The K\"ahler Ricci flow around complete bounded curvature K\"ahler metrics

TL;DR

This paper develops existence and stability results for the Kähler Ricci flow on complete manifolds with bounded curvature, including non-smooth and degenerate initial conditions, advancing understanding of geometric evolution in complex geometry.

Contribution

It introduces new existence time estimates for the Kähler Ricci flow starting from various initial conditions, including non-smooth and degenerate metrics.

Findings

Established existence of solutions with bounded curvature under minimal initial assumptions

Derived stability results for complex space forms under the flow

Extended flow results to non-smooth and degenerate initial data

Abstract

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow.

Equations

$$\left\{\begin{array}[]{ll}\displaystyle\frac{\partial g_{{i\bar{j}}}}{\partial t}=-R_{{i\bar{j}}}\\ g(0)=g_{0}.\end{array}\right.$$

$$S(c_1,c_2,h):=\{g_0:g_0\text{ is smooth and Ka¨hler on M and }{c}_{1}h\le g_0\le c_{2}h\}$$

$$\begin{array}{rl}{T}_{g_0}=\sup \{T:& \text{eqrefkrf has a solution g(t) on M×[0,T) which has}\\ & \text{bounded curvature and is equivalent to g0 for all t>0}\}\end{array}$$

$$\dot{\phi }=\log \frac{({\omega }_0-t\text{Ric}({\omega }_h)+\sqrt{-1}\partial \bar{\partial }\phi {)}^n}{{\omega }_h^n}$$

$$\phi (t)={\int }_0^t\log \frac{{\omega }^n(s)}{{\omega }_h^n}ds.$$

$$\omega (t)={\omega }_0-t\text{Ric}(h)+\sqrt{-1}\partial \bar{\partial }\phi .$$

$$\begin{array}{rl}\left(\frac{\partial }{\partial t}-\Delta \right)t{r}_{g}h& =-g^{i\bar{j}}{g}^{p\bar{q}}{h}_{k\bar{l}}{\Psi }_{pi}^k{\Psi }_{\bar{q}\bar{j}}^{\bar{l}}+g^{i\bar{j}}{g}^{p\bar{q}}{R}_{i\bar{j}k\bar{l}}^h;\\ \left(\frac{\partial }{\partial t}-\Delta \right)|\Psi {|}^2& =-|\nabla \Psi {|}^2-|\bar{\nabla }\Psi {|}^2-2\mathbf{Re}\left(g^{r\bar{s}}{g}^{i\bar{j}}{g}^{p\bar{q}}{g}_{k\bar{l}}{\Psi }_{\bar{j}\bar{q}}^{\bar{l}}{\nabla }_r{\hat{R}}_{i\bar{s}p}^k\right);\\ \left(\frac{\partial }{\partial t}-\Delta \right)|\text{Rm}{|}^2& \le -|\nabla \text{Rm}{|}^2-|\bar{\nabla }\text{Rm}{|}^2+C_n|\text{Rm}{|}^3.\end{array}$$

$$C_0^{-1}{\lambda }^{-1}h\le g(t)\le C_0{\lambda }^{c_1}h.$$

$$B_{g(t)}(p,1-{\beta }_n\sqrt{at})\subset B_{g_0}(p,1)$$

$$\left(\frac{\partial }{\partial t}-\Delta \right)\left[d_t(x,p)+c_1(n)\sqrt{at}\right]\ge 0$$

$$\frac{1}{C(n)\lambda }h\le g(t).$$

$$F(x,t)=\log {\mathrm{tr}}_{g}h+2\log \Phi -L\phi +nLt(\log t-2)$$

$$\omega (t)={\omega }_0-t\text{Ric}(h)+\sqrt{-1}\partial \bar{\partial }\phi \ge \frac{1}{2\lambda }{\omega }_h+\sqrt{-1}\partial \bar{\partial }\phi ,$$

$$\begin{array}{rl}\left(\frac{\partial }{\partial t}-\Delta \right)F& \le {\mathrm{tr}}_{g}h+\frac{C_n}{{\Phi }^2}+L\log \frac{\det h}{\det g}+L\Delta \phi +nL\log t-nL\\ & \le {\mathrm{tr}}_{g}h\cdot \left(1-\frac{1}{2\lambda }L\right)+nL\log (t{\mathrm{tr}}_{g}h)+\frac{C_n}{{\Phi }^2}\\ & \le -{\mathrm{tr}}_{g}h+\frac{C_n}{{\Phi }^2}\end{array}$$

$$F(x_0,t_0)\le C_n+aL{t}_0+nL{t}_0(\log t_0-2)\le \tilde{C}(n)$$

$$F(x,t)\le \max \{{\tilde{C}}_n,\log \lambda +\log n\}$$

$$\dot{\phi }=\log \frac{{\omega }^n(t)}{{\omega }_h^n}\le c(n)\log \lambda .$$

$$G=\tilde{\Phi }\frac{(\dot{\phi }{)}^2}{1+L\phi }$$

$$\begin{array}{rl}& \left(\frac{\partial }{\partial t}-\Delta \right)\frac{(\dot{\phi }{)}^2}{1+L\phi }\\ & =-\frac{L(\dot{\phi }{)}^2}{(1+L\phi {)}^2}\left(\dot{\phi }-n+{\mathrm{tr}}_g\hat{g}\right)-\frac{2L^2{\dot{\phi }}^2}{(1+L\phi {)}^3}{g}^{i\bar{j}}{\phi }_i{\phi }_{\bar{j}}\\ & -\frac{2\dot{\phi }}{1+L\phi }{\mathrm{tr}}_g(\text{Ric}(h))-\frac{2g^{i\bar{j}}{\dot{\phi }}_i{\dot{\phi }}_{\bar{j}}}{1+L\phi }+\frac{4L\dot{\phi }}{(1+L\phi {)}^2}\mathbf{Re}\left(g^{i\bar{j}}{\dot{\phi }}_i{\phi }_{\bar{j}}\right)\\ & \le -\frac{L(\dot{\phi }{)}^2}{(1+L\phi {)}^2}\left(\dot{\phi }-n+{\mathrm{tr}}_g\hat{g}\right)+\frac{2|\dot{\phi }|}{1+L\phi }{\mathrm{tr}}_{g}h\\ & \le -\frac{L{\dot{\phi }}^3}{(1+L\phi {)}^2}+\frac{Ln{\dot{\phi }}^2}{(1+L\phi {)}^2}+\left[-\frac{L{\dot{\phi }}^2}{2\lambda (1+L\phi {)}^2}+\frac{2|\dot{\phi }|}{1+L\phi }\right]{\mathrm{tr}}_{g}h.\end{array}$$

