Regularizing properties of Complex Monge-Amp\`ere flows II: Hermitian manifolds

TL;DR

This paper proves that complex Monge-Ampère flows on Hermitian manifolds can start from arbitrary initial conditions and confirms a conjecture that the Chern-Ricci flow performs canonical surgical contractions, also exploring a twisted version.

Contribution

It establishes the ability to run Monge-Ampère flows from any initial condition and confirms a key conjecture about the Chern-Ricci flow's behavior on Hermitian manifolds.

Findings

Flow can start from arbitrary initial conditions with zero Lelong number.

Confirmed that Chern-Ricci flow performs canonical surgical contraction.

Studied a generalized twisted Chern-Ricci flow.

Abstract

We prove that a general complex Monge-Amp\`ere flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow.

Equations

$$\frac{\partial }{\partial t}\omega =-Ric(\omega ),\omega {|}_{t=0}={\omega }_0,$$

$$Ric(\omega ):=-d{d}^c\log {\omega }^n:=-\frac{\sqrt{-1}}{\pi }\partial \bar{\partial }\log {\omega }^n.$$

$${\omega }_0-TRic({\omega }_0)+d{d}^c\rho ={\pi }^{\ast }{\omega }_Y,$$

$$(CMAF)\frac{\partial {\phi }_t}{\partial t}=\log \frac{({\theta }_t+d{d}^c{\phi }_t{)}^n}{\Omega }-F(t,x,{\phi }_t),$$

$${\phi }_{t,j}\to {\phi }_t\text{ in }{C}^{\infty }([\epsilon ,T]\times X)\text{ as }j\to +\infty .$$

$$||\phi -\psi |{|}_{C^k([\epsilon ,T]\times X)}\le C(k,\epsilon )||{\phi }_0-{\psi }_0|{|}_{L^{\infty }(X,\omega )}.$$

$$\frac{\partial {\omega }_t}{\partial t}=-Ric({\omega }_t)+\eta ,\omega {|}_{t=0}={\omega }_0$$

$$\frac{\partial \phi }{\partial t}=\log \frac{({\hat{\omega }}_t+d{d}^c\phi {)}^n}{{\omega }_0^n},$$

$$T:=\sup \{t\ge 0|\exists \psi \in C^{\infty }(X)\text{ with }{\hat{\omega }}_t+d{d}^c\psi >0\}.$$

$$\{\eta \}:=\{\alpha \text{ is a real (1,1)-form }|\exists f\in C^{\infty }(X)\text{ with }\alpha =\eta +d{d}^{c}f\},$$

$$\frac{\partial {\omega }_t}{\partial t}=-Ric({\omega }_t)-{\omega }_t+\eta .$$

$$Ric({\omega }_{\infty })=\eta -{\omega }_{\infty }.$$

$$Ric(\omega )=\eta -\omega .$$

$$(\omega +d{d}^c\phi {)}^n=e^{\phi }\Omega .$$

$${\nabla }_i{X}^k={\partial }_i{X}^j+{\Gamma }_{ij}^k{X}^j,{\nabla }_i{a}_j={\partial }_i{a}_j-{\Gamma }_{ij}^k{a}_k,$$

$${\Gamma }_{ij}^k=g^{\bar{l}k}{\partial }_i{g}_{j\bar{l}}.$$

$$T$$

$$\bar{T}$$

$$T_{ij\bar{k}}={\partial }_i{g}_{j\bar{k}}-{\partial }_j{g}_{i\bar{k}},\text{ and }{\bar{T}}_{\bar{i}\bar{j}k}={\partial }_{\bar{j}}{g}_{k\bar{i}}-{\partial }_{\bar{i}}{g}_{k\bar{j}}.$$

$$T_{ij}^k={\Gamma }_{ij}^k-{\Gamma }_{ji}^k=g^{\bar{l}k}{T}_{ij\bar{l}}.$$

$$R_{k\bar{l}}(g):=R_{k\bar{l}}(\omega ):=g^{\bar{j}i}{R}_{k\bar{l}i\bar{j}}=-{\partial }_k{\partial }_{\bar{l}}\log \det g,$$

$$Ric(g):=Ric(\omega ):=\frac{\sqrt{-1}}{\pi }{R}_{k\bar{l}}^{C}d{z}_k\wedge d{\bar{z}}_l=-d{d}^c\log \det g,$$

$$d:=\partial +\bar{\partial },d^c:=\frac{1}{2i\pi }(\partial -\bar{\partial }).$$

$$H_{BC}^{1,1}(X,\mathbb{R}):=\frac{\{\text{closed real (1,1)-forms}\}}{\{\sqrt{-1}\partial \bar{\partial }\psi ,\psi \in C^{\infty }(X,\mathbb{R})\}}$$

$$\omega +d{d}^c\phi \ge 0$$

$$\nu (\phi ,x):=\underset{z\to x}{\mathrm{liminf}}\frac{\phi (z)}{\log |z-x|}.$$

$$\frac{\partial {\phi }_t}{\partial t}=\log \frac{({\theta }_t+d{d}^c{\phi }_t{)}^n}{\Omega }-F(t,z,\phi )(CMAF)$$

$$\frac{\partial F}{\partial s}\ge 0\text{ and }\frac{\partial F}{\partial t}>B,$$

$$\frac{\omega }{2}\le {\theta }_t\le 2\omega \text{ and }{\delta }^{-1}\Omega \le {\theta }_t^n\le \delta \Omega ,\forall t\in [0,T]\text{ for some }\delta >0,$$

$${\theta }_t-t{\dot{\theta }}_t\ge 0\text{ for }0\le t\le T.$$

$${\phi }_t\ge \underset{X}{\inf }{\phi }_0-Ct,\forall (t,x)\in [0,T]\times X.$$

Full text

Introduction

Let $(X,g,J)$ be a compact Hermitian manifold of complex dimension $n$, that is a compact complex manifold such that $J$ is compatible with the Riemannian metric $g$. Recently a number of geometric flows have been introduced to study the structure of Hermitian manifolds. Some flows which do preserve the Hermitian property have been proposed by Streets-Tian [ST10, ST11, ST13], Liu-Yang [LY12] and also anomaly flows due to Phong-Picard-Zhang [PPZ15, PPZ16a, PPZ16b] which moreover preserve the conformally balanced condition of Hermitian metrics. Another such flow, namely the Chern-Ricci flow, was introduced by Gill [Gil11] and has been further developed by Tosatti-Weinkove in [TW15]. The Chern-Ricci flow is written as

[TABLE]

where $Ric(\omega)$ is the Chern-Ricci form which is defined locally by

[TABLE]

This flow specializes the Kähler-Ricci flow when the initial metric is Kähler. In [TW15, TW13] Tosatti and Weinkove have investigated the flow on arbitrary Hermitian manifolds, notably in complex dimention 2 (see also [TWY15, FTWT, GS15, Gil15, LV13, Zhe15, Yan16] for more recent works on the Chern-Ricci flow).

For the Kähler case, running the Kähler-Ricci flow (or complex Monge-Ampère flows) from a rough initial data has been studied by several recent works [CD07], [ST16], [SzTo11], [GZ13], [BG13], [DNL17]. In [ST16], [SzTo11] the authors succeeded to run certain complex Monge-Ampère flows from continuous initial data, while [DNL17] and [GZ13] are running a simplified flow starting from an initial current with zero Lelong numbers. Recently, we extended these latter works to deal with general complex Monge-Ampère flows and arbitrary initial condition (cf. [Tô16]). One of the motivations for this problem comes from the Analytic Minimal Model Program proposed by Song-Tian [ST16]. For the Chern-Ricci flow, the same question was asked recently by Tosatti-Weinkove [TW13, TW15] related to the classification of non-Kähler complex surfaces.

Assume that there exists a holomorphic map between compact Hermitian manifolds $\pi:X\rightarrow Y$ blowing down an exceptional divisor $E$ on $X$ to one point $y_{0}\in Y$. In addition, assume that there exists a smooth function $\rho$ on $X$ such that

[TABLE]

with $T<+\infty$. Tosatti and Weinkove proved:

Theorem.([TW15, TW13])The solution $\omega_{t}$ to the Chern-Ricci flow (0.1) converges in $C_{loc}^{\infty}(X\setminus E)$ to a smooth Hermitian metric $\omega_{T}$ on $X\setminus E$.

*Moreover, there exists a distance function $d_{T}$ on $Y$ such that $(Y,d_{T})$ is a compact metric space and $(X,g(t))$ converges in the Gromov-Hausdorff sense $(Y,d_{T})$ as $t\rightarrow T^{-}$. *

Observe that $\omega_{T}$ induces a singular metric $\omega^{\prime}$ on $Y$ which is smooth in $Y\setminus\{y_{0}\}$. Tosatti and Weinkove conjectured that one can continue the Chern-Ricci flow on $Y$ with initial data $\omega^{\prime}$. This is an open question in [TW13, Page 2120] in which they conjectured that the Chern-Ricci flow performs a canonical surgical contraction:

Conjecture. (Tosatti-Weinkove [TW13, Page 2120])

1. (1)

There exists a smooth maximal solution $\omega_{t}$ of the Chern-Ricci on $Y$ for $t\in(T,T_{Y})$ with $T<T_{Y}\leq+\infty$ such that $\omega_{t}$ converges to $\omega^{\prime}$, as $t\rightarrow T^{+}$, in $C^{\infty}_{loc}(Y\setminus\{y_{0}\})$. Furthermore, $\omega_{t}$ is uniquely determined by $\omega_{0}$. 2. (2)

The metric space $(Y,g(t))$ converges to $(Y,d_{T})$ as $t\rightarrow T^{+}$ in the Gromov-Hausdorff sense.

