{U}\to\mathcal{U}$ is a differentiable $L^{2}$ symmetry at $\phi\in\mathcal{U}$ if $F^{2}=Id$, $F(\phi)=\phi$, $F_{*}|_{\phi}=-Id$ and

[TABLE]

If $F:\mathcal{U}\to\mathcal{V}$ is $C^{1}$, satisfies (3) and it is bijective, then it is called a differentiable $L^{2}$ isometry. Due to infinite dimensionality, it is not yet known if differentiable $L^{2}$ isometries are automatically smooth [20], hence the isometries we consider in this work are possibly more general than the ones in [2, 21].

A small class of global $L^{2}$ isometries has been previously known in the literature [20, p. 16]. One of them is the so called Monge–Ampère flip $\mathcal{I}:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$, and is defined by the formula $\mathcal{I}(u)=u-2I(u)$, where $I:\mathcal{H}_{\omega}\to\mathbb{R}$ is the Monge–Ampère energy:

[TABLE]

The map $\mathcal{I}$ is involutive and its name is inspired by the fact that it flips the sign of $I$. Indeed, $I(\mathcal{I}(u))=-I(u)$.

We say that a biholomorphism $f:X\to X$ preserves the Kähler class $[\omega]$ if $[f^{*}\omega]=[\omega]$. Similarly, an anti-biholomorphism $g:X\to X$ flips the the Kähler class $[\omega]$ if $[g^{*}\omega]=-[\omega]$. Such maps also induce a class of global $L^{2}$ isometries, and we refer to Section 2.3 for the detailed construction.

In our first main result we point out that these maps and their compositions are the only global differentiable $L^{2}$ isometries:

Theorem 1.1**.**

*Let $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ be a differentiable $L^{2}$ isometry. Then exactly one of the following holds:

(i) $F$ is induced by a biholomorphism or anti-biholomorphism $f:X\to X$ that preserves or flips $[\omega]$, respectively.

(ii) $F\circ\mathcal{I}$ is induced by a biholomorphism or anti-biholomorphism $f:X\to X$ that preserves or flips $[\omega]$, respectively.*

The space of potentials $\mathcal{H}_{\omega}$ admits a Riemannian splitting $\mathcal{H}_{\omega}=\mathcal{H}\oplus\mathbb{R}$, via the Monge–Ampère energy $I$. As the fixed point set of $\mathcal{I}$ is exactly $\mathcal{H}=I^{-1}(0)$, we obtain the following corollary regarding isometries of $\mathcal{H}$:

Corollary 1.2**.**

Let $F:\mathcal{H}\to\mathcal{H}$ be a differentiable $L^{2}$ isometry. Then $F$ is induced by a biholomorphism or anti-biholomorphism $f:X\to X$ that preserves or flips $[\omega]$, respectively.

The above results answer explicitly questions raised by Lempert regarding the extension property of local isometries [20, p. 3], though questions surrouding the isometry group of $(\mathcal{H}_{\omega},\langle\cdot,\cdot\rangle)$ go back to early work of Semmes [26, 27].

Lastly, via the classification theorem of Lempert (recalled in Theorem 2.1), we will see that neither of the maps in the statement of Theorem 1.1 are symmetries, immediately giving the following non-existence result for differentiable $L^{2}$ symmetries:

Corollary 1.3**.**

There exists no differentiable $L^{2}$ symmetry $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ at any $\phi\in\mathcal{H}_{\omega}$.

Non-existence of local $L^{2}$ symmetries on the completions.

It was shown in [7] that (1) induces a path length metric space $(\mathcal{H}_{\omega},d_{2})$. By $(\mathcal{E}^{2}_{\omega},d_{2})$ we denote the $d_{2}$-metric completion of this space, that can identified with a class of finite energy potentials [11].

Using density, any differentiable $L^{2}$ isometry $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ extends to a unique metric $d_{2}$-isometry $F:\mathcal{E}^{2}_{\omega}\to\mathcal{E}^{2}_{\omega}$. The proof of Theorem 1.1 consists of showing that contradictions arise in this extension process, unless $F$ is very special. With this and the above results in mind, one may hope that the isometry group of the metric space $(\mathcal{E}^{2}_{\omega},d_{2})$ could possibly admit elements beyond the ones that arise from the global differentiable $L^{2}$ isometries of $\mathcal{H}_{\omega}$. Though this may be true, we point out below that even local symmetries fail to exist in the context of the completion, further elaborating on phenomenon related to Corollary 1.3.

Before stating our result, we recall some facts about the $d_{2}$-geodesics of $\mathcal{E}^{2}$. For more details we refer to Section 2.2 and the recent survey [13]. Let $\mathcal{V}\subset\mathcal{E}^{2}_{\omega}$ be $d_{2}$-open with $\phi\in\mathcal{V}\cap\mathcal{H}_{\omega}$. Given a $d_{2}$-geodesic $[0,1]\ni t\to\phi_{t}\in\mathcal{V}$ with $\phi_{0}=\phi$, since $t\to\phi_{t}(x)$ is $t$-convex for almost every $x\in X$, it is possible to introduce $\dot{\phi}_{0}=\frac{d}{dt}|_{t=0}\phi_{t}$. Moreover, due to [12, Theorem 2], it follows that $\dot{\phi}_{0}\in L^{2}(\omega_{\phi}^{n}).$

Let $G:\mathcal{V}\to G(\mathcal{V})\subset\mathcal{E}^{2}_{\omega}$ be an $L^{2}$ isometry, i.e, a bijective map satisfying $d_{2}(v_{1},v_{2})=d_{2}(G(v_{1}),G(v_{2})),\ v_{1},v_{2}\in\mathcal{V}$. It is clear that in this case $t\to G(\phi_{t})$ is also a $d_{2}$-geodesic. Furthermore, we say that $G$ is a metric $L^{2}$ symmetry at $\phi$ if $G^{2}=Id$, $G(\phi)=\phi$ and $\dot{G(\phi_{0})}=-\dot{\phi}_{0},$ i.e., $G$ “reverses” $d_{2}$-geodesics at $\phi$.

Unfortunately, metric $L^{2}$ symmetries actually do not exist, implying that the analog of [2, Theorem 1.2] does not hold in the context of the metric completion, answering questions of Berndtsson and Rubinstein [25]:

Theorem 1.4**.**

Let $\mathcal{V}\subset\mathcal{E}^{2}_{\omega}$ be a $d_{2}$-open set and $\phi\in\mathcal{V}\cap\mathcal{H}_{\omega}$. There exists no metric $L^{2}$ symmetry $F:\mathcal{V}\to\mathcal{V}$ at $\phi$.