$$\dot{\phi }=\log \frac{\det g}{\det h}\ge -n\log (2n\lambda ).$$

$$\begin{array}{rl}0& \le \left(\frac{\partial }{\partial t}-\Delta \right)G\\ & \le -\frac{L{\dot{\phi }}^3}{(1+L\phi {)}^2}\tilde{\Phi }+\frac{Ln{\dot{\phi }}^2}{(1+L\phi {)}^2}\tilde{\Phi }+C_{n}G+2\frac{|\partial \tilde{\Phi }{|}^2}{{\tilde{\Phi }}^2}G\\ & \le -G^{3/2}{\tilde{\Phi }}^{-1/2}+C_{n}G+C_n{\tilde{\Phi }}^{-1/2}G\end{array}$$

$$\dot{\phi }\le C_n\log \lambda$$

$${\mathrm{tr}}_{h}g\le \frac{\det g}{\det h}({\mathrm{tr}}_{h}g{)}^{n-1}.$$

$$t|\text{Rm}(x,t)|+t^{3/2}|\nabla \text{Rm}|\le C_1.$$

$$\begin{array}{rl}\left(\frac{\partial }{\partial t}-\Delta \right)F& =\Phi Q+t\left[\Phi \left(\frac{\partial }{\partial t}-\Delta \right)Q+Q\left(\frac{\partial }{\partial t}-\Delta \right)\Phi \right]\\ & -2t\mathbf{Re}\left(g^{i\bar{j}}{\Phi }_i{Q}_{\bar{j}}\right)+L\left(\frac{\partial }{\partial t}-\Delta \right){\mathrm{tr}}_{g}h\\ & \le \Phi Q+t\Phi \left[-|D\Psi {|}^2+C_{2}Q\right]\\ & +100t\Phi Q+C_{n}t{Q}^{\frac{1}{2}}|D\Psi ||D\Phi |+L\left(-{\Lambda }^{-1}Q+C_2\right)\\ & \le Q\left(-L{\Lambda }^{-1}+C_3\right)+C_3\\ & \le -Q+C_3.\end{array}$$

$$tQ(x,t)$$

$$G=\frac{\tilde{\Phi }{t}^2|\text{Rm}{|}^2}{A-\tilde{F}}$$

$$\begin{array}{rl}\left(\frac{\partial }{\partial t}-\Delta \right)\tilde{F}& \le -t|\nabla \Psi {|}^2-t|\bar{\nabla }\Psi {|}^2+C_4.\end{array}$$

$$\begin{array}{rl}& \left(\frac{\partial }{\partial t}-\Delta \right)\left[t^2|\text{Rm}{|}^2(A-\tilde{F}{)}^{-1}\right]\\ & \le (A-\tilde{F}{)}^{-1}\left[C_n{t}^2|\text{Rm}{|}^3-t^2|\nabla \text{Rm}{|}^2-t^2|\bar{\nabla }\text{Rm}{|}^2+2t|\text{Rm}{|}^2\right]\\ & +t^2|\text{Rm}{|}^2(A-\tilde{F}{)}^{-2}\left[-t|\nabla \Psi {|}^2-t|\bar{\nabla }\Psi {|}^2+C_5\right]\\ & -2(A-\tilde{F}{)}^{-3}{t}^2|\text{Rm}{|}^2|\nabla \tilde{F}{|}^2-2t^2(A-\tilde{F}{)}^{-2}\mathbf{Re}\left(g^{i\bar{j}}|\text{Rm}{|}_i^2\cdot {\tilde{F}}_{\bar{j}}\right)\\ & \le -C_6^{-1}{t}^3|\text{Rm}{|}^4+C_6{t}^{-1}.\end{array}$$

$$\begin{array}{r}0\le \left(\frac{\partial }{\partial t}-\Delta \right)G\le C_7{\tilde{\Phi }}^{-1}G+C_7{t}^{-1}-C_7^{-1}\tilde{\Phi }{t}^3|\text{Rm}{|}^4\end{array}$$

Full text

Keywords: Kähler Ricci flow, complete non-compact Kähler manifolds.

1Research partially supported by NSERC grant no. #327637-06

1. Introduction

Let $M^{n}$ be a non-compact complex manifold. The Kähler Ricci flow on $M^{n}$ starting from an initial Kähler metric $g_{0}$ is the evolution equation

[TABLE]

By a solution to (1.1) we mean a smooth family of Kähler metrics $g(t)$ satisfying (1.1) on $M\times[0,T)$ for some $T>0$. A classical theorem of W.X. Shi [21] says that if $g_{0}$ is complete with bounded curvature then (1.1) has a solution $g(t)$ which has bounded curvature and is equivalent to $g_{0}$ for $t>0$. In this paper we show that the same result holds assuming only that $g_{0}$ is equivalent to a complete Kähler metric with bounded curvature. In particular, $g_{0}$ may have unbounded curvature.

There have been many other works on the existence of solutions to Kähler Ricci flow and real (Riemannian) Ricci flow when the initial metric $g_{0}$ has possibly unbounded curvature (see for example [2] [3], [8], [4], [5], [11], [16], [22], [13] and references therein). In particular, Simon [22] proved the real Ricci flow has a complete solution starting from any Riemannian metric $g_{0}$ inside an $\epsilon$ neighborhood of a complete bounded curvature Riemannian metric $h$ in the $C^{0}$ topology induced by $h$, provided $\epsilon$ is sufficiently small depending on $n$. A solution to Kähler Ricci flow is a special solution to the real Ricci flow, thus our result shows the smallness condition on $\epsilon$ can be removed in Simon’s theorem when $g_{0}$ and $h$ are smooth Kähler .

Fix a complete Kähler metric $h$ on $M$ with curvature bound $|Rm(h)|\leq K$. For $c_{2}>c_{1}>0$, define the following space of Kähler metrics

[TABLE]

and for each $g_{0}\in S(c_{1},c_{2},h)$, define

[TABLE]

We will refer to a solution as in the definition of $T_{g_{0}}$ as a bounded curvature solution equivalent to $g_{0}$. Then our main existence result for (1.1) can be stated as follows.

Theorem 1.1**.**

If $g_{0}\in S(1,c,h)$ then (1.1) has a bounded curvature solution $g(t)$ on $M\times[0,T_{h})$ which is equivalent to $g_{0}$. Moreover, there exists positive constants $a(n,c,K),T(n,c,K),C_{2}(n,c,K),C_{1}(n)>0$ such that for all $t\in[0,T)$ we have

1. (1)

$g(t)\in S((e^{-C_{1}Kt}),C_{2},h)$** 2. (2)

$\sup_{M}\|Rm(t)\|_{g(t)}\leq a/t$**

Remark 1.1*.*

The existence part also holds when the reference metric $h$ is only Hermtian (not Kähler) and with bounded torsion and Chern curvature in addition. We leave the verification of this to interested readers.

When $g_{0}$ is Hermitan but not Kähler on $M$ then by a smooth solution to (1.1) on $M\times(0,T)$ we mean a smooth family of Kähler metrics $g(t)$ on $M\times(0,T)$ solving the first equation in (1.1) such that $g(t)\to g_{0}$ pointwise on $M$ as $t\to 0$. In this sense, we may use Theorem 1.1 to solve (1.1) assuming $g_{0}$ is a $C^{2}_{loc}(M)$ limit of smooth Kähler metrics $g_{k}\in S(c,c_{k},h)$ for some $c,c_{k}>0$ where $c_{k}$ is not assumed uniformly bounded above. On the other hand, if $g_{0}$ is a $C^{0}_{loc}(M)$ limit of smooth Kähler metrics $g_{k}\in S(c,\tilde{c},h)$ for some $c,\tilde{c}>0$ we may have a solution to (1.1) satisfying the same conclusions as in Theorem 1.1.