In this note, we confirm this conjecture111After this paper was completed, the author learned that Xiaolan Nie proved the first statement of the conjecture for complex surfaces (cf. [Nie17]). She also proved that the Chern-Ricci flow can be run from a bounded data. The author would like to thank Xiaolan Nie for sending her preprint. . An essential ingredient of its proof is to prove that the Monge-Ampère flow corresponding to the Chern-Ricci flow can be run from a rough data. By generalizing a result of Székelyhidi-Tosatti [SzTo11], Nie [Nie14] has proved this property for compact Hermitian manifolds of vanishing first Bott-Chern class and continous initial data. In this paper, we generalize the previous results of Nie [Nie14] and the author [Tô16] by considering the following complex Monge-Ampère flow:

[TABLE]

where $(\theta_{t})_{t\in[0,T]}$ is a family of Hermitian forms with $\theta_{0}=\omega$ and $F$ is a smooth function on ${\mathbb{R}}\times X\times{\mathbb{R}}$.

Theorem A. Let $\varphi_{0}$ be a $\omega$-psh function with zero Lelong number at all points. Let $(t,z,s)\mapsto F(t,z,s)$ be a smooth function on $[0,T]\times X\times{\mathbb{R}}$ such that $\partial F/\partial s\geq 0$ and $\partial F/\partial t$ is bounded from below.

Then there exists a family of smooth strictly $\theta_{t}-psh$ functions $(\varphi_{t})$ satisfying $(CMAF)$ in $(0,T]\times X,$ with $\varphi_{t}\rightarrow\varphi_{0}$ in $L^{1}(X),$ as $t\searrow 0^{+}$ and $\varphi_{t}$ converges to $\varphi_{0}$ in $C^{0}(X)$ if $\varphi_{0}$ is continuous. This family is moreover unique if $\partial F/\partial t$ is bounded and $\partial F/\partial s\geq 0$.

The following stability result is a straighforward extension of [Tô16, Theorem 4.3, 4.4].

Theorem B. Let $\varphi_{0},\varphi_{0,j}$ be $\omega$-psh functions with zero Lelong number at all points, such that $\varphi_{0,j}\rightarrow\varphi_{0}$ in $L^{1}(X)$. Denote by $\varphi_{t,j}$ and $\varphi_{t}$ the corresponding solutions of $(CMAF)$ with initial condition $\varphi_{0,j}$ and $\varphi_{0}$ respectively. Then for each $\varepsilon\in(0,T)$

[TABLE]

Moreover, if $\varphi_{0}$ and $\psi_{0}$ are continuous, then for any $k\geq 0$, for any $0<\varepsilon<T$, there exists a positive constant $C(k,\varepsilon)$ depending only on $k$ and $\varepsilon$ such that

[TABLE]

As a consequence of Theorem A and Theorem B, the Chern-Ricci flow on any Hermitian manifold can be run from rough data. Using this result and a method due to Song-Tosatti-Weinkove [SW13, TW13] we prove the conjecture. The proof is given in Section 4.

The second purpose of this paper is to study a generalization of the Chern-Ricci flow, namely the twisted Chern-Ricci flow:

[TABLE]

where $Ric(\omega_{t})$ is the Chern-Ricci form of $\omega_{t}$, $\omega_{0}$ is a Hermitian metric on $X$ and $\eta$ is a smooth $(1,1)$-form. In general, we do not assume $\eta$ is closed. This flow also generalizes the twisted Kähler-Ricci flow which has been studied recently by several authors (see for instance [CS12, GZ13]).

We show that the twisted Chern-Ricci flow starting from a Hermitian metric $\omega_{0}$ is equivalent to the following complex Monge-Ampère flow

[TABLE]

where $\hat{\omega}_{t}=\omega_{0}+t(\eta-Ric(\omega_{0}))$. We first prove the following, generalizing [TW15, Theorem 1.2]:

Theorem C. There exists a unique solution to the twisted Chern-Ricci flow on $[0,T)$, where

[TABLE]

When the twisted Chern-Ricci flow has a long time solution, it is natural to study its behavior at infinity. When the Bott-Chern class vanishes and $\eta=0$, Gill has proved that the flow converges to a Chern-Ricci flat Hermitian metric (cf. [Gil11]).

Denote by

[TABLE]

the equivalence class of $\eta$. Suppose that $c^{BC}_{1}(X)-\{\eta\}$ is negative. Consider the normalized twisted Chern-Ricci flow

[TABLE]

Then we have the following result for the long time behavior of the flow generalizing [TW15, Theorem 1.7]:

Theorem D. Suppose $c^{BC}_{1}(X)-\{\eta\}<0$. The normalized twisted Chern-Ricci flow smoothly converges to a Hermitian metric $\omega_{\infty}=\eta-Ric(\Omega)+dd^{c}\varphi_{\infty}$ which satisfies

[TABLE]

Observe that $\omega_{\infty}$ satisfies the twisted Einstein equation:

[TABLE]

We can prove the existence of a unique solution of (0.5) using a result of Monge-Ampère equation due to Cherrier [Che87] (see Theorem 5.2). Theorem D moreover gives an alternative proof of the existence of the twisted Einstein metric $\omega_{\infty}$ in $-c^{BC}_{1}(X)+\eta$. This is therefore a generalization of Cao’s approach [Cao85] by using Kähler-Ricci flow to prove the existence of Kähler-Einstein metric on Kähler manifold of negative first Chern class. In particular, when $\eta=0$, we have $c_{1}^{BC}(X)<0$ hence we have $c_{1}(X)<0$ and $X$ is a Kähler manifold, this is [TW15, Theorem 1.7].

Note that in general, one cannot assume $\eta$ to be closed, in contrast with the twisted Kähler-Ricci flow. Let us stress also that the limit of the normalized twisted Chern-Ricci flow exists without assuming that the manifold is Kähler (a necessary assumption when studying the long term behavior of the Chern-Ricci flow). Therefore the twisted Chern-Ricci flow is somehow more natural in this context.

As an application of Theorem D, we give an alternative proof of the existence of a unique smooth solution for the following Monge-Ampère equation

[TABLE]

We show that the solution is the limit of the potentials of a suitable twisted normalized Chern-Ricci flow. Cherrier [Che87] proved this result by generalizing the elliptic approach of Aubin [Aub78] and [Yau78].

The paper is organized as follows. In Section 1, we recall some notations in Hermitian manifolds. In Section 2 we prove various a priori estimates following our previous work [Tô16]. The main difference is that we will use the recent result of Kołoziedj’s uniform type estimates for Monge-Ampère on Hermitian manifolds (cf. [DK12, Bł11, Ngu16]) instead of the one on Kähler manifolds to bound the oscillation of the solution. The second arises when estimating the gradient and the Laplacian: we use a special local coordinate system due to Guan-Li [GL10, Lemma 2.1] instead of the usual normal coordinates in Kähler geometry. In Section 3 we prove Theorem B and Theorem C. In Section 4, we prove the conjecture. In Section 5 we define the twisted Chern-Ricci flow and prove the existence of a unique maximal solution using the estimates in Section 2. The approach is different from the one for the Chern-Ricci flow due to Tosatti-Weinkove [TW15]. We also show that the twisted Chern-Ricci flow on negative twisted Bott-Chern class smoothly converges to the unique twisted Einstein metric.

Acknowledgement. The author is grateful to his supervisor Vincent Guedj for support, suggestions and encouragement. The author thanks Valentino Tosatti and Ben Weinkove for their interest in this work and helpful comments. We also thank Thu Hang Nguyen, Van Hoang Nguyen and Ahmed Zeriahi for very useful discussions. The author would like to thank the referee for useful comments and suggestions. This work is supported by the Jean-Pierre Aguilar fellowship of the CFM foundation.

1. Preliminaries

1.1. Chern-Ricci curvature on Hermitian manifold

Let $(X,g)$ be a compact Hermitian manifold of complex dimension $n$. In local coordinates, $g$ is determined by the $n\times n$ Hermitian matrix $(g_{i\bar{j}})=g(\partial_{i},\partial_{\bar{j}})$. We write $\omega=\sqrt{-1}g_{i\bar{j}}dz_{i}\wedge d\bar{z}_{j}$ for its associated $(1,1)$-form.

We define the Chern connection $\nabla$ associated to $g$ as follows. If $X=X^{j}\partial_{j}$ is a vector field and $\alpha=a_{i}dz_{i}$ is a $(1,0)$-form then theirs covariant derivatives have components

[TABLE]

where the Christoffel symbols $\Gamma^{k}_{ij}$ are given by

[TABLE]

We define the torsion tensors $T$ and $\bar{T}$ of $\omega$ as follows

[TABLE]

where

[TABLE]

Then the torsion tensor of $\omega$ has component

[TABLE]

Definition 1.1**.**

The Chern-Ricci curvature of $g$ is the tensor

[TABLE]

and the Chern-Ricci form is

[TABLE]

where

[TABLE]

It is a closed real $(1,1)$-form and its cohomology class in the Bott-Chern cohomology group

[TABLE]

is the first Bott-Chern class of $X$, denoted by $c^{BC}_{1}(X)$, which is independent of the choice of Hermitian metric $g$. We also write $R=g^{k\bar{l}}R_{k\bar{l}}$ for the Chern scalar curvature.