Given that $(\mathcal{E}^{2}_{\omega},d_{2})$ is CAT(0), the group of isometries of this metric space has special structure [6], as pointed by B. McReynolds during the Ph.D. thesis defense of the author. In light of the above result, we expect that the group of metric isometries can be characterized as in Theorem 1.1, though this remains an open question.

The extension property of geodesic segments.

As an intermediate step in the proof of Theorem 1.4 we show that a large class of $d_{2}$-geodesic segments inside $\mathcal{E}^{2}_{\omega}$ can not be extended at one of the endpoints. Previously no such examples were known.

Theorem 1.5**.**

Let $\phi_{0}\in\mathcal{H}_{\omega}$ and $\phi_{1}\in\mathcal{E}^{2}_{\omega}\setminus L^{\infty}$. Then the $d_{2}$-geodesic $t\to\psi_{t}$ connecting these potentials can not be extended to a $d_{2}$-geodesic $(-\varepsilon,1]\ni t\to\phi_{t}\in\mathcal{E}^{2}_{\omega}$ for any $\varepsilon>0$.

For finite dimensional manifolds, topological and geodesical completeness are equivalent due to the classical Hopf–Rinow theorem. According to the above result, this is not the case for the completion $(\mathcal{E}^{2}_{\omega},d_{2})$, despite the fact that this space it is non-positively curved [8, 11].

It will be interesting to see if a similar property holds for the $C^{1,1}$-geodesics of Chen and Chu–Tosatti–Weinkove, joining the potentials of $\mathcal{H}_{\omega}$ [7, 9].

Relation to the $L^{p}$ geometry of $\mathcal{H}_{\omega}$.

In [12] the author introduced a family of $L^{p}$ Finsler metrics on $\mathcal{H}_{\omega}$ for any $p\geq 1$, generalizing (1):

[TABLE]

These induce path length metric spaces $(\mathcal{H}_{\omega},d_{p})$, and in [12] the author computed the corresponding metric completions, that later found applications to existence of canonical metrics (for a survey see [13]). Though this more general context lacks the symmetric space interpretation, all of our above results can be considered in the $L^{p}$ setting as well.

As the reader will be able to deduce from our arguments below, the $L^{p}$ version of Theorem 1.4 holds for any $p>1$. Our proof does not work when $p=1$, since the class of finite energy geodesics may not be stable under isometries in this case (see [14, Theorem 1.2]). On the other hand, the $L^{p}$ version of Theorem 1.5 does hold for all $p\geq 1$. Lastly, our argument for Theorem 1.1 would most likely go through in the $L^{p}$ context in case one could obtain the analog of Theorem 2.1 for differentiable $L^{p}$ isometries.

Acknowledgements.

We thank L. Lempert for extensive feedback on our manuscript, and for generously explaining to us details about his paper [20]. We also thank B. Berndtsson and Y.A. Rubinstein for suggestions on how to improve the presentation. This work was partially supported by NSF grant DMS 1610202.

2 Preliminaries

For simplicity we assume throughout the paper the the Kähler metric $\omega$ satisfies the following volume normalization:

[TABLE]

Using a dilation of $\omega$ this can always be achieved and does not represent loss of generality.

2.1 The classification theorem of Lempert

In this short section we recall the particulars of a result due to Lempert on the classification of local $C^{1}$ isometries on $\mathcal{H}_{\omega}$ ([20, Theorem 1.1]), tailored to our global setting:

Theorem 2.1**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry. Then for $u\in\mathcal{H}_{\omega}$ there exists a unique $C^{\infty}$ diffeomorphism $G_{u}:X\to X$ such that $G_{u}^{*}\omega_{u}=\pm\omega_{F(u)}$ and

[TABLE]

where $a=1$, or $a=-1$, or $b=0$, or $b=2a$.

In the particular case of the (local) $L^{2}$ symmetries constructed in [2], formula (4) is a consequence of [2, Theorem 5.1, Theorem 6.1, Proposition 7.1] with $a=-1$ and $b=0$.

Remark 2.2**.**

It follows from the proof of [20, Theorem 1.1] that the integers $a$ and $b$ in the statement depend continuously on $u\in\mathcal{H}_{\omega}$ (as does $G_{u}$), hence in our case they are independent of $u$, as $\mathcal{H}_{\omega}$ is connected. This was pointed out to us by L. Lempert [23].

From the classification theorem we obtain the following simple monotonicity result:

Proposition 2.3**.**

*Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $b=0$. Let $c\in\mathbb{R}$ and $u,v\in\mathcal{H}_{\omega}$ with $u\leq v$. Then the following hold:

(i) if $a=1$ then $F(u)\leq F(v)$ and $F(u+c)=F(u)+c$.

(ii) if $a=-1$ then $F(u)\geq F(v)$ and $F(u+c)=F(u)-c$.*

Proof.

We only address (ii), as the proof of (i) is analogous. Let $[0,1]\ni t\to\gamma_{t}:=v+t(u-v)\in\mathcal{H}_{\omega}$. Then $t\to F(\gamma_{t})$ is a $C^{1}$ curve connecting $F(v)$ and $F(u)$. Moreover, Theorem 2.1 implies that

[TABLE]

The fact that $F(u+c)=F(u)-c$, follows after another application of Theorem 2.1 to the curve $[0,1]\ni t\to\eta_{t}:=u+tc\in\mathcal{H}_{\omega}$. ∎

Corollary 2.4**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $b=0$. Then, in the language of Theorem 2.1 applied to $F$, we have that $G_{u+c}=G_{u}$ for all $u\in\mathcal{H}_{\omega}$ and $c\in\mathbb{R}$.

Proof.

We only address the case $a=1$, as the argument for $a=-1$ is identical. Let $\xi\in C^{\infty}(X)$. By Proposition 2.3(i) and Theorem 2.1 we have that

[TABLE]

Since $\xi\in C^{\infty}(X)$ is arbitrary, we obtain that $G_{u+c}=G_{u}$. ∎

2.2 The complete metric space $(\mathcal{E}^{2}_{\omega},d_{2})$

In this short subsection we recall aspects from the work of the author related to the metric completion of $(\mathcal{H}_{\omega},d_{2})$. For details we refer to the survey [13].