Theorem 1.2**.**

We have the following

1. (1)

If $g_{0}\in Cl^{2}_{loc}\left(\bigcup_{c>0}S(1,c,h)\right)$, then there is $T(n,K),C_{1}(n)>0$ such that (1.1) has a solution on $M\times(0,T)$ in the sense above with $g(t)\geq e^{-C_{1}Kt}h$ on $(0,T]$. 2. (2)

If $g_{0}\in Cl^{0}_{loc}S(1,c,h)$ for some $c>1$, then (1.1) has a bounded curvature solution $g(t)$ on $M\times(0,T_{h})$ satisfying the same conclusions as in Theorem 1.1.

As another application, we have the following stability result for (1.1) around complex spaceforms. In the Riemannian case, the stability of spaceforms was first studied in [23, 24]. We show that in the Kähler category, we do not require the $\epsilon$-fairness required in those works.

Theorem 1.3**.**

Suppose $(M,h)$ is a complete noncompact Kähler manifold with constant holomorphic sectional curvature $H_{h}=2k\leq 0$. Then if $g_{0}\in S(c_{1},c_{2},h)$ for some $c_{2}>c_{1}>0$, then $T_{g_{0}}=+\infty$. Moreover,

1. (1)

If $k<0$, then $t^{-1}g(t)$ converges to $h$ in $C^{\infty}_{loc}$ as $t\rightarrow\infty$; 2. (2)

If $k=0$, then $g(t)$ converges sub-sequentially in $C^{\infty}_{loc}$ to a complete flat metric as $t\rightarrow\infty$.

On the other hand, if $g_{0}$ is assumed only degenerate Kähler, in other words the corresponding (1,1) form $\omega_{0}$ is closed and nonnegative but not necessarily positive, then under certain condtitions we may also produce a solution to (1.1) with exsitence time estimates in the sense of the following Theorem

Theorem 1.4**.**

Suppose $(M,\omega_{h})$ is a complete noncompact Kähler manifold with bounded curvature. If $\omega_{0}$ is a closed nonnegative $(1,1)$ form such that

1. (1)

$\omega_{h}\geq\omega_{0}$* on $M$;* 2. (2)

$\omega_{0}-s\text{\rm Ric}(\omega_{h})+s\sqrt{-1}\partial\bar{\partial}f>{\beta}\omega_{h}$* for some ${\beta},s>0$ and $f\in C^{\infty}(M)\cap L^{\infty}(M)$.*

Then there is a smooth solution to

[TABLE]

on $M\times(0,s)$ with $\varphi\rightarrow 0$ in $L^{\infty}(M)$ and $\omega(t)=\omega_{0}-t\text{\rm Ric}(\omega_{h})+\sqrt{-1}\partial\bar{\partial}\varphi$ is a solution to the Kähler-Ricci flow (1.1) on $M\times(0,s)$ which is uniformly equivalent to $\omega_{h}$ on $M$. Moreover $\varphi$ is smooth up to $t=0$ on $U=\{x:\omega_{0}>0\}$.

In particular, if $h$ is a Kähler-Einstein metric with bounded curvature and negative Einstein constant, then any Kähler metric bounded from above by some multiples of $h$ can be deformed to $h$ along the normalized Kähler-Ricci flow. When $n=1$, the existence of Ricci flow starting from incomplete metric has been studied in details by Giesen and Topping [6, 7, 8] where they do not require any boundedness on $\omega_{0}$. In contrast with [10, Theorem 1.1], we remove the boundedness of $|\nabla^{h}g_{0}|$ when $h$ has bounded curvature and $g_{0}$ is Kähler.

Our approach in proving Theorem 1.1 is similar in spirit to that taken in [16] (see also [25, 13]). We make use of an iterative process using the Chern-Ricci flow to produce a solution $g_{R}(t)$ to (1.1) on $B_{h}(p,R)\times[0,T)$ with $g_{R}(0)=g_{0}$. Estimates for $g_{R}(t)$ over a compact subset $S\subset\subset B_{h}(p,R)$ are then established which depend only on $n,c_{1},c_{2},K,S$ thus allowing us to let $R\to\infty$ to obtain a smooth limit solution $g(t)$ on $M\times[0,T)$ with the desired properties.

The paper is orgainzed as follows. In §2 we prove our main local estimates for Kähler Ricci flow. In §3 we recall some basic results for Chern Ricci flow from [16]. Then in §4, 5, 6, 7 we prove Theorems 1.1, 1.2, 1.3 and 1.4 respectively.

Acknowledgement: The authors would like to thank Luen-Fai Tam and Xiangwen Zhang for the interest in this work.

2. estimates for Kähler Ricci flow

In this section we establish basic estimates for solutions to (1.1). Here, $(M,g_{0})$ is a smooth Kähler manifold and $h$ is another smooth Kähler metric on $M$ with bisectional curvatures bounded by $K=1$. Theorem 2.1, Lemma 2.1 and Lemma 2.2 are purely local in nature and $B_{g_{0}}(p,r)$ there refers to an arbitrary ball of radius $r$ relative to $g_{0}$. Then in Proposition 2.1 we establish a global estimate for smooth solutions to (1.1) assuming a priori they are uniformly equivalent to $h$ for all $t$.

Given a solution $g(t)$ to (1.1) we will denote by $\omega(t)$ the corresponding family of Kähler forms. We will also let $\omega_{h}$ denote the Kähler form corresponding to $h$. We will let

[TABLE]

In particular, applying $\sqrt{-1}\partial\bar{\partial}$ to both sides and using (1.1) and the local formula for the Ricci tensor of a Kähler metric allows us to write

[TABLE]

We also recall the following evolution equations for (1.1) (see for example [28]). Here $\Psi_{ij}^{k}=\Gamma(g(t))_{ij}^{k}-\Gamma(h)_{ij}^{k}$.

[TABLE]

For notational convenience, we adopt the notation $a\wedge b=\min\{a,b\}$ for any real numbers $a,b$ below. The main estimate in this section is the following

Theorem 2.1**.**

For any $a,\lambda>1$, there is $\tilde{T}(n,a,\lambda)>0$ and $C_{0}(n),c_{1}(n)>1$ such that the following is true. Let $g(t)$ be a solution to the Kähler-Ricci flow on $B_{g_{0}}(p,1)\times[0,T]$ so that

1. (1)

$|\text{\rm Rm}(h)|\leq 1$* on $B_{g_{0}}(p,1)$;* 2. (2)

$\lambda^{-1}h\leq g_{0}\leq\lambda h$* on $B_{g_{0}}(p,1)$.* 3. (3)

$|\mathrm{Rm}(g(t))|\leq at^{-1}$* on $B_{g_{0}}(p,1)\times(0,T]$;* 4. (4)

$|\varphi|\leq at$* on $B_{g_{0}}(p,1)\times[0,T]$;*

Then for all $t\in[0,T\wedge\tilde{T}]$ we have $B_{t}(p,1/4)\subset B_{0}(p,1)$, and for all $x\in B_{t}(p,1/4)$ we have

[TABLE]

Proof.