1.2. Plurisubharmonic functions and Lelong number

Let $(X,\omega)$ be a compact Hermitian manifold.

Definition 1.2**.**

We let $PSH(X,\omega)$ denote the set of all $\omega$-plurisubharmonic functions ($\omega$-psh for short), i.e the set of functions $\varphi\in L^{1}(X,{\mathbb{R}}\cup\{-\infty\})$ which can be locally written as the sum of a smooth and a plurisubharmonic function, and such that

[TABLE]

in the weak sense of positive currents.

Definition 1.3**.**

Let $\varphi$ be a $\omega$-psh function and $x\in X$. The Lelong number of $\varphi$ at $x$ is

[TABLE]

We say $\varphi$ has a logarithmic pole of coefficient $\gamma$ at $x$ if $\nu(\varphi,x)=\gamma$.

2. A priori estimates for complex Monge-Ampère flows

In this section we prove various a priori estimates for $\varphi_{t}$ which satisfies

[TABLE]

with a smooth strictly $\omega$-psh initial data $\varphi_{0}$, where $\Omega$ is a smooth volume form, $(\theta_{t})_{t\in[0,T]}$ is a family of Hermitian forms on $X$ and $(t,z,s)\mapsto F(t,z,s)$ is a smooth function on $[0,T]\times X\times{\mathbb{R}}$ with

[TABLE]

for some $B\in{\mathbb{R}}$.

Since we are interested in the behavior near 0 of $(CMAF)$, we can further assume that

[TABLE]

[TABLE]

The assumption (2.3) will be used to bound $\dot{\varphi}_{t}$ from above.

2.1. Bounds on $\varphi_{t}$ and $\dot{\varphi}_{t}$

As in the Kähler case, the upper bound of $\varphi$ is a simple consequence of the maximal principle (see [Tô16, Lemma 2.1]).

For a lower bound of $\varphi_{t}$, we have

Lemma 2.1**.**

There is a constant $C>0$ depending only on $\inf_{X}\varphi_{0}$ such that,

[TABLE]

Proof.

Set

[TABLE]

where $C$ will be chosen hereafter. Since we assume that $2\theta_{t}\geq\omega$,

[TABLE]

Combine with $\omega^{n}\geq 2^{-n}\theta^{n}_{t}\geq\Omega/(2^{n}\delta)$, we have

[TABLE]

We now choose $C>0$ satisfying

[TABLE]

hence

[TABLE]

It follows from the maximum principle [Tô16, Proposition 1.5] that

[TABLE]

as required. ∎

For another lower bound, we follow the argument in [GZ13], replacing the uniform a priori bound of Kołodziej [Koł98] by its Hermitian version (see for instance [Ngu16, Theorem 2.1]). First, we assume that $\theta_{t}\geq\omega+t\chi,\,\forall t\in[0,T],$ for some smooth $(1,1)$-form $\chi$. Let $0<\beta<+\infty$ be such that

[TABLE]

It follows from Kołodziej’s uniform type estimate for Monge-Ampère equation on Hermitian manifolds (cf. [Ngu16, Theorem 2.1]) that the exists a continuous $\omega$-psh solution $u$ of the equation

[TABLE]

which satisfies

[TABLE]

where $C$ only depends on $||e^{-2\beta\varphi_{0}}||_{L^{p}(X)}$, for some $p>1$.

Remark 2.2*.*

Latter on we will replace $\varphi_{0}$ by smooth approximants $\varphi_{0,j}$ of initial data. Since the latter one has zero Lelong numbers, Skoda’s integrability theorem [Sko72] will provide a uniform bound for $||e^{-2\beta\varphi_{0}}||_{L^{p}(X)}$ and $||u||_{L^{\infty}(X)}$.

Lemma 2.3**.**

For all $z\in X$ and $0<t<\min(T,(2\beta)^{-1})$, we have

[TABLE]

where $A$ depends on $\sup_{X}\varphi_{0}$. In particular, there exists $c(t)\geq 0$ such that

[TABLE]

with $c(t)\searrow 0$ as $t\searrow 0$.

Proof.

Set

[TABLE]

where $A:=\sup_{[0,T]\times X}F(t,z,C_{0})$ with $(1-2\beta t)\varphi_{0}+tu+n(t\log t-t)\leq C_{0}$ for all $t\in[0,\min(T,(2\beta)^{-1})]$.

By our choice of $\beta$ we have

[TABLE]

Moreover

[TABLE]

hence $\phi_{t}$ is a subsolution to $(CMAF)$. Since $\phi_{0}=\varphi_{0}$ the conclusion follows from the maximum principle [Tô16, Proposition 1.5]. ∎

The lower bound for $\dot{\varphi}$ comes from the same argument in [Tô16, Proposition 2.6]:

Proposition 2.4**.**

Assume $\varphi_{0}$ is bounded. There exist constants $A>0$ and $C=C(A,Osc_{X}\varphi_{0})>0$ such that for all $(x,t)\in X\times(0,T]$,

[TABLE]

We now prove a crucial estimate for $\dot{\varphi}_{t}$ which allows us to use the uniform version of Kolodziej’s uniform type estimates in order to get the bound of $Osc_{X}\varphi_{t}$. The proof is the same in [GZ13, Tô16], but we include a proof for the reader’s convenience.

Proposition 2.5**.**

There exists $0<C=C(\sup_{X}\varphi_{0},T)$ such that for all $0<t\leq T$ and $z\in X$,

[TABLE]

Proof.

We consider $G(t,z)=t\dot{\varphi_{t}}-\varphi_{t}-nt+Bt^{2}/2$, with $B$ is the constant in (2.1). We obtain

[TABLE]

Since we assume that $\theta_{t}-t\dot{\theta}_{t}\geq 0$ (see (2.3)), we get

[TABLE]

If $G$ attains its maximum at $t=0$, we have the result. Otherwise, assume that $G$ attains its maximum at $(t_{0},z_{0})$ with $t_{0}>0$, then at $(t_{0},z_{0})$ we have

[TABLE]

Since $\frac{\partial F}{\partial s}\geq 0$ by the hypothesis, we obtain $\dot{\varphi}(t_{0},z_{0})<0$ and

[TABLE]

Using Lemma 2.3 we get $\varphi_{t_{0}}\geq\varphi_{0}-C_{1}$, where $C_{1}$ only depends on $\sup_{X}\varphi_{0}$ and $T$, hence there is a constant $C_{2}$ depending on $\sup_{X}\varphi_{0}$ and $T$ such that

[TABLE]

Since $\varphi_{t}\leq C_{3}(\sup\varphi_{0},T)$, so

[TABLE]

where $C$ only depends on $\sup_{X}\varphi_{0}$ and $T$. ∎

2.2. Bounding the oscillation of $\varphi_{t}$

Once we get an upper bound for $\dot{\varphi_{t}}$ as in Proposition 2.5, we can bound the oscillation of $\varphi_{t}$ by using the following uniform version of Kolodziej’s estimates due to Dinew- Kołodziej [DK12, Theorem 5.2].

Theorem 2.6**.**

Let $(X,\omega)$ be a compact Hermitian manifold. Assume $\varphi\in C^{2}(X)$ is such that $\omega+dd^{c}\varphi\geq 0$ and

[TABLE]

Then for $p>1$,

[TABLE]

where $C$ only depends on $\omega,p,||f||_{L^{p}(X)}$.

Indeed, observe that $\varphi_{t}$ satisfies

[TABLE]

then by Proposition 2.5, for any $\varepsilon\in(0,T)$,

[TABLE]

for all $t\in[\varepsilon,T]$. Fix $p>1$ and $\mathcal{F}$ a compact family of $\omega$-psh functions with zero Lelong numbers, and assume that $\varphi_{0}\in\mathcal{F}$. It follows from the uniform version of Skoda’s integrability theorem (cf. [Sko72, Proposition 7.1] and [Zer01, Theorem 3.1]) that there exists $C_{\varepsilon}>0$ such that

[TABLE]

for all $\psi\in\mathcal{F},t\in[\varepsilon,T]$. We thus write for short $||H_{t}||_{L^{p}(\Omega)}\leq C(t)$ for some $C(t)>0$.

Remark 2.7*.*

Later on we will replace $\varphi_{0}$ by smooth approximants $\varphi_{0,j}$ of initial data. We can thus apply the previous estimate with $\mathcal{F}=\{\varphi_{0}\}\cup\{\varphi_{0,j},j\in{\mathbb{N}}\}$, where $\varphi_{0}$ is now the initial data. This yields

[TABLE]

Now, thanks to Theorem 2.6, we infer that the oscillation of $\varphi_{t}$ is uniformly bounded:

Theorem 2.8**.**

Fix $0<t\leq T$. There exist $C(t)>0$ independent of $\inf_{X}\varphi_{0}$ such that

[TABLE]

2.3. Bounding the gradient of $\varphi$

In this section we bound the gradient of $\varphi$ using the same technique as in [Tô16] (see also [SzTo11]) which is a parabolic version of Błocki’s estimate [Bł09] for Kähler manifolds. In these articles we used the usual normal coordinates in Kähler geometry. For Hermitian manifolds, we need to use the following local coordinate system due to Guan-Li [GL10, Lemma 2.1] (see also [ST11] for a similar argument), which is also essential for our second order estimate. We also refer the reader to [Ha96, Lemma 6] for a gradient estimate for the elliptic Complex Mong-Ampère equation in the Hermitian case without using the local coordinate system. We thank Valentino Tosatti for indicating the reference [Ha96]. We remark that similar arguments of the proof below can be found in [Nie14, Lemma 3.3].