As conjectured by V. Guedj [18], $\overline{(\mathcal{H}_{\omega},d_{2})}$ can be identified with $(\mathcal{E}^{2}_{\omega},d_{2})$, where $\mathcal{E}^{2}_{\omega}\subset\textup{PSH}(X,\omega)$ is an appropriate subset of $\omega$-plurisubharmonic potentials [11, Theorem 1]. Moreover, $(\mathcal{E}^{2}_{\omega},d_{2})$ is a non-positively curved complete metric space, whose points can be joined by unique $d_{2}$-geodesics.

Given $u_{0},u_{1}\in\mathcal{E}^{2}_{\omega}$, the unique $d_{2}$-geodesic $[0,1]\ni t\to u_{t}\in\mathcal{E}^{2}_{\omega}$ connecting these points has special properties. To start, we recall that this curve arises as the following envelope:

[TABLE]

Here a subgeodesic $(0,1)\ni t\to v_{t}\in\textup{PSH}(X,\omega)$ is a curve satisfying $\limsup_{t\to 0,1}v_{t}\leq u_{0,1}$ and $u(s,x):=u_{\textup{Re }s}(x)\in\textup{PSH}(S\times X,\omega)$, where $S=\{0<\textup{Re }s<1\}\subset\mathbb{C}$.

It follows from (5) that $t\to u_{t}(x),t\in(0,1)$ is convex for all $x\in X$ away from a set of measure zero. On the complement we have that $u_{t}(x)=-\infty,\ t\in(0,1)$. Moreover, due to [11, Corollary 7], we also have that

[TABLE]

for all $x\in X$ away from a set of measure zero. In the particular case when $u_{0},u_{1}\in\mathcal{H}_{\omega}$, the curve $t\to u_{t}$ is $C^{1,1}$ on $[0,1]\times X$ [7, 4, 9].

By $\mathcal{C}_{\omega}$ we denote the set of continuous potentials in $\textup{PSH}(X,\omega)$. As pointed out previously, a differentiable $L^{2}$ isometry $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ induces a unique $d_{2}$-isometry $F:\mathcal{E}^{2}_{\omega}\to\mathcal{E}^{2}_{\omega}$, extending the original map (using density). Going forward, we do not distinguish $F$ from its unique extension. Moreover, if $F$ is an isometry with $b=0$ (see Theorem 2.1), we point out that $\mathcal{C}_{\omega}$ is stable under the extension:

Proposition 2.5**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $b=0$. Then $F(\mathcal{C}_{\omega})\subset\mathcal{C}_{\omega}$. More importantly, $\sup_{X}\|u_{j}-u\|\to 0$ implies $\sup_{X}\|F(u_{j})-F(u)\|\to 0$ for any $u_{j},u\in\mathcal{C}_{\omega}$.

Proof.

We only argue the case when $a=1$, as the proof is analogous in case $a=-1$. Since $d_{2}$-convergence implies pointwise a.e. convergence (see [12, Theorem 5]), Proposition 2.3(i) holds for the extension $F:\mathcal{E}^{2}_{\omega}\to\mathcal{E}^{2}_{\omega}$ and $u,v\in\mathcal{E}^{2}_{\omega}$ satisfying $u\leq v$.

Let $u\in\mathcal{C}_{\omega}$. Then [5] implies existence of $u_{k}\in\mathcal{H}_{\omega}$ such that $u_{k}\searrow u$. In fact, due to Dini’s lemma, the convergence is uniform. From Proposition 2.3 it follows that $\{F(u_{k})\}_{k}\subset\mathcal{H}_{\omega}$ is monotone decreasing. Due to uniform convergence, we have that for any $\varepsilon>0$ there exists $k_{0}$ such that $u\leq u_{k}\leq u+\varepsilon$ for $k\geq k_{0}$. Then Proposition 2.3 implies that $F(u)\leq F(u_{k})\leq F(u)+\varepsilon,\ k\geq k_{0}$. This gives that $F(u_{k})$ converges to $F(u)$ uniformly, in particular $F(u)\in\mathcal{C}_{\omega}$.

Lastly, we can essentially repeat the above argument for continuous potentials $u_{j}$ converging uniformly to $u$, concluding the last statement of the proposition. ∎

2.3 Examples of differentiable $L^{2}$ isometries on $\mathcal{H}_{\omega}$

In this short subsection we describe three examples of global differentiable $L^{2}$ isometries on $\mathcal{H}_{\omega}$. Later we will argue that in fact all isometries arise as compositions of these examples.

$\bullet$ First we take a closer look at the Monge–Ampère flip $\mathcal{I}:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$, defined in Section 1, perhaps first introduced in [20]. Let $[0,1]\ni t\to\gamma_{t}\in\mathcal{H}_{\omega}$ be a smooth curve. Since $\frac{d}{dt}I(\gamma_{t})=\int_{X}\dot{\gamma}_{t}\omega_{\gamma_{t}}^{n}$, we obtain that

[TABLE]

hence $\mathcal{I}$ is indeed an involutive $L^{2}$ isometry, with $a=1$ and $b=2$ (see Theorem 2.1). This simple map has the following intriguing property, that will help to adjust the $b$ parameter of arbitrary isometries without changing the $a$ parameter:

Lemma 2.6**.**

*Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry. The $a$ parameter of $F$ and $F\circ\mathcal{I}$ is always the same. Regarding the $b$ parameter the following hold:

(i) If $b=0$ for $F$, then $b=2a$ for $F\circ\mathcal{I}$.

(ii) If $b=2a$ for $F$, then $b=0$ for $F\circ\mathcal{I}$.*

Proof.

Let $[0,1]\ni t\to\gamma_{t}\in\mathcal{H}_{\omega}$ be a smooth curve. Then we have that

[TABLE]

If $a=1$ and $b=0$ for $F$, then we get that $\frac{d}{dt}F(\mathcal{I}(\gamma_{t}))=\dot{\gamma}_{t}\circ G_{u}-2\int_{X}\dot{\gamma}_{t}\omega_{\gamma_{t}}^{n}$. If $a=-1$ and $b=0$ for $F$, then $\frac{d}{dt}F(\mathcal{I}(\gamma_{t}))=-\dot{\gamma}_{t}\circ G_{u}+2\int_{X}\dot{\gamma}_{t}\omega_{\gamma_{t}}^{n}$, addressing (i).