We begin by constructing appropriate barrier functions for our arguments. By the shrinking ball lemma [26, Lemma 3.2], for $t\in[0,T]$

[TABLE]

for some constant ${\beta}_{n}$. By [19, Lemma 8.3] with $\displaystyle K=\frac{a}{t}$ and $\displaystyle r=\sqrt{\frac{t}{a}}$ there, we may infer that

[TABLE]

in the sense of barriers whenever $d_{t}(x,p)\geq\sqrt{a^{-1}t}$. Here $d_{t}(x,p)=dist_{g(t)}(x,p)$. Denote $\eta(x,t)=d_{t}(x,p)+\tilde{\beta}_{n}\sqrt{at}$ where $\tilde{\beta}_{n}\geq\max\{c_{1}(n),{\beta}_{n}\}$. Let $\phi$ be a smooth function on $[0,+\infty)$ so that $\phi\equiv 1$ on $[0,\frac{3}{4}]$, vanishes outside $[0,1]$ and satisfies $-100{\phi}^{3/4}\leq\phi^{\prime}\leq 0;\phi^{\prime\prime}\geq-100\phi$.

We will make use of the evolving barrier $\log\Phi(x,t):=\log\phi\left(\eta(x,t)\right)$. By (2.3) and (2.4), we may choose $\tilde{T}$ sufficiently small depending on $n,a$ such that for all $t\in T\wedge\tilde{T}$,

1. (1)

$Domain(\log\Phi(x,t))\subset B_{g_{0}}(p,1)$ 2. (2)

$B_{g(t)}(p,1/2)\subset\{x:\log\Phi(x,t)=0\}$ 3. (3)

$\left(\frac{\partial}{\partial t}-\Delta\right)(\log\Phi)\leq C_{n}/\Phi^{2}$

We will also make use of the cut off function $\widetilde{\Phi}(x,t)=\phi\left(2\eta(x,t)\right)$. Then similarly we may choose $\tilde{T}$ sufficiently small depending on $n,a$ such that for all $t\in T\wedge\tilde{T}$,

1. (1)

$B_{g(t)}(p,1/2)\subset Domain(\widetilde{\Phi}(x,t))\subset\{x:\log\Phi(x,t)\neq-\infty\}$ 2. (2)

$B_{g(t)}(p,1/4)\subset\{x:\widetilde{\Phi}(x,t)=1\}$ 3. (3)

$\left(\frac{\partial}{\partial t}-\Delta\right)(\widetilde{\Phi})\leq C_{n}\widetilde{\Phi}$ 4. (4)

$\frac{|\partial\widetilde{\Phi}|^{2}}{\widetilde{\Phi}^{2}}\leq C_{n}\widetilde{\Phi}^{-1/2}$

The function $d_{t}(x,p)$ and hence barrier functions above are in general only Lipschitz continuous on $M$, though by Calabi’s trick (see [26, Section 7] for a detailed exposition) we may assume these to be smooth, and the differential inequalitites above to hold in the usual sense, at a given point where we argue by the maximum principle below.

Claim 2.1**.**

If $\tilde{T}(n,a,\lambda)$ is sufficiently small, then for all $x\in B_{t}(p,1/2),\;t\leq T\wedge\tilde{T}$,

[TABLE]

proof of Claim 2.1.

Consider the function

[TABLE]

for times $t\in[0,T\wedge\tilde{T}\wedge(nL)^{-1}]$ where $\Phi,\tilde{T}$ is as above and $L=6\lambda$. By the properties of the barrier $\log\Phi$, $F(x,t)$ attains a maximum value at some point $(x_{0},t_{0})$ in its domain. If $t_{0}=0$, then the Claim immeditely follows from the definition of $F$ and the hypothesis of the Theorem. Now suppose $t_{0}>0$. Assuming that $\tilde{T}\leq 1/2\lambda$, we have

[TABLE]

and then the evolution equation of $\log\operatorname{tr}_{h}g$ from (2.2) gives the following at $(x_{0},t_{0})$

[TABLE]

Here we have assumed that $tr_{g}h(x_{0},t_{0})\geq 1$ (without loss of generality) in the second inequality, and we have used the elementary inequality $\log x\leq x$ for all $x>0$ and $L=6\lambda$ in the last inequality. By the maximum principle we then conclude that $\operatorname{tr}_{g}h\leq\frac{C_{n}}{\Phi^{2}}$ at $(x_{0},t_{0})$ and thus

[TABLE]

for some constant $\tilde{C}(n)$ provided we further shrink $\tilde{T}$ if necessary depending only on $n,a,\lambda$.

On the other hand, we also have $F(0)\leq\log\lambda+\log n$. Thus in summary, we conclude that for $t\in[0,T\wedge\tilde{T}\wedge(nL)^{-1}]$ we have

[TABLE]

and the claim follows from the defninition of $F$ and the properties of the barrier $\log\Phi$. ∎

To prove the Theorem, it suffices to obtain a local upper bound for the volume form of $g(t)$ in view of Claim 2.1. We do this in the following

Claim 2.2**.**

There exists $c(n),\tilde{T}(n,a,\lambda)>0$ such that for all $x\in B_{t}(p,1/4),\;t\leq T\wedge\tilde{T}$ we have

[TABLE]

proof of Claim 2.2.

Consider the function

[TABLE]

for $t\leq T\wedge\tilde{T}$ where $L>0$ is a constant to be chosen later. Due to the cutoff function, $G$ attains its maximum at some point $(x_{0},t_{0})$ with $t_{0}\leq T\wedge\tilde{T}$. If $t_{0}=0$, then the upper bound in the claim is trivial. Now suppose $t_{0}>0$. Assume $\tilde{T}\leq 1/(2\lambda)$ so that $\eqref{ref-ineq}$ gives $\hat{g}(t_{0}):=g_{0}-t_{0}\text{\rm Ric}(h)\leq h/(2\lambda)$. We calculate at $(x_{0},t_{0})$ that

[TABLE]

By Claim 2.1 and the properties of the cut off function $\widetilde{\Phi}$ we have the lower bound

[TABLE]

on the support of $G$. Therefore, we will assume that $\dot{\varphi}(x_{0},t_{0})\geq 1$ as the otherwise the Claim follows. We may then choose $L=16\lambda$ and assume $\tilde{T}$ is sufficiently small depending only on $n,a,\lambda$ so that the last term in (2.12) will be bounded above by [math], and moreover, using $\nabla G(x_{0},t_{0})=0$ and the properties of $\widetilde{\Phi}$, that at $(x_{0},t_{0})$

[TABLE]

which in turn implies $G(x_{0},t_{0})\leq C(n)$ for some constant $C(n)$. Thus $G(x,t)\leq C(n)(\log\lambda)^{2}$ on $M\times[0,T\wedge\tilde{T}]$, and by assuming $\tilde{T}$ is sufficiently smaller still depending on $n,a,\lambda$, by the properties of the cut off $\widetilde{\Phi}$ we may conclude that for $x\in B_{t}(p,1/4)$ and $t\leq T\wedge\tilde{T}$ we have

[TABLE]

∎

The Theorem follows from combining Claim 2.1 and Claim 2.2 together with the following elementary inequality

[TABLE]

∎

The next lemma shows that we have a local curvature estimate provided $g(t)$ stays uniformly equivalent to a good reference metric. In [27], Sherman-Weinkove showed essentially the same estimate but with less detail on the dependence of the various constants in the Lemma.