Lemma 2.9**.**

At any point $x\in X$ there exists a local holomorphic coordinate system centered at $x$ such that for all $i,j$

[TABLE]

We now prove

Proposition 2.10**.**

Fix $\varepsilon\in[0,T]$. There exists $C>0$ depending on $\sup_{X}\varphi_{0}$ and $\varepsilon$ such that for all $\varepsilon<t\leq T$

[TABLE]

Proof.

Since the bound on $Osc_{X}\varphi_{\varepsilon}$ only depends on $\sup_{X}\varphi_{0}$ and $\varepsilon$ (see Theorem 2.8), we can consider the flow starting from $\varphi_{\varepsilon}$, i.e $\varphi(0,x)=\varphi_{\varepsilon}$. Then we need to show that there exists a constant $C$ depending on $Osc_{X}\varphi_{0}$ and $\varepsilon$ such that

[TABLE]

for all $t\in[0,T-\varepsilon]$.

Define

[TABLE]

for $(t,x)\in[0,T-\varepsilon]\times X$ where, $\beta=|\nabla\varphi|^{2}_{\omega}$ and $\gamma\in C^{\infty}({\mathbb{R}},{\mathbb{R}})$ will be chosen hereafter.

If $K(t,z)$ attains its maximum for $t=0$, $\beta$ is bounded in terms of $\sup_{X}\varphi_{0}$ and $\varepsilon$, since $|\varphi_{t}|$ is bounded by a constant depending on $\sup_{X}\varphi_{0}$ and $\varepsilon$ for all $t\in[0,T-\varepsilon]$ (see Section 2.1).

We now assume that $K(t,z)$ attains its maximum at $(t_{0},z_{0})$ in $[0,T-\varepsilon]\times X$ with $t_{0}>0$. Near $z_{0}$ we have $\omega=\sqrt{-1}g_{i\bar{j}}dz_{i}\wedge d\bar{z}_{j}$ for some and $\theta_{t}=\sqrt{-1}h_{i\bar{j}}dz_{i}\wedge d\bar{z}_{j}$. We take the local coordinates (2.5) for $\omega$ at $z_{0}$ such that

[TABLE]

here for convenience we denote in local coordinate, $u_{p}:=\frac{\partial u}{\partial z_{p}},u_{j\bar{k}}:=\frac{\partial^{2}u}{\partial z_{j}\partial\bar{z}_{k}}$, and $g_{i\bar{j}k}:=\frac{\partial g_{i\bar{j}}}{\partial z_{k}}$.

We now compute $K_{p},K_{p\bar{p}}$ at $(t_{0},z_{0})$ in order to use the maximum principle. At $(t_{0},z_{0})$ we have $K_{p}=0$ hence

[TABLE]

or

[TABLE]

Therefore,

[TABLE]

Now we compute $\beta_{p},\beta_{p\bar{p}}$ at $(t_{0},z_{0})$ with $\beta=g^{j\bar{k}}\varphi_{j}\varphi_{\bar{k}}$ where $(g^{j\bar{k}})=[(g_{j\bar{k}})^{t}]^{-1}$. We have

[TABLE]

Since

[TABLE]

[TABLE]

and

[TABLE]

Therefore, at $(t_{0},x_{0})$,

[TABLE]

[TABLE]

and

[TABLE]

Now at $(t_{0},z_{0})$

[TABLE]

Since $u_{p\bar{p}}=\varphi_{p\bar{p}}+h_{p\bar{p}}$ near $(t_{0},z_{0})$, then at $(t_{0},z_{0})$

[TABLE]

with $\lambda\leq h_{p\bar{p}}$, on $[0,T]\times X$ for all $p=1,\ldots,n$.

Moreover, assume that the holomorphic bisectional curvature of $\omega$ is bounded from below by a constant $-B\in{\mathbb{R}}$ on X, then at $(t_{0},z_{0})$

[TABLE]

therefore

[TABLE]

By the maximum principle, at $(t_{0},z_{0})$

[TABLE]

hence,

[TABLE]

We will simplify (2.3) to get a bound for $\beta$ at $(t_{0},z_{0})$.

Claim 1. There exist $C_{1}>0$ depending on $\sup|\varphi_{0}|$ and $C_{2},C_{3},C_{4}>0$ only depending on $h$ such that

[TABLE]

and

[TABLE]

Proof of Claim 1.

For the first one, we note that near $(t_{0},z_{0})$

[TABLE]

hence using

[TABLE]

we have at $(t_{0},z_{0})$

[TABLE]

Therefore

[TABLE]

In addition, at $(t_{0},z_{0})$

[TABLE]

we infer that

[TABLE]

We may assume that $\log\beta>1$ so that

[TABLE]

Since $\frac{\partial F}{\partial s}\geq 0$, there exist $C_{1}>0$ depending on $\sup|\varphi_{0}|$ and $C_{2}$ depending on $h$ such that

[TABLE]

We now estimate

[TABLE]

It follows from (2.9) and (2.10) that

[TABLE]

then,

[TABLE]

Hence at $(t_{0},z_{0})$, using $\log\beta>1$, we have

[TABLE]

here $C_{3}$ and $C_{4}$ only depend on $h$. This completes Claim 1. ∎

We now choose

[TABLE]

with $A$ so large that $\gamma^{\prime}=A-2\left(\frac{1}{A}+C_{4}\right)s>0$ and $\gamma^{\prime\prime}=-2/A-2C_{4}<0$ for all $s\leq\sup_{[0,T]\times X}\varphi_{t}$. From Lemma 2.4 we have $\dot{\varphi}\geq C_{0}+n\log t$, where $C_{0}$ depends on $Ocs_{X}\varphi_{0}$. Combining this with (2.3) and Claim 1, we obtain

[TABLE]

where $C_{1},C_{2},C^{\prime}$ depend on $\sup_{X}|\varphi_{0}|$ and $h$. If A is chosen sufficiently large, we have a constant $C_{5}>0$ such that

[TABLE]

since otherwise (2.13) implies that $\beta$ is bounded. So we get $(u_{p\bar{p}})^{-1}\leq C_{5}\log\beta$ for $1\leq p\leq n$. It follows from Lemma 2.5 we have at $(t_{0},z_{0})$

[TABLE]

where $C_{6}$ depends on $\sup_{X}|\varphi_{0}|,\varepsilon$. Then we get

[TABLE]

so from (2.14) we have

[TABLE]

hence $\log\beta<C_{7}$ at $(t_{0},z_{0})$. This shows that $\beta=|\nabla\varphi(z)|^{2}_{\omega}<e^{C/t}$ for some $C$ depending on $\sup|\varphi_{0}|$ and $\varepsilon$. ∎

2.4. Bounding $\Delta\varphi_{t}$

We now estimate $\Delta\varphi$. The estimate on $|\nabla\varphi|^{2}_{\omega}$ is needed here. The argument follows from [GZ13, Tô16] but there are difficulties in using this approach because of torsion terms that need to be controlled (see also [TW10] for similar computation for the elliptic Monge-Ampère equation).

Lemma 2.11**.**

Fix $\varepsilon>0$. There exist constants $A$ and $C$ only depending on $\varepsilon$ and $\sup_{X}\varphi_{0}$ such that for all $0\leq t\leq T-\varepsilon$,

[TABLE]

Proof.

We first denote by $C$ a uniform constant only depending on $\varepsilon$ and $\sup_{X}\varphi_{0}$.

Define

[TABLE]

and

[TABLE]

with $A>0$ to be chosen latter. Set $\Delta_{t}:=\Delta_{\omega_{t+\varepsilon}}$, then

[TABLE]

hence

[TABLE]

First, we have

[TABLE]

Suppose $P$ attains its maximum at $(t_{0},z_{0})$. If $t_{0}=0$, we get the desired inequality. We now assume that $P(t,x)$ attains its maximum at $(t_{0},x_{0})$ with $t_{0}>0$.

It follows from Proposition 2.4, Proposition 2.5 and Theorem 2.8 that $\dot{\varphi}_{t+\varepsilon}$ depends on $\varepsilon$ and $\sup_{X}\varphi_{0}$, hence

[TABLE]

Combine with the inequality

[TABLE]

we infer that

[TABLE]

[TABLE]

Denoting $\tilde{g}(t,x)=g_{t+\varepsilon}(x)$ and using the local coordinate system (2.5) at $(t_{0},x_{0})$, we have

[TABLE]

where the last inequality comes from $(\ref{lower bound of trace})$. Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Therefore

[TABLE]

Now we have

[TABLE]

It follows that

[TABLE]

We now claim that

[TABLE]

By computation,

[TABLE]

where $T_{ij\bar{j}}=h_{j\bar{j}i}-h_{i\bar{j}j}$.