In case $a=1$ and $b=2a$ for $F$, then $\frac{d}{dt}F(\mathcal{I}(\gamma_{t}))=\dot{\gamma}_{t}\circ G_{u}$. Similarly, if $a=-1$ and $b=2a$ for $F$, then $\frac{d}{dt}F(\mathcal{I}(\gamma_{t}))=-\dot{\gamma}_{t}\circ G_{u}$, addressing (ii). ∎

$\bullet$ Now let $f:X\to X$ be a biholomorphism preserving the Kähler class $[\omega]$. Then $f$ induces a map $L_{f}:\mathcal{H}\to\mathcal{H}$ via pullbacks: $\omega_{L_{f}(u)}:=f^{*}\omega_{u}$, where we made the identification $\mathcal{H}\simeq I^{-1}(0)$. Using this identification it is possible to describe the action of $F$ on the level of potentials in the following manner [15, Lemma 5.8]:

[TABLE]

where $0\in I^{-1}(0)$ is simply the zero Kähler potential. More importantly, $L_{f}$ further extends to a map $L_{f}:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ in the following manner:

[TABLE]

It is well known that $L_{f}$ thus described gives a differentiable $L^{2}$ isometry of $\mathcal{H}_{\omega}$ with $a=1$ and $b=0$. Actually, using the language of Theorem 2.1 applied to $L_{f}$, we obtain that $G_{u}=f$ for all $u\in\mathcal{H}_{\omega}$. We leave the related simple computation to the reader.

$\bullet$ Now let $g:X\to X$ be an anti-biholomorphism that flips the Kähler class $[\omega]$. By definition, such a map is a diffeomorphism satisfying $\frac{\partial g_{j}}{\partial z_{k}}=0$ for all $j,k\in\{1,\ldots,n\}$ in any choice of local coordinates. For example, the map $g(z)=\bar{z}$ is an anti-biholomorphism of the unit torus $\mathbb{C}/\mathbb{Z}[i]$ that flips that class of the flat Kähler metric.

Such a map $g$ induces another map $N_{g}:\mathcal{H}\to\mathcal{H}$ via pullbacks: $\omega_{N_{g}(u)}:=-g^{*}\omega_{u}$. Here we used again the identification $\mathcal{H}\simeq I^{-1}(0)$. Similar to (7), it is possible to describe the action of $N_{g}$ on the level of potentials in the following manner:

[TABLE]

To show this, we have to go through the proof of [15, Lemma 5.8] in the anti-holomorphic context. As a beginning remark, we notice that $g^{*}\partial\bar{\partial}v=-\partial\bar{\partial}v\circ g$ for all smooth functions $v$. With this in mind, we have that

[TABLE]

In particular, $N_{g}(0)+u\circ g-N_{g}(u)$ is a constant. To show that this constant is equal to zero, we only need to argue that $I(N_{g}(0)+u\circ g)=0=I(N_{g}(u))$. But this holds because of the following computation:

[TABLE]

As above, $N_{g}$ extends to a map $N_{g}:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ in the following manner:

[TABLE]

We point out that $N_{g}$ thus described gives a differentiable $L^{2}$ isometry of $\mathcal{H}_{\omega}$ with $a=1$ and $b=0$. To see this, let $[0,1]\ni\to\gamma_{t}\in\mathcal{H}_{\omega}$ be a smooth curve. Using (8) we can write the following

[TABLE]

In the language of Theorem 2.1 applied to $N_{g}$, we actually obtained that $G_{u}=g$ for all $u\in\mathcal{H}_{\omega}$.

3 Proof of Theorem 1.1

The argument of Theorem 1.1 is split into two parts. First we show that there exist no global differentiable isometries with $a=-1$. Later we will classify all global differentiable isometries with $a=1$.

Before we go into specific details, we recall the following simple lemma that will be used numerous times in our arguments:

Lemma 3.1**.**

[10, Lemma 3.1]* Suppose that $u_{0},u_{1}\in\mathcal{C}_{\omega}$ and $[0,1]\ni t\to u_{t}\in\mathcal{E}^{2}_{\omega}$ is the $d_{2}$-geodesic connecting these potentials. Then we have that*

[TABLE]

Proof.

First we argue that $\inf_{X}\dot{u}_{0}=\inf_{X}(u_{1}-u_{0})$. From (5) we obtain the estimate $u_{t}\geq u_{0}+t\inf_{X}(u_{1}-u_{0}),\ t\in[0,1]$. In particular, $\dot{u}_{0}\geq\inf_{X}(u_{1}-u_{0})$. Using $t$-convexity it follows that $u_{t}(y)=u_{0}(y)+t\inf_{X}(u_{1}-u_{0})$ for $y\in X$ such that $u_{1}(y)-u_{0}(y)=\inf_{X}(u_{1}-u_{0})$. This implies that $t\to u_{t}(y)$ is linear, implying that $\inf_{X}\dot{u}_{0}=\inf_{X}(u_{1}-u_{0})$.

For the second identity, we notice that $t$-convexity implies $\sup_{X}\dot{u}_{0}\leq\sup_{X}(u_{1}-u_{0})$. In addition, (5) implies that $u_{1}-(1-t)\sup_{X}(u_{1}-u_{0})\leq u_{t},\ t\in[0,1]$. Relying on $t$-convexity again, we obtain that $\dot{u}_{0}(z)=u_{1}(z)-u_{0}(z)=\sup_{X}(u_{1}-u_{0})$, for $z\in X$ with $u_{1}(z)-u_{0}(z)=\sup_{X}(u_{1}-u_{0})$. Summarizing, we obtain that $\sup_{X}\dot{u}_{0}=\sup_{X}(u_{1}-u_{0})$, as desired. ∎

3.1 Isometries with $a=-1$

We start with a lemma:

Lemma 3.2**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $a=-1$ and $b=0$. Let $\phi\in\mathcal{H}_{\omega}$ and $u\in\mathcal{H}_{\omega}$ with $u\leq\phi$. Then we have that $F(u)\geq F(\phi)$ and

[TABLE]

Proof.

That $F(u)\geq F(\phi)$ follows from Proposition 2.3(ii). As it is pointed out on [20, p.2], Theorem 2.1 implies that $F$ is a $d_{p}$-isometry for any $p\geq 1$. This implies that $d_{p}(\phi,u)=d_{p}(F(\phi),F(u))$ for any $p\geq 1$.

Let $[0,1]\ni t\to u_{t},v_{t}\in\mathcal{H}^{1,1}_{\omega}$ be the $C^{1,1}$ geodesic connecting $u_{0}:=\phi,u_{1}:=u$, respectively $v_{0}:=F(\phi)$ and $v_{1}:=F(u)$. By the comparison principle for weak geodesics (see for example [4, Proposition 2.2]) it follows that $v_{t}\geq F(\phi)$ and $u_{t}\leq\phi$ for any $t\in[0,1]$. In particular, $\dot{v}_{0}\geq 0$ and $\dot{u}_{0}\leq 0$.