Lemma 2.1**.**

For any $a,\Lambda>1$, there is $C_{1}(n,\Lambda),\hat{T}(n,a,\Lambda)>0$ so that the following holds. Suppose $g(t)$ is a solution to the Kähler-Ricci flow on $B_{g_{0}}(p,1)\times[0,T]$ so that

1. (1)

$|\nabla Rm(h)|+|\text{\rm Rm}(h)|^{3/2}\leq 1$* on $B_{g_{0}}(p,1)$;* 2. (2)

$|\mathrm{Rm}(g(t))|\leq at^{-1}$* on $B_{g_{0}}(p,1)\times(0,T]$;* 3. (3)

$\Lambda^{-1}h\leq g(t)\leq\Lambda h$* on $B_{g_{0}}(p,1)$, $t\in[0,T]$.*

Then for all $x\in B_{t}(p,1/4),\;t\leq\hat{T}\wedge T$,

[TABLE]

Proof.

We will consider the cutoff functions $\Phi(x,t)=\phi(\eta(x,t))$ and $\tilde{\Phi}=\phi(2\eta(x,t))$ as in the proof of Theorem 2.1, and we may choose $\hat{T}(n,a,\Lambda)$ sufficiently small so that $\Phi$ and $\tilde{\Phi}$ satisfy the same conditions there for all $t\leq T\wedge\hat{T}$. The dependence of $a$ will in fact only appear in $\Phi$ and $\tilde{\Phi}$ and hence $\hat{T}$.

We follow the proof in [27]. We will use $C_{i}$ to denote constants depending only on $n,\Lambda$ but not $a$.

Denote $\Psi=\Gamma_{g(t)}-\Gamma_{h}$. Consider the function $F=t\Phi Q+L\cdot\operatorname{tr}_{g}h$ where $L$ is a constant to be chosen later and $Q=|\Psi|^{2}$. Using the evolution equation (2.2), we have

[TABLE]

where the last inequality holds provided we choose $L=\Lambda(C_{3}+1)$. By the maximum principle, we conclude that $Q\leq C_{3}$ at a point where $F$ is maximal for $t\leq T\wedge\hat{T}$, and we conclude from this, the defintion of $F$ and the propoerties of the cutoff $\Phi$ that we have

[TABLE]

on $B_{t}(p,\frac{1}{2})$ for all $t\leq T\wedge\hat{T}$

Now we consider the function

[TABLE]

where $\tilde{F}:=tQ+L\operatorname{tr}_{g}h$. By (2.17) and our previous estimates, we may assume $A$ is sufficiently large so that $2A\geq A-\tilde{F}\geq\frac{1}{2}A$ on the support of $\widetilde{\Phi}$. On the support of $\widetilde{\Phi}$ we have

[TABLE]

Thus we have

[TABLE]

Now suppose $(x_{0},t_{0})$ is the point where $G$ is maximal for $t\leq T\wedge\tilde{T}$. Then either $t_{0}=0$ and then $G(t)$ thus $|Rm(g(t))|$ vanishes for all $t$, or else $t_{0}>0$ and from (2.19) and the properties of $\widetilde{\Phi}$ and the fact that $\nabla G(x_{0},t_{0})=0$

[TABLE]

and hence $G(x_{0},t_{0})\leq C_{8}$. In particular, we have shown that if $t\leq T\wedge\hat{T}$, $x\in B_{t}(p,1/4)$, then

[TABLE]

The first order estimate $|\nabla Rm(g(t))|$ follows from Shi’s derivative estimate, see for example [1]. ∎

We also need the following local estimate from [17, Propositon A.1].

Lemma 2.2**.**

For any $A_{1},n>0$, there exists $B(n,A_{1})$ depending only on $n,A_{1}$ such that the following holds: For any Kähler manifold $(N^{n},g_{0})$ (not necessarily complete), suppose $g(t),t\in[0,T]$ is a solution of Kähler Ricci flow on $B_{0}(x_{0},r)\subset N$ such that

[TABLE]

on $B_{0}(x_{0},r)$ and

[TABLE]

on $B_{0}(x_{0},r)]\times[0,T]$. Then we have

[TABLE]

on $B_{0}(x_{0},\frac{r}{8})\times[0,T]$.

The next proposition shows that the global lower bound of a complete solution is continuous in the following sense

Proposition 2.1**.**

Let $g(t)$ be a smooth solution to (1.1) on $M\times[0,T]$ such that $g(t)\in S(c_{1},c_{2},h)$ for all $t\in[0,T]$and some constants $c_{i}>0$. If $g(0)\geq h$ then we have

[TABLE]

on $M\times[0,T]$.

Proof.

Suppose the curvatures of $h$ are bounded by $K$ in absolute values. In particular, there exits an exhaustion function $\tilde{\rho}$ of $M$ with $\tilde{\rho}\geq 1$ where $|\partial\tilde{\rho}|+|\sqrt{-1}\partial\bar{\partial}\tilde{\rho}|$ bounded on $M$. We will let $\rho:=e^{At}\tilde{\rho}$ where the constant $A$ is large enough such that

[TABLE]

on $M\times[0,T]$. Let $\epsilon>0$ be given and consider the family of tensors

[TABLE]

on $M$ where $c=n/c_{1}$. Clearly, ${\alpha}(0)>0$ and ${\alpha}(t)\rightarrow+\infty$ as $x\rightarrow\infty$. We will show that ${\alpha}(t)>0$ on $M\times[0,T]$. The proposition will then follow by letting $\epsilon\to 0$.

Suppose on the contrary that ${\alpha}$ is not positive on $M\times[0,T]$. Due to the presence of $\rho$, there is $(x_{0},t_{0})\in M\times(0,T]$ and $X\in T^{1,0}_{x_{0}}M$ with $|X|_{g(t_{0})}=1$ such that

[TABLE]

We may choose $t_{0}>0$ such that for all $t<t_{0}$, ${\alpha}>0$ on $M$. Extend $X$ by parallel transport using metric $g(t_{0})$ so that at $(x_{0},t_{0})$, $\partial_{t}X=\nabla X=\Delta X=0$. At $(x_{0},t_{0})$, we have

[TABLE]

On the other hand, using the fact that ${\alpha}_{X\bar{Y}}=0$ for all $Y\in T^{1,0}M$, we have

[TABLE]

where we denote $\tilde{R}$ to be the curvature of $h$. Hence, we have

[TABLE]

But this contradicts (2.23).