It follows from the Cauchy-Schwarz inequality that

[TABLE]

For the second term, we have

[TABLE]

Now at the maximum point $(t_{0},z_{0}),t_{0}>0$, we have $\nabla P=0$, hence

[TABLE]

Since $\tilde{g}_{k\bar{i}\bar{k}}=\tilde{g}_{k\bar{k}\bar{i}}-\bar{T}_{ik\bar{k}}$, we have

[TABLE]

Combining all of these inequalities we obtain (2.21).

It now follows from (2.20) and (2.21) that

[TABLE]

Moreover,

[TABLE]

It follows from Proposition 2.10, $\operatorname{tr}_{\omega_{t+\varepsilon}}\omega\operatorname{tr}_{\omega}\omega_{t+\varepsilon}\geq n$ and $\operatorname{tr}_{\omega}\omega_{t+\varepsilon}\geq C^{-1}$ that

[TABLE]

Now

[TABLE]

Therefore

[TABLE]

Then we infer

[TABLE]

so from Proposition 2.10 and (2.23) we have

[TABLE]

Moreover, the inequalities $(n-1)\log x\leq x+C_{n}$ and

[TABLE]

for any two positive $(1,1)$-froms $\omega$ and $\tilde{\omega}$, imply that

[TABLE]

It follows from (2.15), (2.16), (2.24) and (2.25) that

[TABLE]

We choose $A$ sufficiently large such that $Ct+1-A/2<0$. Applying Proposition 2.4,

[TABLE]

Now suppose $P$ attains its maximum at $(t_{0},z_{0})$. If $t_{0}=0$, we get the desired inequality. Otherwise, at $(t_{0},z_{0})$

[TABLE]

Hence we get

[TABLE]

as required. ∎

2.5. Higher order estimates

For the higher order estimates, one can follow [SzTo11] (see [Nie14] for its version on Hermitian manifolds) by bounding

[TABLE]

then using the parabolic Schauder estimates to obtain higher order estimates for $\varphi$. Additionally, we can also combine previous estimates with Evans-Krylov and Schauder estimates [Tô16, Theorem 1.7] to get the $C^{k}$ estimates for all $k\geq 0$.

Theorem 2.12**.**

For each $\varepsilon>0$ and $k\in{\mathbb{N}}$, there exists $C_{k}(\varepsilon)$ such that

[TABLE]

3. Proof of Theorem A and B

We now consider the complex Monge-Ampère flow

[TABLE]

starting from a $\omega$-psh function $\varphi_{0}$ with zero Lelong numbers at all points, where $F(t,z,s)\in C^{\infty}([0,T]\times X\times{\mathbb{R}},{\mathbb{R}})$ with $\frac{\partial F}{\partial s}\geq 0$ and $\frac{\partial F}{\partial t}$ is bounded from below.

3.1. Convergence in $L^{1}$

We fisrt approximate $\varphi_{0}$ by a decreasing sequence $\varphi_{0,j}$ of smooth $\omega$-psh fuctions (see [BK07]). Denote by $\varphi_{t,j}$ the smooth family of $\theta_{t}$-psh functions satisfying on $[0,T]\times X$

[TABLE]

with initial data $\varphi_{0,j}$.

It follows from the maximum principle [Tô16, Proposition 1.5] that $j\mapsto\varphi_{j,t}$ is non-increasing. Therefore we can set

[TABLE]

Thanks to Lemma 2.3 the function $t\mapsto\sup_{X}\varphi_{t,j}$ is uniformly bounded, hence $\varphi_{t}$ is a well-defined $\theta_{t}$-psh function. Moreover, it follows from Theorem 2.12 that $\varphi_{t}$ is also smooth in $(0,T]\times X$ and satisfies

[TABLE]

Observe that $(\varphi_{t})$ is relatively compact in $L^{1}(X)$ as $t\rightarrow 0^{+}$, we now show that $\varphi_{t}\rightarrow\varphi_{0}$ in $L^{1}(X)$ as $t\searrow 0^{+}$.

First, let $\varphi_{t_{k}}$ is a subsequence of $(\varphi_{t})$ such that $\varphi_{t_{k}}$ converges to some function $\psi$ in $L^{1}(X)$ as $t_{k}\rightarrow 0^{+}$. By the properties of plurisubharmonic functions, for all $z\in X$

[TABLE]

with equality almost everywhere. We infer that for almost every $z\in X$

[TABLE]

by continuity of $\varphi_{t,j}$ at $t=0$. Thus $\psi\leq\varphi_{0}$ almost everywhere.

Moreover, it follows from Lemma 2.3 that

[TABLE]

with $u$ continuous, so

[TABLE]

Since $\psi\leq\varphi_{0}$ almost everywhere, we get $\psi=\varphi_{0}$ almost everywhere, so $\varphi_{t}\rightarrow\varphi_{0}$ in $L^{1}$.

3.2. Uniform convergence

If the initial condition $\varphi_{0}$ is continuous then by [Tô16, Proposition 1.5] we infer that $\varphi_{t}\in C^{0}([0,T]\times X)$, hence $\varphi_{t}$ uniformly converges to $\varphi_{0}$ as $t\rightarrow 0^{+}$.

3.3. Uniqueness and stability of solution

We now study the uniqueness and stability for the complex Monge-Ampère flow

[TABLE]

where $F(t,z,s)\in C^{\infty}([0,T]\times X\times{\mathbb{R}},{\mathbb{R}})$ satisfies

[TABLE]

for some constant $C^{\prime}>0$.

The uniqueness of solution follows directly from the same result in the Kähler setting [Tô16]

Theorem 3.1**.**

Suppose $\psi$ and $\varphi$ are two solutions of (3.1) with $\varphi_{0}\leq\psi_{0}$, then $\varphi_{t}\leq\psi_{t}.$ In particular, the equation (3.1) has a unique solution.

The stability result also comes from the same argument as in [Tô16]. The difference is that we use Theorem 2.6 instead of the one in Kähler manifolds.

Theorem 3.2**.**

Fix $\varepsilon>0$. Let $\varphi_{0,j}$ be a sequence of $\omega$-psh functions with zero Lelong number at all points, such that $\varphi_{0,j}\rightarrow\varphi_{0}$ in $L^{1}(X)$. Denote by $\varphi_{t,j}$ and $\varphi_{j}$ the solutions of (3.1) with the initial condition $\varphi_{0,j}$ and $\varphi_{0}$ respectively. Then

[TABLE]

Moreover, if $\varphi,\psi\in C^{\infty}((0,T]\times X)$ are solutions of $(CMAF)$ with continuous initial data $\varphi_{0}$ and $\psi_{0}$, then

[TABLE]

Proof.

We use the techniques in Section 2 to obtain estimates of $\varphi_{t,j}$ in $C^{k}([\varepsilon,T]\times X)$ for all $k\geq 0$. In particular, for the $C^{0}$ estimate, we need to have uniform bound for $H_{t,j}=\exp(\dot{\varphi}_{t,j}+F)$ in order to use Theorem 2.6. By Lemma 2.5 we have

[TABLE]

where $C,C^{\prime}$ depend on $\varepsilon,\sup_{X}{\varphi_{0,j}}$. Since $\varphi_{0,j}$ converges to $\varphi_{0}$ in $L^{1}$, we have the $\sup_{X}\varphi_{0,j}$ is uniformly bounded in term of $\sup_{X}\varphi_{0}$ for all $j$ by the Hartogs lemma, so we can choose $C,C^{\prime}$ independently of $j$. It follows from [DK01, Theorem 0.2 (2)] that there is a constant $A(t,\varepsilon)$ depending on $t$ and $\varepsilon$ such that $||H_{t,j}||_{L^{2}(X)}$ is uniformly bounded by $A(t,\varepsilon)$ for all $t\in[\varepsilon,T]$. The rest of the proof is now siminar to [Tô16, Theorem 4.3]. ∎

4. Chern-Ricci flow and canonical surgical contraction

In this section, we give a proof of the conjecture of Tosatti and Weinkove. Let $(X,\omega_{0})$ be a Hermitian manifold. Consider the Chern-Ricci flow on $X$,

[TABLE]

Denote

[TABLE]

where $\hat{\omega}_{X}=\omega_{0}+t\chi,$ with $\chi$ is a smooth $(1,1)$-form representing $-c_{1}^{BC}(X)$.

Assume that there exists a holomorphic map between compact Hermitian manifolds $\pi:X\rightarrow Y$ blowing down an exceptional divisor $E$ on $X$ to one point $y_{0}\in Y$. In addition, assume that there exists a smooth function $\rho$ on $X$ such that

[TABLE]

with $T<+\infty$, where $\omega_{Y}$ is a Hermitian metric on $Y$. In [TW15, TW13], Tosatti and Weinkove proved that the solution $\omega_{t}$ to the Chern-Ricci flow (4.1) converges in $C^{\infty}_{loc}(X\setminus E)$ to a smooth Hermitian metric $\omega_{T}$ on $X\setminus E$. Moreover, there exists a distance function $d_{T}$ on $Y$ such that $(Y,d_{T})$ is a compact metric space and $(X,g(t))$ converges in the Gromov-Hausdorff sense $(Y,d_{T})$ as $t\rightarrow T^{-}$. Denote by $\omega^{\prime}$ the push-down of the current $\omega_{T}$ to $Y$. They conjectured that:

Conjecture 4.1**.**

[TW13, Page 2120]**

1. (1)

There exists a smooth maximal solution $\omega_{t}$ of the Chern-Ricci flow on $Y$ for $t\in(T,T_{Y})$ with $T<T_{Y}\leq+\infty$ such that $\omega_{t}$ converges to $\omega^{\prime}$, as $t\rightarrow T^{+}$, in $C^{\infty}_{loc}(Y\setminus\{y_{0}\})$. Furthermore, $\omega_{t}$ is uniquely determined by $\omega_{0}$. 2. (2)

The metric space $(Y,\omega_{t})$ converges to $(Y,d_{T})$ as $t\rightarrow T^{+}$ in the Gromov-Hausdorff sense.