Using [12, Theorem 1] we arrive at:

[TABLE]

Raising to the $\frac{1}{p}$-power, and letting $p\to\infty$ gives that

[TABLE]

From Lemma 3.1 we get that $\inf_{X}\dot{u}_{0}=\inf_{X}(u-\phi)$ and $\sup_{X}\dot{v}_{0}=\sup_{X}(F(u)-F(\phi))$. Putting this together with (10), we obtain (9), as desired. ∎

Theorem 3.3**.**

There exists no differentiable $L^{2}$ isometry $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ with $a=-1$.

We note that this result already implies Corollary 1.3.

Proof.

Due to Lemma 2.6, after possibly composing $F$ with $\mathcal{I}$, we only need to worry about the case $a=-1$ and $b=0$.

Since $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$-isometry, it is also a $d_{2}$-isometry, hence it extends to a unique $d_{2}$-isometry $F:\mathcal{E}^{2}_{\omega}\to\mathcal{E}^{2}_{\omega}$.

Let $\phi\in\mathcal{H}_{\omega}$. Let $u\in\mathcal{E}^{2}_{\omega}\setminus L^{\infty}$ with $u\leq\phi-1$, and we choose $u_{k}\in\mathcal{H}_{\omega}$ such that $u_{k}\searrow u$ and $u_{k}\leq\phi$. Such a sequence can always be found [5].

Due to our choice of $u$ we have that $\inf_{X}(u_{k}-\phi)\searrow-\infty$. From Lemma 3.2 it follows that $\sup_{X}F(u_{k})=\sup_{X}(F(u_{k})-F(\phi))\nearrow+\infty$. Since $F$ is a $d_{2}$-isometry, we have that $d_{2}(F(u_{k}),F(u))=d_{2}(u,u_{k})\to 0$. However [12, Theorem 5(i)] gives that $\sup_{X}F(u_{k})\to\sup_{X}F(u)<+\infty$, which is a contradiction. ∎

3.2 Isometries with $a=1$

To start, we point out an important relationship between $d_{2}$-geodesics and differentiable $L^{2}$ isometries with $a=1$ and $b=0$:

Proposition 3.4**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $a=1$ and $b=0.$ Let $[0,1]\ni t\to u_{t}\in\mathcal{E}^{2}_{\omega}$ be the $d_{2}$-geodesic connecting $u_{0}\in\mathcal{H}_{\omega}$ and $u_{1}\in\mathcal{C}_{\omega}$. Then

[TABLE]

Here and below \dot{u}_{0}:=\frac{d}{dt}\big{|}_{t=0}F(u_{t}) and \dot{F(u_{0})}:=\frac{d}{dt}\big{|}_{t=0}F(u_{t}) are the initial tangent vectors of the $d_{2}$-geodesics $t\to u_{t}$ and $t\to F(u_{t})$, interpreted according to the discussion preceding Theorem 1.4.

Proof.

There exists a constant $c\in\mathbb{R}$ such that $u_{0}>u_{1}+c$. Since $F(u_{t}+tc)=F(u_{t})+tc$ (Proposition 2.3(i)), we can assume without loss of generality that $u_{0}>u_{1}$.

First, we show (11) in case $u_{1}\in\mathcal{H}_{\omega}$. Let $[0,1]\ni u^{\varepsilon}_{t}\in\mathcal{H}_{\omega}$ be the smooth $\varepsilon$-geodesics of X.X. Chen, connecting $u_{0}$ and $u_{1}$ [7]. It is well known that $u^{\varepsilon}_{t}\nearrow u_{t}$ as $\varepsilon\to 0$, where $t\to u_{t}$ is the $C^{1,1}$-geodesic joining $u_{0}$ and $u_{1}$. Due to Proposition 2.3 and Proposition 2.5, for the curves $t\to F(u^{\varepsilon}_{t}),F(u_{t})$ we obtain that $F(u^{\varepsilon}_{t})\nearrow F(u_{t})$. Since $t\to F(u^{\varepsilon}_{t})$ is a $C^{1}$ curve, via Theorem 2.1, we obtain that

[TABLE]

Taking the limit $\varepsilon\to 0$, since $u^{\varepsilon}\to_{C^{1,\alpha}}u$, we arrive at $\dot{u}_{0}\circ G_{u_{0}}\leq\dot{F(u_{0})}\leq 0$. By Theorem 2.1 we have that $G_{u_{0}}^{*}\omega_{u_{0}}^{n}=\pm\omega_{F(u_{0})}^{n}$. Using this and [7] (see also [12, Theorem 1]) we obtain that

[TABLE]

Due to continuity we conclude that $\dot{u}_{0}\circ G_{u_{0}}=\dot{F(u_{0})}$, as desired.

Now we treat the general case. Let $u^{k}_{1}\in\mathcal{H}_{\omega},\ k\in\mathbb{N}$ such that $u_{0}>u^{k}_{1}$ and $u^{k}_{1}\searrow u_{1}\in\mathcal{C}_{\omega}$. Also, by $[0,1]\ni t\to u_{t},u^{k}_{t}\in\mathcal{E}^{2}_{\omega}$ we denote the $d_{2}$-geodesics connecting $u_{0}$ and $u_{1}$, respectively $u_{0}$ and $u^{k}_{1}$. Since $F$ is a $d_{2}$-isometry, we obtain that $[0,1]\ni t\to F(u_{t}),F(u^{k}_{t})\in\mathcal{E}^{2}_{\omega}$ are the $d_{2}$-geodesics connecting $F(u_{0})$ and $F(u_{1})$, respectively $F(u_{0})$ and $F(u^{k}_{1})$. Due to $t$-convexity, $k$-monotonicity and Proposition 2.3, we obtain that $\dot{u}_{0}^{k}\searrow\dot{u}_{0}$ and $\dot{F(u_{0}^{k})}\searrow\dot{F(u_{0})}$. Letting $k\to\infty$ we arrive at the desired conclusion: $\dot{u}_{0}\circ G_{u_{0}}=\lim_{k}(\dot{u}_{0}^{k}\circ G_{u_{0}})=\lim_{k}\dot{F(u_{0}^{k})}=\dot{F(u_{0})}$. ∎

This result together with Lemma 3.1 gives the following corollary, paralleling Lemma 3.2:

Corollary 3.5**.**

Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $a=1$ and $b=0$. Suppose that $u,v\in\mathcal{C}_{\omega}$. Then we have that $F(u),F(v)\in\mathcal{C}_{\omega}$ and

[TABLE]

By the switching the role of $u$ and $v$, we obtain that the above identity holds for the suprema as well.