∎

3. A global existence result for Chern Ricci flow

We recall here a global existence result for the Chern Ricci flow from [16]. A family of hermitian metrics $g(t)$ on $M$ is said to be a solution to the Chern Ricci flow if

[TABLE]

where $R^{C}_{i\bar{j}}(g(t))$ is the Chern-Ricci curvature of $g(t)$. If $g(t)$ is a family of Kähler metrics, then (3.1) and (1.1) coincide since in this case we have the Chern-Ricci curvature $R^{C}_{i\bar{j}}$ of $g(t)$ is the same as the Ricci curvature $R_{i\bar{j}}$ of $g(t)$. In a local holomorphic coordinate, the Chern Ricci curvature is given by

[TABLE]

where $R^{C}$ denotes the curvature tensor with respect to the Chern connection. Similar to the Kähler-Ricci flow, if $g(t)$ is a Hermitian family solving (3.1) then we may write the corresponding family of forms $\omega(t)$ as

[TABLE]

where $h$ is any Hermitian background metric on $M$ and the evolving potential $\varphi$ is given by

[TABLE]

In particular, the Chern-Ricci flow will preserve Kählerity on open set $U$ provided $g_{0}$ is initially Kähler on $U$. From now on we will simply use Ric to denote the Chern-Ricci curvature of a Hermitian metric $g$, which will coincide with the Levi Civita Ricci curvature at any point $g$ is Kähler . We have the following fundamental existence theorem for the Chern-Ricci flow.

Theorem 3.1**.**

[16, Theorem 4.2]** Let $(M^{n},g_{0})$ be a complete noncompact Hermitian manifold and let $T$ be the torsion of $g_{0}$. Suppose the Chern curvature of $g_{0}$, $|T|^{2}_{g_{0}}$ and $|\bar{\partial}T|_{g_{0}}$ are uniformly bounded by $K>0$. Suppose also the Riemannian curvature of $g_{0}$ is bounded. Then there is a constant ${\alpha}(n)>0$ depending only on $n$ such that the Chern-Ricci flow has a smooth solution $g(t)$ on $M\times[0,{\alpha}K^{-1}]$ such that ${\alpha}g_{0}\leq g(t)\leq{\alpha}^{-1}g_{0}$ on $M\times[0,{\alpha}K^{-1}]$.

4. Proof of Main Theorem 1.1

We begin by proving a result on the existence of local solutions to (1.1) satisfying certain estimates (Lemma 4.1). For our purposes later on, it will be convenient to state the result in terms of the following constants.

Definition 4.1**.**

Let $h$ be a complete Kähler metric satisfying $|\nabla\text{\rm Rm}(h)|+|Rm(h)|^{3/2}\leq 1$ on $M$ and let $g_{0}$ be a Kähler metric with $\lambda^{-1}h\leq g_{0}\leq\lambda h$. Define the following numbers depending only on $n,\lambda$:

• •

Let $\Lambda(n,\lambda)=\max\{\lambda,2C_{0}\lambda^{c_{1}}\}$ where $C_{0}(n)$ and $c_{1}(n)$ be the constants obtained from Theorem 2.1;

• •

Let $C_{1}(n,\Lambda)$ be the constant from Lemma 2.1 with $\Lambda$ from above;

• •

Let $B(n,C_{1}+{\alpha}(n)^{-1})$ be the constant obtained from Lemma 2.2 with $A_{1}=C_{1}+{\alpha}(n)^{-1}$;

• •

Let $\mu(n,\lambda)=\sqrt{\left(1+\frac{{\alpha}(n)}{C_{1}+{\beta}_{n}\Lambda^{4}b}\right)}-1$ where ${\alpha}(n)$ is from Lemma 3.1, $C_{1}$ is from above, $b$ is from Lemma 4.2 with $\kappa=0.1$ and ${\beta}_{n}$ is a large dimensional constant to be specified in the proof;

• •

Let $a(n,\lambda)=\max\left\{2B(1+\mu)^{2},n\log\frac{\Lambda}{{\alpha}}\right\}$ ;

• •

Let $\tilde{T}(n,a,\lambda)$ be the constant from Theorem 2.1;

• •

Let $\hat{T}(n,a,\Lambda)$ be the constant from Lemma 2.1;

• •

Let $\sigma(n,\lambda)=\max\{1,\tilde{T}^{-1/2},\hat{T}^{-1/2}\}$.

By the bound on the curvature of $h$ and its gradient in Definition 4.1, by [29] there is an exhaustion function $\rho$ on $M$ with

[TABLE]

Let

[TABLE]

Our result on existence of local solutions can then be stated as

Lemma 4.1**.**

Under Definition 4.1, given any $R>>1$ there is a solution $g(t)$ to Kähler-Ricci flow (1.1) defined on $U_{R-4\Lambda\mu(1+\mu)^{-1}}\times[0,\sigma^{-2}(1+\mu)^{-2}]$ with

1. (1)

$|\text{\rm Rm}(g(t))|\leq at^{-1}$; 2. (2)

$|\varphi(t)|\leq at$.

Proof.

We begin with the following

Claim 4.1**.**

There is a short-time solution to the Kähler-Ricci flow on $U_{R}\times[0,t_{0}]$ for some $t_{0}>0$ sufficiently small and possibly depending on $R$, so that

1. (1)

$|\text{\rm Rm}(x,t)|\leq at^{-1}$; 2. (2)

$|\varphi(x,t)|\leq at$.

proof of Claim 4.1.

Let $F(x)=\mathfrak{F}(\frac{\rho(x)}{R+1})$ with $0<\kappa<1/8$ such that $(1+R)(1-\kappa)\geq R$ where $\mathfrak{F},\kappa$ are from Lemma 4.2. Consider the complete Hermitian metric $\tilde{g}_{0}=e^{2F}g_{0}$ on $U_{R+1}$. It follows from Lemma 4.2 that $\tilde{g}_{0}$ will satisfy the hypothesis of Theorem 3.1 (see [16]) and so we obtain a solution $\tilde{g}(t)$ to Chern Ricci flow (3.1) on $U_{R+1}\times[0,t_{0}]$ for some $t_{0}>0$ with initial condition $\tilde{g}_{0}$. Moreover, the restriction $g(t)=\tilde{g}(t)|_{U_{R}}$ solves Kähler-Ricci flow (1.1) with initial condition $g_{0}$ because $\tilde{g}_{0}=g_{0}$ on $U_{R}$. Let $\tilde{h}=e^{2F}h$ and define

[TABLE]

Then we may write $\tilde{\omega}(t)=\tilde{\omega}_{0}-t\text{\rm Ric}(\tilde{h})+\sqrt{-1}\partial\bar{\partial}\varphi$.