We now prove this conjecture using Theorem A, Theorem B and some arguments in [SW13, TW13].

4.1. Continuing the Chern-Ricci flow

We prove the first claim in the conjecture showing how to continue the Chern-Ricci flow.

Write $\hat{\omega}=\pi^{*}\omega_{Y}=\omega_{0}-TRic(\omega_{0})+dd^{c}\rho$. Then there is a positive $(1,1)$-current $\omega_{T}=\hat{\omega}+dd^{c}\varphi_{T}$ for some bounded function $\varphi_{T}$. By the same argument in [SW13, Lemma 5.1] we have

[TABLE]

Hence there exists a bounded function $\phi_{T}$ on $Y$ that is smooth on $Y\setminus\{y_{0}\}$ with $\varphi_{T}=\pi^{*}\phi_{T}$.

We now define a positive current $\omega^{\prime}$ on $Y$ by

[TABLE]

which is the push-down of the current $\omega_{T}$ to $Y$ and is smooth on $Y\setminus\{y_{0}\}$. By the same argument in [SW13, Lemma 5.2] we have $\omega^{\prime n}/\omega_{Y}^{n}\in L^{p}(Y)$. It follows from [DK12, Theorem 5.2] that $\phi_{T}$ is continuous.

We fix a smooth $(1,1)$ form $\chi\in-c_{1}^{BC}(Y)$ and a smooth volume form $\Omega_{Y}$ such that $\chi=dd^{c}\log\Omega_{Y}$. Denote

[TABLE]

Fix $T^{\prime}\in(T,T_{Y})$, we have:

Theorem 4.2**.**

There is a unique smooth family of Hermitian metrics $(\omega_{t})_{T<t\leq T^{\prime}}$ on $Y$ satisfying the Chern-Ricci flow

[TABLE]

with $\omega_{t}=\omega_{Y}+(t-T)\chi+dd^{c}\phi_{t}$. Moreover, $\phi_{t}$ uniformly converges to $\phi_{T}$ as $t\rightarrow T^{+}$.

Proof.

We can rewrite the flow as the following complex Monge-Ampère flow

[TABLE]

where $\hat{\omega}_{Y}:=\omega_{Y}+(t-T)\chi$ and $\phi_{T}$ is continuous.

It follows from Theorem A and Theorem B that there is a unique solution $\phi$ of (4.4) in $C^{\infty}((T,T^{\prime}]\times Y)$ such that $\phi_{t}$ uniformly converges to $\phi_{T}$ as $t\rightarrow T^{+}$. ∎

4.2. Backward convergences

Once the Chern-Ricci flow can be run from $\omega^{\prime}$ on $Y$, we can prove the rest of Conjecture 4.1 following the idea in [SW13, Section 6].

We keep the notation as in [TW15]. Let $h$ be a Hermitian metric on the fibers of the line bundle $[E]$ associated to the divisor $E$, such that for $\varepsilon_{0}>0$ sufficiently small, we have

[TABLE]

Take $s$ a holomorphic section of $[E]$ vanishing along $E$ to order 1. We fix a a coordinate chart $U$ centered at $y_{0}$, which identities with the unit ball $B\subset{\mathbb{C}}^{n}$ via coordinates $z_{1},\ldots,z_{n}$. Then the function $|s|^{2}_{h}$ on $X$ is given on $\pi^{-1}\left(B(0,1/2)\right)$ by

[TABLE]

Hence, the curvature $R(h)$ of $h$ is given by

[TABLE]

The crucial ingredient of the proof of the conjecture is the following proposition:

Proposition 4.3**.**

The solution $\omega_{t}$ of (4.3) is in $C^{\infty}([T,T^{\prime}]\times Y\setminus\{y_{0}\})$ and there exists $\eta>0$ and a uniform constant $C>0$ such that for $t\in[T,T^{\prime}]$

1. (1)

$\omega_{t}\leq C\frac{\omega_{Y}}{\pi_{*}|s|^{2}_{h}},$** 2. (2)

$\omega_{t}\leq C\pi_{*}\left(\frac{\omega_{0}}{|s|^{2(1-\eta)}}\right)$.

In order to prove this propositon, we use the method in [SW13] to construct a smooth approximant of the solution $\phi_{t}$ of (4.4). Denote by $f_{\varepsilon}$ a family of positive smooth functions $f_{\varepsilon}$ on $Y$ such that it has the form

[TABLE]

on $B$, hence $f_{\varepsilon}(z)\rightarrow f(z)=r^{2(n-1)}$ as $\varepsilon\rightarrow 0$. Moreover, there is a smooth volume form $\Omega_{X}$ on $X$ with $\pi^{*}\Omega_{Y}=(\pi^{*}f)\Omega_{X}$.

Observe that $\hat{\omega}_{Y}(t)-\frac{\varepsilon}{T}\omega_{Y}$ is Hermitian on $Y$ for $t\in[T,T^{\prime}]$ if $\varepsilon$ is sufficiently small. Therefore

[TABLE]

is Hermitian for $\varepsilon>0$ sufficiently small.

We denote $\psi^{\varepsilon}_{t}$ the unique smooth solution of the following Monge-Ampère flow on $X$:

[TABLE]

Define Kähler metrics $\omega^{\varepsilon}$ on $[T,T^{\prime}]\times X$ by

[TABLE]

then

[TABLE]

where $\eta=-dd^{c}\log((\pi^{*}f_{\varepsilon})\Omega_{X})+\pi^{*}\chi=-dd^{c}\log(\pi^{*}f_{\varepsilon})\Omega_{X})+dd^{c}\log((\pi^{*}f)\Omega_{X}).$

We claim that $\pi_{*}\psi^{\varepsilon}_{t}$ converges to the solution $\phi_{t}$ of the equation (4.4) in $C^{\infty}([T,T^{\prime}]\times Y\setminus\{y_{0}\})$, then $\pi_{*}\omega^{\varepsilon}$ smoothly converges to $\omega_{t}$ on $[T,T^{\prime}]\times Y\setminus\{y_{0}\}$.

Lemma 4.4**.**

There exists $C>0$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ such that on $[T,T^{\prime}]\times X$ we have

1. (i)

$\eta\leq C\omega_{0}$, 2. (ii)

$Osc_{X}\psi^{\varepsilon}\leq C$; 3. (iii)

$\frac{(\omega^{\varepsilon})^{n}}{\Omega_{X}}\leq C.$**

Proof.

By straightforward calculation, in $\pi^{-1}(B(0,1/2))$, we have

[TABLE]

for some constant $C>0$. This proves (i). Using the same argument in Section 2 (see Theorem 2.8) we get (ii). Finally, the estimate (iii) follows from the same proof for the Kähler-Ricci flow (cf. [SW13, Lemma 6.2]) ∎

Two following lemmas are essential to prove Proposition 4.3.

Lemma 4.5**.**

There exists $\eta>0$ and a uniform constant $C>0$ such that

[TABLE]

Proof.

We first denote by $C>0$ a uniform constant which is independent of $\varepsilon$. Set $\hat{\omega}=\pi^{*}\omega_{Y}$ and fix $\delta>0$ a small constant. Following the same method in [TW13, Lemma 3.4] (see [PS10] for the original idea), we consider

[TABLE]

where $\tilde{\psi}^{\varepsilon}:=\psi^{\varepsilon}-\frac{1+\delta}{A}\log|s|^{2}_{h}$ and $C_{0}>0$ satisfies $\tilde{\psi}^{\varepsilon}+C_{0}\geq 1$.

It follows from [TW13, (3.17)] and [SW13, Lemma 2.4] that

[TABLE]

hence $|s|^{2}_{h}\operatorname{tr}_{\hat{\omega}}\omega^{\varepsilon}\leq C\operatorname{tr}_{\omega_{0}}\omega^{\varepsilon}$. Therefore $H_{\delta}$ goes to negative infinity as $x$ tends to $E$. Suppose that $H_{\delta}$ attains its maximum at $(t_{0},x_{0})\in[T,T^{\prime}]\times X\setminus E$. Without loss of generality, we assume that $\operatorname{tr}_{\hat{\omega}}\omega^{\varepsilon}\geq 1$ at $(t_{0},x_{0})$.

The condition (4.2) implies that $\pi^{*}\omega_{Y}-\omega_{0}$ is a $d$-closed form, so that $d\omega_{0}=\pi^{*}(d\omega_{Y})$. Therefore we have

[TABLE]

The condition (4.2) moreover implies that

[TABLE]

where $\beta(t)=(1-\varepsilon/T)(-TRic(\omega_{0})+dd^{c}\psi)+(t-T)\pi^{*}\chi$ is a closed $(1,1)$-form.