Proof.

That $F(u),F(v)\in\mathcal{C}_{\omega}$, follows from Proposition 2.5. First we deal with the case when $u,v\in\mathcal{H}_{\omega}$. If $[0,1]\ni t\to h_{t}\in\mathcal{H}_{\omega}$ is the $C^{1,1}$-geodesic connecting $h_{0}:=u$ and $h_{1}:=v$, then Lemma 3.1 gives that

[TABLE]

Putting this together with (11), we obtain that $\inf_{X}(v-u)=\inf_{X}(F(v)-F(u))$, as desired.

When $u,v\in\mathcal{C}_{\omega}$, by [5] one can find $u^{k},v^{k}\in\mathcal{H}_{\omega}$ such that $\sup_{X}|u^{k}-u|\to 0$ and $\sup_{X}|v^{k}-v|\to 0$. Then Proposition 2.5 implies that $\sup_{X}|F(u^{k})-F(u)|\to 0$ and $\sup_{X}|F(v^{k})-F(v)|\to 0$.

By uniform convergence we have $\inf_{X}(u^{k}-v^{k})\to\inf_{X}(u-v)$ and $\inf_{X}(F(u^{k})-F(v^{k}))\to\inf_{X}(F(u)-F(v))$. The conclusion follows after taking the $k$-limit of $\inf_{X}(u^{k}-v^{k})=\inf_{X}(F(u^{k})-F(v^{k}))$. ∎

To continue, we need an an auxiliary construction. Fixing $x\in X$ and a small enough coordinate neighborhood $O_{x}\subset X$, we can find a function $\rho_{x}\in C^{\infty}(X)$ such that $\rho_{x}(y)=e^{\frac{-1}{\|y-x\|^{2}}}$ for all $y\in O_{x}$, and there exists $\beta>0$ such that $\beta\leq\rho_{x}(y)\leq 1$ for all $y\in X\setminus O_{x}$.

Proposition 3.6**.**

For $u\in\mathcal{H}_{\omega}$ and $x\in X$ there exists $\delta>0$ such that $[0,1]\ni t\to u_{t}:=u+\delta(t+\frac{t^{2}}{2})\rho_{x}\in\mathcal{H}_{\omega}$ is a subgeodesic.

Proof.

Let $U(s,y)=u_{\textup{Re }s}(y)\in C^{\infty}(S\times X)$, where $S=\{0\leq\textup{Re }z\leq 1\}\subset\mathbb{C}$. It is clear that for small enough $\delta>0$ we have that $u_{t}\in\mathcal{H}_{\omega},\ t\in[0,1]$. More precisely, there exists $\alpha>0$ such that $\omega_{u_{t}}\geq\alpha\omega,\ t\in[0,1]$.

This implies that $\omega+i\partial_{S\times X}\bar{\partial}_{S\times X}U$ has at least $n$ non-negative eigenvalues for all $(s,y)\in S\times X$. To conclude that $\omega+i\partial_{S\times X}\bar{\partial}_{S\times X}U\geq 0$ it is enough to show that the determinant of this Hermitian form is non-negative. This is equivalent with $\ddot{u}_{t}-\langle\partial\dot{u}_{t},\bar{\partial}\dot{u}_{t}\rangle_{\omega_{u_{t}}}\geq 0$ on $[0,1]\times X.$ To show this, we start the following sequence of estimates:

[TABLE]

After possibly shrinking $\delta\in(0,1)$, we obtain that it is enough to conclude that the last expression is non-negative on the neighborhood $O_{x}$, where know that $\rho_{x}(y)=e^{\frac{-1}{\|y-x\|^{2}}},\ y\in O_{x}$. In particular, on $O_{x}\setminus\{x\}$ we have that $\langle\partial\rho_{x},\bar{\partial}\rho_{x}\rangle_{\omega}/\rho_{x}\simeq e^{\frac{-1}{\|y-x\|^{2}}}\frac{1}{\|y-x\|^{6}}$, which is uniformly bounded. In particular, after possibly further shrinking $\delta\in(0,1)$ we obtain that

[TABLE]

what we desired to prove. ∎

Theorem 3.7**.**

*Suppose that $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is a differentiable $L^{2}$ isometry with $a=1$. Then exactly one of the following holds:

(i) $F$ is induced by a biholomorphism or anti-biholomorphism $f:X\to X$ that preserves or flips the Kähler class $[\omega]$, respectively.

(ii) $F\circ\mathcal{I}$ is induced by a biholomorphism or anti-biholomorphism $f:X\to X$ that preserves or flips the Kähler class $[\omega]$, respectively.*

Proof.

Due to Lemma 2.6, after possibly composing $F$ with $\mathcal{I}$, we only need to worry about the case $a=1$ and $b=0$. In this case we will show that $F$ is induced by a biholomorphism or anti-biholomorphism $g:X\to X$ that preserves or flips the Kähler class $[\omega]$.

In the language of Theorem 2.1 applied to $F$, the first step is to show that $G_{u}=G_{v}$ for all $u,v\in\mathcal{H}_{\omega}$.

We fix $x\in X$ and $u,v\in\mathcal{H}_{\omega}$. We will show that $G_{u}^{-1}(x)=G_{v}^{-1}(x)$. Since $G_{u+c}=G_{u}$ for any $c\in\mathbb{R}$ (Corollary 2.4), we can assume that $u(x)=v(x)$. First we prove that $G_{u}^{-1}(x)=G_{v}^{-1}(x)$ under the extra non-degeneracy condition $\nabla u(x)\neq\nabla v(x)$.

Let $\eta>0$ be such that $w:=\max(u,v)+\eta\rho_{x}\in\mathcal{C}_{\omega}$. From our setup it is clear that $w\geq\max(u,v)$, and the graphs of $w$, $u$ and $v$ only meet at $x$. Extending the isometry $F$ to the metric completion, Proposition 2.3 and Proposition 2.5 implies that $F(w)\geq\max(F(u),F(v)),\ F(w)\in\mathcal{C}_{\omega}$ and $F(u),F(v)\in\mathcal{H}_{\omega}$. Below we will show that $F(w)$ and $F(u)$ only meet at $G^{-1}_{u}(x)$, moreover $F(w)$ and $F(v)$ only meet at $G^{-1}_{v}(x)$. Finally, we will show that the graphs of $F(w)$, $F(u)$ and $F(v)$ have to meet at some point of $X$, implying that $G^{-1}_{u}(x)=G^{-1}_{v}(x)$, as desired.