By smoothness of the solution, the curvature estimate in the claim follows by shrinking $t_{0}$ if necessary to be sufficiently small. Moreover by (LABEL:initial-equi), by further shrinking $t_{0}$ if necessary we may have

[TABLE]

on $U_{R}\times[0,t_{0}]$ which implies the second conclusion in the claim. ∎

Now we provide the construction of an auxilliary function used in our constructions of local solutions to Kähler-Ricci flow.

For $\kappa\in(0,1)$, let $f:(-\infty,1)\to[0,\infty)$ be the function:

[TABLE]

Let $\varphi\geq 0$ be a smooth function on $\mathbb{R}$ such that $\varphi(s)=0$ if $s\leq 1-\kappa+\kappa^{2}$, $\varphi(s)=1$ for $s\geq 1-\kappa+2\kappa^{2}$

[TABLE]

such that $\displaystyle{\frac{2}{\kappa^{2}}}\geq\varphi^{\prime}\geq 0$. Define

[TABLE]

Now we have the following Lemma from [16].

Lemma 4.2**.**

[16]** Suppose $0<\kappa<\frac{1}{8}$. Then the function $\mathfrak{F}\geq 0$ defined above is smooth and satisfies the following:

1. (i)

$\mathfrak{F}(s)=0$* for $0\leq s\leq 1-\kappa$.* 2. (ii)

$\mathfrak{F}^{\prime}\geq 0$* and $\sum_{k=1}^{2}\exp(-k\mathfrak{F})|\mathfrak{F}^{(k)}|\leq b(\kappa)$. *

Now define $t_{k}$ and $R_{k}$ inductively as follows

1. (a)

$t_{0}$ is from claim 4.1 and $R_{0}=R$. 2. (b)

$t_{k}=t_{k-1}(1+\mu)^{2}=t_{0}(1+\mu)^{2k}$; 3. (c)

$R_{k}=R_{k-1}-4\Lambda\sigma\sqrt{t_{k-1}}=R-4\Lambda\sigma(1+\mu)\mu^{-1}\sqrt{t_{k}}\left[1-(1+\mu)^{-k}\right]$.

Consider the following statement

$P(k):$ There is a solution of the Kähler-Ricci flow $g(t)$ on $U_{R_{k}}\times[0,t_{k}]$ with $g(0)=g_{0}$ such that

1. (1)

$|\text{\rm Rm}(g(t))|\leq at^{-1}$; 2. (2)

$|\varphi(t)|\leq at$

on $U_{R_{k}}\times[0,t_{k}]$ with $t_{k}\leq\sigma^{-2}$.

Clearly we see that $P(0)$ is true, while $t_{k}\rightarrow+\infty$ and $R_{k}\rightarrow-\infty$ as $k\rightarrow\infty$. Let $k$ be the largest integer such that $P(k)$ is true. Then at $k+1$, we have the following possibilities.

Case 1: $t_{k+1}=(1+\mu)^{2}t_{k}\geq\sigma^{-2}>t_{k}$. Then

[TABLE]

and the Lemma holds in this case.

Case 2: $t_{k+1}<\sigma^{-2}$. We will show that this is not possible. Let $g(t)$ be the solution to (1.1) on $U_{R_{k}}\times[0,t_{k}]$ from the statement $P(k)$. Using $\lambda^{-1}h\leq g_{0}\leq\lambda h$, for $x\in U_{R_{k}-\Lambda\sigma\sqrt{t_{k}}}$ we have

[TABLE]

Let $r=\sigma\sqrt{t_{k}}<1$. By the choice of $\sigma$ from definition 4.1, we can apply Theorem 2.1 to Kähler-Ricci flow $\tilde{g}(t)=r^{-2}g(r^{2}t)$ on $B_{\tilde{g}(0)}(x,1)\times[0,t_{k}r^{-2}]$ with reference metric $\tilde{h}=r^{-2}h$ to deduce that for all $(x,t)\in U_{R-\Lambda\sigma\sqrt{t_{k}}}\times[0,t_{k}]$,

[TABLE]

Here we use the fact that $r<1$ so that the rescaled reference metric $\tilde{h}$ satisfies the curvature assumptions made on $h$ in definition 4.1. Thus for $x\in U_{R_{k}-2\Lambda\sigma\sqrt{t_{k}}}$, we apply Lemma 2.1 to $\tilde{g}(t)=r^{-2}g(r^{2}t)$ on $B_{\tilde{g}(0)}(x,1)\times[0,t_{k}r^{-2}]$ with reference metric $\tilde{h}$ to show that for $(x,t)\in U_{R-2\Lambda\sigma\sqrt{t_{k}}}\times[0,t_{k}]$,

[TABLE]

At $t=t_{k}$, we extend the Kähler-Ricci flow as follows. Fix $\kappa\in(0,\frac{1}{10})$, let

[TABLE]

and consider the conformal change of metric $\hat{g}_{0}=e^{2F}g(t_{k})$ and $\hat{h}=e^{2F}h$. By [16, Lemma 4.3], $\hat{g}_{0}$ and $\hat{h}$ are complete Hermitian metric with bounded geometry of $\infty$ order on $U_{R_{k}-2\Lambda\sigma\sqrt{t_{k}}}$ with $\hat{g}_{0}=g(t_{k})$ on $U_{R_{k}-3\Lambda\sigma\sqrt{t_{k}}}$. Moreover, by [16, Lemma B1], (4.1), (4.6) and the fact that $r<1<\Lambda$, there is $C^{\prime}(n)>0$ such that

[TABLE]

We choose ${\beta}(n)$ in definition 4.1 to be $C^{\prime}(n)$ here. By Theorem 3.1, there is a solution to the Kähler-Ricci flow $\hat{g}(t)$ starting from $\hat{g}_{0}$ on $U_{R_{k}-2\Lambda\sigma\sqrt{t_{k}}}\times[0,{\alpha}(n)K^{-1}]$ with

[TABLE]

Restrict $\hat{g}(t)$ to $U_{R_{k}-3\Lambda\sigma\sqrt{t_{k}}}$ provides solution $g(t)=\hat{g}(t-t_{k})$ to Kähler-Ricci flow on $U_{R_{k}-3\Lambda\sigma\sqrt{t_{k}}}\times[0,t_{k+1}]$.

Moreover using (4.9), (4.7) we apply Lemma 2.2 on $B_{g(t_{k})}(x,\sqrt{t_{k}})$ where $x\in U_{R_{k}-4\Lambda\sigma\sqrt{t_{k}}}$ to deduce that for all $t\in[0,t_{k+1}]$,

[TABLE]

The last inequality is by our choice of $a$ in definition 4.1 and properties from $P(k)$. On the other hand, using (4.6) and (4.9) for $t\in[t_{k},t_{k+1}]$,

[TABLE]

We have thus shown that in this case $P(k+1)$ is true, which contradicts the maximality of $k$.