Combining (4.12), (4.11) and the calculation of [TW15, Proposition 3.1], at $(t_{0},x_{0})$ we get

[TABLE]

where $\hat{T}:=\pi^{*}T_{Y}$.

It follows from Lemma 4.4 and (4.10) that

[TABLE]

Moereover, we may assume without loss of generality that

[TABLE]

for some uniform constant $C^{\prime}$, since otherwise $H_{\delta}$ is already uniformly bounded. Therefore, we get

[TABLE]

Since at $(t_{0},z_{0})$ we have $\nabla H_{\delta}(t_{0},x_{0})=0$,

[TABLE]

hence

[TABLE]

We also have

[TABLE]

Combining all inequalities above and Lemma 4.4 (iii), at $(t_{0},x_{0})$, we obtain

[TABLE]

Since $\pi^{*}\omega_{Y}-\varepsilon_{0}R_{h}>0$ , we have

[TABLE]

for $A$ sufficiently large. Combining with $\operatorname{tr}_{\omega^{\varepsilon}}\hat{\omega}\leq C\operatorname{tr}_{\omega^{\varepsilon}}\omega_{0}$, we can choose $A$ sufficiently large so that at $(t_{0},x_{0})$

[TABLE]

Therefore, at $(t_{0},x_{0})$

[TABLE]

Since $\omega^{\varepsilon}/\Omega_{X}\leq C$ (Lemma 4.4 (iii)) and $x\rightarrow x\log|x|^{n-1}$ is bounded from above for $x$ close to zero, we get

[TABLE]

This implies that $H_{\delta}$ is uniformly bounded from above at its maximum. Hence we obtain the estimate (4.9). ∎

Lemma 4.6**.**

There exists a uniform $\lambda>0$ and $C>0$ such that

[TABLE]

Proof.

Following the method in [TW13, Lemma 3.5] (see also [PS10]), we consider for each $\delta>0$,

[TABLE]

where $\tilde{\varphi}^{\varepsilon}:=-\log(\operatorname{tr}_{\hat{\omega}}\omega^{\varepsilon}|s|^{2(1+\delta)})+A\tilde{\psi}^{\varepsilon}$ and $\tilde{C}$ is chosen so that $\tilde{\varphi}^{\varepsilon}+\tilde{C}>1$ and $\tilde{\psi}^{\varepsilon}+\tilde{C}>1$. The constant $A>0$ will be chosen hereafter. Lemma 4.5 and Lemma 4.4 (ii) imply that $H_{\delta}$ goes to negative infinity as $x$ tends to $E$. Hence we can assume that $H_{\delta}$ attains its maximum at $(t_{0},x_{0})\in[T,T^{\prime}]\times X\setminus E$. Without loss of generality, let’s assume further that $\operatorname{tr}_{\omega_{0}}\omega^{\varepsilon}\geq 1$ at $(t_{0},x_{0})$.

As in Lemma 4.5 we have

[TABLE]

and

[TABLE]

It follows from Lemma 4.4 and (4.10) that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

For the last term, we may assume without of generality that

[TABLE]

since otherwise $H_{\delta}$ is already uniformly bounded. Using

[TABLE]

and $\operatorname{tr}_{\omega_{0}}\omega^{\varepsilon}\geq 1$ at $(t_{0},x_{0})$, we get

[TABLE]

for $A>n-1$.

It follows from (4.5) that

[TABLE]

for all $A$ sufficiently large. Therefore we can choose $A$ sufficiently large such that

[TABLE]

Compute at $(t_{0},x_{0})$, using (4.16), (4.17), $\tilde{\psi}^{\varepsilon}+\tilde{C}\geq 1$ and $\tilde{\varphi}^{\varepsilon}+\tilde{C}\geq 1$,

[TABLE]

for $B$ is a constant in $[A^{2},A^{2}+A+1]$.

By the same the argument in [TW13, Lemma 3.5], we get, at $(t_{0},x_{0})$,

[TABLE]

As in the proof of Lemma 4.5, we infer that $\operatorname{tr}_{\omega_{0}}\omega^{\varepsilon}$ is bounded from above at $(t_{0},x_{0})$. Therefore, it follows from Lemma 4.4 and Lemma 4.5 that $H_{\delta}$ is bounded from above uniformly in $\delta$. Let $\delta\rightarrow 0$, we get

[TABLE]

Since $\operatorname{tr}_{\omega_{0}}\omega^{\varepsilon}\leq C\operatorname{tr}_{\hat{\omega}}\omega^{\varepsilon}$, we have

[TABLE]

and the desired inequality follows with $\lambda=1/(A+1)>0$. ∎

Proof of Proposition 4.3.

On $Y$, the function $\phi^{\varepsilon}:=\pi_{*}\psi^{\varepsilon}_{t}$ satisfies

[TABLE]

Since $\alpha^{\varepsilon}_{t}=\hat{\omega}_{Y}+\frac{\varepsilon}{T}(\omega_{Y}-\pi_{*}\omega_{0})$ is uniformly equivalent to $\hat{\omega}_{Y}$ for all $\varepsilon\in[0,\varepsilon_{0}]$ and $t\in[T,T^{\prime}]$, we can follow the same argument as in Section 2 to obtain the $C^{k}$-estimates for $\phi^{\varepsilon}_{t}$ which are independent of $\varepsilon$, for all $t\in(T,T^{\prime}]$. By Azela-Ascoli theorem, after extracting a subsequence, we can assume that $\phi^{\varepsilon}$ converges to $\tilde{\phi}$, as $\varepsilon\rightarrow 0^{+}$, in $C^{\infty}([\delta,T^{\prime}]\times Y)$ for all $\delta\in(T,T^{\prime})$. Moreover $\tilde{\phi_{t}}$ uniformly converges to $\phi_{T}$, hence $\tilde{\phi}$ satisfies (4.4). Thanks to Theorem 3.2, $\tilde{\phi}$ is equal to the solution $\phi$ of (4.4). Using Lemma 4.6 and the standard local parabolic theory, we obtain the $C^{\infty}$ estimates $\omega^{\varepsilon}$ on compact sets away from $E$. Hence $\phi$ is the smooth solution of 4.1 on $[T,T^{\prime}]\times Y\setminus\{y_{0}\}$. Finally, Proposition 4.3 follows directly from Lemma 4.5 and Lemma 4.6. ∎

Finally, we get the following:

Theorem 4.7**.**

The solution $\omega_{t}$ of (4.3) smoothly converges to $\omega^{\prime}$, as $t\rightarrow T^{+}$, in $C_{loc}^{\infty}(Y\setminus\{y_{0}\})$ and $(Y,\omega_{t})$ converges in the Gromov-Hausdorff sense to $(Y,d_{T})$ as $t\rightarrow T^{+}$.

Proof.

It follows from the proof of Proposition 4.3 that $\phi\in C^{\infty}([T,T^{\prime}]\times Y\setminus\{y_{0}\})$, hence $\omega_{t}$ smoothly converges to $\omega|_{t=T}=\omega^{\prime}$ in $C^{\infty}_{loc}(Y\setminus\{y_{0}\})$.

Denote by $d_{\omega_{t}}$ the metric induced from $\omega_{t}$ and $S_{r}$ the $2n-1$ sphere of radius $r$ in $B$ centered at the origin. Then it follows from Lemma 4.5 and the argument of [SW13, Lemma 2.7(i)] that:

(a) There exists a uniform constant $C$ such that

[TABLE]

Following the same argument of [SW13, Lemma 2.7 (ii)], we have

(b) For any $z\in B(0,\frac{1}{2})\setminus\{0\}$, the length of a radial path $\gamma(s)=sz$ for $s\in(0,1]$ with respect to $\omega_{t}$ is uniformly bounded from above by $C|z|^{\lambda}$, where $C$ is a uniformly constant and $\lambda$ as in Lemma 4.6.

Given (a) and (b), the Gromov-Hausdorff convergence follows exactly as in [SW13, Section 3]. This completes the proof of Theorem 4.7 and Conjecture 4.1. ∎

5. Twisted Chern-Ricci flow

5.1. Maximal existence time for the twisted Chern-Ricci flow

Let $(X,g)$ be a compact Hermitian manifold of complex dimension $n$. We define here the twisted Chern-Ricci flow on $X$ as

[TABLE]

where $\eta$ is a smooth $(1,1)$-form. Set $\hat{\omega}_{t}=\omega_{0}+t\eta-tRic(\omega_{0})$. We now define

[TABLE]

We now prove the following theorem generalizing the same result due to Tosatti-Weinkove [TW15, Theorem 1.2]. We remark that our ingredients for the proof come from a priori estimates proved in Section 2 which are different from the approach of Tosatti and Weinkove.

Theorem 5.1**.**

There exists a unique maximal solution to the twisted Chern-Ricci flow on $[0,T)$.

Proof.