Let us denote by $[0,1]\ni t\to u_{t},v_{t}\in\mathcal{E}^{2}_{\omega}$ the $d_{2}$-geodesics joining $u_{0}:=u$ with $u_{1}:=w$, respectively $v_{0}:=v$ with $v_{1}:=w$. From Proposition 3.4 it follows that

[TABLE]

Using (5) there exists a small enough $\delta>0$ in the statement of Proposition 3.6 such that $u+\delta(t+\frac{t^{2}}{2})\rho_{x}\leq u_{t}$ and $v+\delta(t+\frac{t^{2}}{2})\rho_{x}\leq v_{t},\ t\in[0,1]$. Using this, $t$-convexity and (13), we obtain that

[TABLE]

Due to (12) these two estimates imply the existence of a unique $y\in X$ and a unique $z\in X$ such that

[TABLE]

In fact, we need to have that $y=G^{-1}_{u}(x)$ and $z=G^{-1}_{v}(x)$. In particular, the graphs of $F(w)$ and $F(u)$ only meet at $y$, and graphs of $F(w)$ and $F(u)$ only meet at $z$.

In case $y\neq z$, uniqueness of $y$ and $z$ implies that $y\in\{F(u)>F(v)\}$ and $y\in\{F(v)>F(u)\}$ (recall that $F(w)\geq\max(F(u),F(v)$). This implies that the graphs of $F(w)$ and $\max(F(u),F(v))$ meet at only two points ($y$ and $z$), away from the compact set $\{F(u)=F(v)\}$. Consequently, using classical Richberg approximation [16, Chapter I, Lemma 5.18], one can take a “regularized maximum” of $F(u)$ and $F(v)$ to obtain $\beta\in\mathcal{H}_{\omega}$ satisfying

[TABLE]

Since $F:\mathcal{H}_{\omega}\to\mathcal{H}_{\omega}$ is surjective, there exists a unique $\alpha\in\mathcal{H}_{\omega}$ s.t. $F(\alpha)=\beta$. Using (12) again, we obtain that

[TABLE]

Since $\nabla u(x)\neq\nabla v(x)$ and $w(x)=\alpha(x)=\max(u,v)(x)$, this is a contradiction with the smoothness of $\alpha$ at $x$. Consequently, we need to have that $G^{-1}_{u}(x)=y=z=G^{-1}_{v}(x)$, as desired.

In case $\nabla u(x)=\nabla v(x)$, one finds $q\in\mathcal{H}_{\omega}$ (via small perturbation) such that $u(x)=v(x)=q(x)$ and $\nabla u(x)\neq\nabla q(x)$ along with $\nabla v(x)\neq\nabla q(x)$. Then by the above we have that $G^{-1}_{u}(x)=G^{-1}_{q}(x)$ and $G^{-1}_{v}(x)=G^{-1}_{q}(x)$, ultimately giving that $G^{-1}_{u}(x)=G^{-1}_{v}(x)$ for any $u,v\in\mathcal{H}_{\omega}$.

Using Theorem 2.1, an integration along the curve $t\to tu$ gives that

[TABLE]

Returning to the statement of Theorem 2.1, we either have $g^{*}\omega_{u}=\omega_{F(u)},\ u\in\mathcal{H}_{\omega}$, or $g^{*}\omega_{u}=-\omega_{F(u)},\ u\in\mathcal{H}_{\omega}$.

Assuming that $g^{*}\omega_{u}=\omega_{F(u)}$, using (15) we arrive at the identity $g^{*}(i\partial\bar{\partial}u)=i\partial\bar{\partial}(u\circ g)$. Since after a dilation all elements of $C^{\infty}(X)$ land in $\mathcal{H}_{\omega}$, we obtain that actually $g^{*}(i\partial\bar{\partial}v)=i\partial\bar{\partial}(v\circ g)$ for all $v\in C^{\infty}(X)$. According to the next lemma $g$ has to be holomorphic, implying that $F=L_{g}$ (see Section 2.3).

In case $g^{*}\omega_{u}=-\omega_{F(u)}$, by a similar calculation we arrive at $g^{*}(i\partial\bar{\partial}v)=-i\partial\bar{\partial}(v\circ g)$ for all $v\in C^{\infty}(X)$. According to the next lemma $g$ has to be anti-holomorphic, giving that $F=N_{g}$ (see Section 2.3), finishing the proof. ∎

Lemma 3.8**.**

*Suppose that $g:X\to X$ is a smooth map.

(i) If $i\partial\bar{\partial}(u\circ g)=g^{*}(i\partial\bar{\partial}u)$ for all $u\in C^{\infty}(X)$ then $g$ is holomorphic.

(ii) If $i\partial\bar{\partial}(u\circ g)=-g^{*}(i\partial\bar{\partial}u)$ for all $u\in C^{\infty}(X)$ then $g$ is anti-holomorphic.*

Proof.

We only show (i) as the proof of (ii) is analogous. We start with the following computations expressed in local coordinates:

[TABLE]

Knowing that $g^{*}(i\partial\bar{\partial}u)$ is a $(1,1)$ form we also have that

[TABLE]

Clearly, it is enough to show that $g$ is holomorphic in local coordinate charts. By linearity we can assume that $i\partial\bar{\partial}(u\circ g)=g^{*}(i\partial\bar{\partial}u)$ holds for complex valued smooth functions $u$.

Let $x\in X$, and we pick $u$ such that in a coordinate neighborhood of $x$ we have that $u(z)=z_{b},\ b\in\{1,\ldots,n\}$. Then $i\partial\bar{\partial}(u\circ g)=g^{*}(i\partial\bar{\partial}u)$ gives that ${\partial^{2}g_{b}}/{\partial z_{j}\partial\overline{z_{k}}}=0$ for all $j,k\in\{1,\ldots,n\}$ at $x$. Similarly, after choosing $u(z)=\overline{z_{b}},\ b\in\{1,\ldots,n\}$ in a coordinate neighborhood of $x$, we obtain that ${\partial^{2}\overline{g_{b}}}/{\partial z_{j}\partial\overline{z_{k}}}=0$ for all $j,k\in\{1,\ldots,n\}$ at $x$. Since $x\in X$ was arbitrary, the terms in the last line of (3.2) vanish for any choice of $u$.