Fix $T^{\prime}<T$. We show that there exists a solution of (5.1) on $[0,T^{\prime}]$. First we prove that the twisted Chern-Ricci flow is equivalent to a Monge-Ampère flow. Indeed, consider the following Monge-Ampère flow

[TABLE]

If $\varphi$ solves (5.2) on $[0.T^{\prime}]$ then taking $\omega_{t}:=\hat{\omega}_{t}+dd^{c}\varphi$, we get

[TABLE]

hence

[TABLE]

Conversely, if $\omega_{t}$ solves (5.1) on $[0,T^{\prime}]$, then we get

[TABLE]

Therefore if $\varphi$ satisfies

[TABLE]

so $\omega_{t}=\hat{\omega}_{t}+dd^{c}\varphi$ and $\varphi$ satisfies (5.2).

By the standard parabolic theory [Lie96], there exists a maximal solution of $(\ref{MAF of CRF})$ on some time interval $[0,T_{\max})$ with $0<T_{\max}\leq\infty$. We may assume without loss of generality that $T_{\max}<T^{\prime}$. We now show that a solution of $(\ref{MAF of CRF})$ exists beyond $T_{\max}$. Indeed, the a priori estimates for more general Monge-Ampère flows in Section 2 gives us uniform estimates for $\varphi$ in $[0,T_{\max})$ (see Theorem 2.12), so we get a solution on $[0,T_{\max}]$. By the short time existence theory the flow (5.2) can go beyond $T_{\max}$, this gives a contradiction. So the twisted Chern-Ricci flow has a solution in $[0,T)$. Finally, the uniqueness of solution follows from Theorem 3.1. ∎

5.2. Twisted Einstein metric on Hermitian manifolds

We fix a smooth $(1,1)$-form $\eta$. A solution of the equation

[TABLE]

with $\mu=1$ or $-1$, is called a twisted Einstein metric. We recall

[TABLE]

the equivalence class of $\eta$.

In the sequel we study the convergence of the normalized twisted Chern-Ricci flow to a twisted Einstein metric $\omega=\chi+dd^{c}\varphi\in-(c_{1}^{BC}-\{\eta\})$ assuming that $c_{1}^{BC}-\{\eta\}<0$ and $\mu=-1$. Note that if $c_{1}^{BC}(X)<0$ (resp. $c_{1}^{BC}(X)>0$) implies that $X$ is a Kähler manifold which admits a Kähler metric in $-c_{1}(X)$ (resp. in $c_{1}(X)$). Therefore the positivity of the twisted Bott-Chern class is somehow more natural in our context.

Assume the twisted first Bott-Chern class $\alpha:=c_{1}^{BC}(X)-\{\eta\}$ is negative. We now use a result in elliptic Monge-Ampère equation due to Cherrier [Che87] to prove the existence of twisted Einstein metric. An alternative proof using the convergence of the twisted Chern-Ricci flow will be given in Theorem 5.3.

Theorem 5.2**.**

There exists a unique twisted Einstein metric in $-\alpha$ satisfying (5.3):

[TABLE]

Proof.

Let $\chi=\eta-Ric(\Omega)$ be a Hermitian metric in $\alpha$, then any Hermitian metric in $\alpha$ can be written as $\omega=\chi+dd^{c}\varphi$ where $\varphi$ is smooth strictly and $\chi$-psh. Since

[TABLE]

we get

[TABLE]

Therefore the equation (5.4) can be written as the following Monge-Ampère equation

[TABLE]

It follows from [Che87] that (5.5) admits an unique smooth $\chi$-psh solution, therefore there exists an unique twisted Einstein metric in $-(c_{1}^{BC}(X)-\{\eta\})$. ∎

5.3. Convergence of the flow when $c_{1}^{BC}(X)-\{\eta\}<0$

We defined the normalized twisted Chern-Ricci flow as follows

[TABLE]

We have (5.6) is equivalent to the following Monge-Ampère flow

[TABLE]

where $\hat{\omega}_{t}=e^{-t}+(1-e^{-t})\left(\eta-Ric(\Omega)\right)$ and $\Omega$ is a fixed smooth volume form on $X$. Since we assume $c_{1}^{BC}(X)-\{\eta\}$ is negative, the flow (5.6) has a longtime solution. The longtime behavior of (5.6) is as follows

Theorem 5.3**.**

Suppose $c_{1}(X)-\{\eta\}<0$. Then the normalized twisted Chern-Ricci flow starting from any initial Hermitian metric $\omega_{0}$ smoothly converges, as $t\rightarrow+\infty$, to a twisted Einstein Hermitian metric $\omega_{\infty}=\eta-Ric(\Omega)+dd^{c}\varphi_{\infty}$ which satisfies

[TABLE]

Proof.

We now derive the uniform estimates for the solution $\varphi$ of the following Monge-Ampère

[TABLE]

where $\hat{\omega}_{t}:=e^{-t}\omega_{0}+(1-e^{-t})\chi$, and $\chi=\eta-Ric(\Omega)>0$.

The $C^{0}$-estimates for $\varphi$ and $\dot{\varphi}$ follow from the same arguments as in [Cao85, TZ06, Tsu88] for Kähler-Ricci flow (see [TW15] for the same estimates for the Chern-Ricci flow). Moreover, since

[TABLE]

and

[TABLE]

therefore

[TABLE]

The maximum principle follows that $(e^{t}-1)\dot{\varphi}-\varphi-nt\leq C$, hence

[TABLE]

For the second order estimate, we follow the method of Tosatti and Weinkove [TW15, Lemma 4.1 (iii)] in which they have used a technical trick due to Phong and Sturn [PS10].

Lemma 5.4**.**

There exists uniform constant $C>1$ such that

[TABLE]

Proof.

Since $\varphi$ is uniformly bounded, we can choose $C_{0}$ such that $\varphi+C_{0}\geq 1$. Set

[TABLE]

where $A>0$ will be chosen hereafter. The idea of adding the third term in $P$ is due to Phong-Sturn [PS10] and was used in the context of Chern-Ricci flow (cf. [TW15], [TW13],[TWY15]).

Assume without loss of generality that $\operatorname{tr}_{\hat{\omega}}\omega_{t}\geq 1$ at a maximum point $(t_{0},x_{0})$ with $t_{0}>0$ of $P$. It follows from the same calculation in Lemma 4.5 that at $(t_{0},x_{0})$, we have

[TABLE]

where $C_{1}>0$ satisfies $\eta\leq C_{1}\hat{\omega}$.

Now at a maximum point $(t_{0},x_{0})$ with $t_{0}>0$ we have $\nabla P=0$, hence

[TABLE]

Therefore

[TABLE]

Moreover, we have

[TABLE]

Combining these inequalities, at $(t_{0},z_{0})$ we have

[TABLE]

We can choose $A$ sufficiently large such that at the maximum of $P$ either $\operatorname{tr}_{\hat{\omega}}\omega_{t}\leq A^{2}(\varphi+C_{0})^{3},$ then we are done, or $\operatorname{tr}_{\hat{\omega}}\omega_{t}\geq A^{2}(\varphi+C_{0})^{3},$ and $A\geq 2C$. For the second case, we obtain at the maximum of $P$, there exists a uniform constant $C_{3}>0$ so that

[TABLE]

Hence combining with the following inequality (see for instance [BG13, Lemma 4.1.1])

[TABLE]

we have

[TABLE]

This implies that $P$ is bounded from above at its maximum, so we complete the proof of the lemma. ∎

It follows from Lemma 5.4 that $\omega_{t}$ is uniformly equivalent to $\hat{\omega}$ independent of $t$, hence

[TABLE]

hence $\dot{\varphi}\geq-C(1+t)e^{-t}$ by the maximum principle. Combining with (5.7), we infer that $\varphi$ converges uniformly exponentially fast to a continuous function $\varphi_{\infty}$. Moreover, by the same argument in Section 2, Evans-Krylov and Schauder estimates give us the uniform higher order estimates for $\varphi$. Therefore $\varphi_{\infty}$ is smooth and $\varphi_{t}$ converges to $\varphi_{\infty}$ in $C^{\infty}$.

Finally, we get the limiting metric $\omega_{\infty}=\eta-Ric(\Omega)+dd^{c}\varphi_{\infty}$ which satisfies the twisted Einstein equation

[TABLE]

This proves the existence of a twisted Einstein metric in $-c_{1}^{BC}(X)+\{\eta\}$. ∎

As an application, we prove the existence of a unique solution of the Monge-Ampère equation on Hermitian manifolds. This result was first proved by Cherrier [Che87, Théorème 1, p. 373].

Theorem 5.5**.**

Let $(X,\omega)$ be a Hermitian manifold, $\Omega$ be a smooth volume form on $X$. Then there exists a unique smooth $\omega$-psh fucntion $\varphi$ satisfying

[TABLE]

Proof.

Set $\eta=\omega+Ric(\Omega)$, then we have $c_{1}^{BC}(X)-\{\eta\}<0$. It follows from Theorem 5.3 that the twisted normalized Chern-Ricci flow

[TABLE]

admits unique solution which smoothly converges to a twisted Einstein Hermitian metric $\omega_{\infty}=\eta-Ric(\Omega)+dd^{c}\varphi_{\infty}=\omega+dd^{c}\varphi_{\infty}$ which satisfies $Ric(\omega_{\infty})=-\omega_{\infty}+\eta=Ric(\Omega)-dd^{c}\varphi_{\infty}$. Therefore $\varphi_{\infty}$ is a solution of the Monge-Ampère equation

[TABLE]

The uniqueness of solution follows from the comparison principle. ∎