Repeating this process for $u(z)=z_{a}z_{b}$ and $u(z)=\bar{z}_{a}\bar{z}_{b}$, we conclude that the terms in the second line of (3.2) vanish as well, for any choice of $u$.

Revisiting the identity $i\partial\bar{\partial}(u\circ g)=g^{*}(i\partial\bar{\partial}u)$ one more time, after picking $u$ such that $i\partial\bar{\partial}u$ is positive definite in a neighborhood of $x\in X$, we obtain that $\partial g_{a}/\partial\overline{z_{j}}=0$ for any $a,j\in\{1,\ldots,n\}$ at $X$, implying that $g$ is indeed holomorphic. ∎

4 Proof of Theorem 1.4 and 1.5

We start with a lemma about the concatenation of geodesics in $\mathcal{E}^{2}_{\omega}$:

Lemma 4.1**.**

Suppose that $[-1,0]\ni t\to v_{t}\in\mathcal{E}^{2}_{\omega}$ and $[0,1]\ni t\to u_{t}\in\mathcal{E}^{2}_{\omega}$ are $d_{2}$-geodesics such that $u_{0}=v_{0}\in\mathcal{H}_{\omega}$ and $\dot{u}_{0}=\dot{v}_{0}\in L^{2}(\omega^{n})$. Then $[-1,1]\ni t\to w_{t}\in\mathcal{E}^{2}_{\omega}$, the concatenation of the curves $t\to u_{t}$ and $t\to v_{t}$, is the $d_{2}$-geodesic joining $v_{-1},u_{1}\in\mathcal{E}^{2}_{\omega}$.

Proof.

By possibly changing the background metric, we can assume that $u_{0}=v_{0}=0$. From the $L^{2}$ version of [1, Lemma 3.4(ii)], (whose proof is identical to the $L^{1}$ version, presented in [1]) we have that

[TABLE]

Next we point out that

[TABLE]

Indeed, from the triangle inequality we have that $d_{2}(v_{-1},u_{1})\leq d_{2}(v_{-1},0)+d_{2}(0,u_{1})$. The reverse inequality follows from (18) and [14, Theorem 3.1]:

[TABLE]

Due to uniqueness of $d_{2}$-geodesic segments, we only need to show that for any $a,b\in[-1,1]$ with $a<b$ we have that

[TABLE]

Since $t\to u_{t}$ and $t\to v_{t}$ are $d_{2}$-geodesics, we only need to treat the case $a\in[-1,0]$ and $b\in[0,1]$. The proof of this is almost identical to that of (19). Indeed after another application of [14, Theorem 3.1] we arrive at

[TABLE]

The reverse inequality follows from the triangle inequality: $d_{2}(v_{a},u_{b})\leq d_{2}(v_{a},0)+d_{2}(0,u_{b})=(b-a)d_{2}(0,u_{1})$. ∎

Proof of Theorem 1.5.

By changing the background metric, we can assume without loss of generality that $\phi_{0}=0$. From (5) it follows that $t\to\phi_{t}+Ct$ is a $d_{2}$-geodesic for any $C\in\mathbb{R}$. As a result, we can also assume that $\phi_{1}\leq 0$.

To derive a contradiction, let us further assume that there exists a $d_{2}$-geodesic $[-\varepsilon,1]\ni t\to\phi_{t}\in\mathcal{E}^{2}_{\omega}$, as described in the statement of the theorem.

First we show that $\phi_{-\varepsilon}\geq 0$. This is a simple consequence of the $t$-convexity. By the results of [11] (see the discussion near (6)) there exists a set $Z\subset X$ of measure zero such that for all $x\in X\setminus Z$ we have that $t\to\phi_{t}(x)$ is convex, $\phi_{0}(x)=0$, $\lim_{t\nearrow 1}\phi_{t}(x)=\phi_{1}(x)\leq 0$, and $\lim_{t\searrow-\varepsilon}\phi_{t}(x)=\phi_{-\varepsilon}(x)$. Due to $t$-convexity, we obtain that $\phi_{-\varepsilon}(x)\geq 0$ away from $Z$. As $\phi_{-\varepsilon}\in\textup{PSH}(X,\omega)$, we obtain that $\phi_{-\varepsilon}\geq 0$.

Since $\phi_{-\varepsilon}$ is usc, it follows that $\sup_{X}\phi_{-\varepsilon}<+\infty$, i.e., $\phi_{-\varepsilon}\in L^{\infty}$. Using (5) for the $d_{2}$-geodesic joining $\phi_{-\varepsilon}$ and $\phi_{0}$, it follows that

[TABLE]

Since $(-\varepsilon,1)\ni t\to\phi_{t}(x)$ is $t$-convex for all $x\in X\setminus Z$, it follows that the above estimate extends to $t\in[-\varepsilon,1]$, contradicting the fact that $\phi_{1}\in\mathcal{E}^{2}_{\omega}\setminus L^{\infty}$. ∎

Proof of Theorem 1.4.

We can assume without loss of generality that $\phi=0$.

To derive a contradiction, we further assume that there exists a metric $L^{2}$ symmetry $F:\mathcal{V}\to\mathcal{V}$, as described in the statement of the theorem.

Since $\mathcal{V}$ is $d_{2}$-open, it follows that $0\in B(0,\delta)\subset\mathcal{V}$ for some $\delta>0$, where $B(0,\delta)$ is the $d_{2}$-ball of radius $\delta$ centered at [math]. As $F$ is a metric $L^{2}$ symmetry it follows that $F:B(0,\delta)\to B(0,\delta)$ is bijective.

Let $\psi_{1}\in B(0,\delta)$ such that $\psi_{1}\in\mathcal{E}^{2}_{\omega}\setminus L^{\infty}$. One can find such $\psi$ as a consequence of [12, Theorem 3]. Let $[0,1]\ni t\to\psi_{t},F(\psi_{t})\in B(0,\delta)$ be the $d_{2}$-geodesics connecting [math] and $\psi_{1}$, respectively [math] and $F(\psi_{1})$.

Since $F$ is a metric $L^{2}$ symmetry, by definition we have that $\dot{\psi}_{0}=-\dot{F(\psi_{0})}$. Consequently, according to Lemma 4.1, the concatenation $[-1,1]\ni t\to w_{t}\in B(0,\delta)$ of the curves $t\to F(\psi_{-t})$ and $t\to\psi_{t}$ is a $d_{2}$-geodesic. But then $t\to w_{t}$ extends $t\to\psi_{t}$ at $t=0$, giving a contradiction with Theorem 1.5. ∎