Thus Case 2 above cannot occur, leaving only Case 1 in which case the Lemma holds. Thuis completes the proof of the Lemma.

∎

We now prove Theorem 1.1 using Lemma 4.1.

Proof of Theorem 1.1.

By the results in [21] and by scaling, there is a Kähler metric $\tilde{h}$ having curvature and gradient of curvature bounded on $M$ by $1$. Also we may have $C^{-1}(n,c,K)\tilde{h}\leq g_{0}\leq C(n,c,K)\tilde{h}$ on $M$ for some constant $C(n,c,K)$ where $c,K$ are from the hypothesis of Theorem 1.1. It follows by Lemma 4.1 that there is a solution $g_{k}(t)$ to Kähler-Ricci flow (1.1) defined on $B_{g_{0}}(k)\times[0,S(n,c,K)]$ for all $k$ sufficiently large satisfying the conditions in the Lemma. Theorem 2.1 and local Evans-Krylov theory [14, 18] or Kähler-Ricci flow local estimates [27] then imply the existence of $\tilde{T}(n,c,K)$ such that some subsequence of $g_{k}(t)$ converges smoothly and locally uniformly to a solution $g(t)$ to (1.1) on $M\times[0,S\wedge\tilde{T}]$ such that $g(t)\in S(c_{1},c_{2})$ for all $t$ and some $c_{i}$ and satisfying the conditions in the claim. The existence of the constants $a,T,C_{2},C_{1}$ in Theorem 1.1 follows from this, and Proposition 2.1. The fact that $g(t)$ extends as a bounded curvature solution to (1.1) on $M\times[0,T_{h})$ follows from condtion (1) in the Theorem and [5, Theorem 2.2] (see also [16, Theorem 1.3]).

∎

5. Proof of Theorem 1.2

Theorem 1.2 (1) will follow from the following slightly more general existence theorem.

Theorem 5.1**.**

Let $g_{0}$ be a continuous Hermitian metric. Suppose there is a sequence of smooth Kähler metric such that

1. (1)

$h_{k,0}\in S(1,c_{k},h)$* for some $c_{k}$;* 2. (2)

for any $\Omega\subset\subset M$, $h_{k,0}\rightarrow g_{0}$ uniformly on $\Omega$; 3. (3)

for any $\Omega\subset\subset M$, the scalar curvature of $h_{k,0}$, $R_{k}\geq-C(\Omega)$ on $\Omega$.

Then there is $T(n,h)>0$ so that (1.1) has a solution $g(t)$ on $M\times(0,T]$ with $g(t)\geq\frac{1}{2n}h$.

We begin by recalling the following Lemma from [4] which basically says that if a local solution $h(t)$ to (1.1) is a priori uniformly equivalent to a fixed metric $\hat{g}$ in space time, and close to $\hat{g}$ at time $t=0$, then it remains close to $\hat{g}$ in a uniform space time region.

Lemma 5.1**.**

Let $h(t)$ be a smooth solution to (1.1) on $B(1)\times[0,T)$ with $h(0)=h_{0}$ where $B(1)$ is the unit Euclidean ball in $\mathbb{C}^{n}$. Let $\hat{g}$ be a smooth Kähler metric on $B(1)$. Suppose

[TABLE]

on $B(1)\times[0,T)$ for some $N>0$, and that

[TABLE]

on $B(1)$. Then there exists a positive continuous function $a(t):[0,T)\to\mathbb{R}$ depending only on $\hat{g},N,d$ and $n$ such that

[TABLE]

on $B(1/2)\times[0,T)$, where $\lim_{t\to 0}a(t)=n\sqrt{2d(1+d)}/(1-d)$.

Proof of Theorem 5.1.

Let $g_{0}$ be as in Theorem 1.2. Thus there is a sequence $\{h_{k}\}\subset S(1,c_{k},h)$ for some sequence $c_{k}$ such that $h_{k}\to g_{0}$ in $C^{0}_{loc}(M)$. By Theorem 1.1, there is a sequence of solutions $h_{k}(t)$ to (1.1) on $M\times[0,T_{h})$ and stays uniformly equivalent to $h$ for $t\leq T(k)$. By applying maximum principle on equation (2.2) of $\operatorname{tr}_{h_{k}(t)}h$, there is $\epsilon_{n}>0$ such that for all $(x,t)\in M\times[0,\epsilon_{n}K^{-1}]$,

[TABLE]

By [4, Lemma 3.3], we also have local uniform upper bound for $h_{k}(t)$ with respect to $h$ on $[0,\epsilon_{n}K^{-1}]$. In particular, by the Evans-Krylov theory [14, 18] or Kähler-Ricci flow local estimates [27], we may conclude subequence convergence $h_{k_{i}}(t)\to g(t)$ in $C^{\infty}_{loc}(M\times(0,\epsilon_{n}K^{-1}]$ where $g(t)$ solves (1.1). By applying Proposition 2.1 on each $h_{k}(t)$, it is easy to see that

[TABLE]

It remains to prove $g(t)\to g_{0}$ in $C^{0}_{loc}(M)$. Fix some holomorphic coordinate ball $B(1)$ on $M$ and let $\delta>0$ be given. Then we may choose some constant $k_{0}$ sufficiently large and some constant $C^{\prime}(n,c_{k},K)>0$ so that on $B(1)$ we have the following

[TABLE]

for some constant $C^{\prime}$ depends only on $n,c,K$. Then by Lemma 5.1 and (5.4) there is a function $a(t):[0,T)\to\mathbb{R}$ depending only on $h_{k_{0}},C^{\prime}(n,c,K),\delta,n$ such that

[TABLE]

and on $B(1)\times[0,T)$ we have

[TABLE]

for all $k\geq k_{0}$.

Using (5.5) and (5.6), we conclude that given any $\epsilon>0$ we may choose $k_{0},t_{0}>0$ so that for all $k\geq k_{0}$ and $0<t<t_{0}$ we have $|h_{k}(t)-g_{0}|_{g_{0}}\leq\epsilon$ on $B(1)$. It follows that $g(t)\to g_{0}$ in $C^{0}_{loc}(M)$ which completes the proof of Theorem 5.1. ∎

Theorem 1.2 (1) follows directly from Theorem 5.1. We now prove (2). Let $g_{0}$ be as in Theorem 1.2 (2). Thus there is a sequence $\{h_{k}\}\subset S(1,c,h)$ such that $h_{k}\to g_{0}$ in $C^{0}_{loc}(M)$, and by Theorem 1.1, there is a sequence of solutions $h_{k}(t)$ to (1.1) on $M\times[0,T_{h})$ each satisyfing the conclusions in Theorem 1.1. In particular, by the uniform estimates in Theorem 1.1, there exists $T(n,c,K)$ such that given any $T<T(n,c,K)$ we may have $C^{-1}h\leq h_{k}(t)\leq Ch$ on $M\times[0,T)$ for some some $C$ independent of $k$. Thus by the Evans-Krylov theory [14, 18] or Kähler-Ricci flow local estimates [27], we may conclude subequence convergence of $h_{k_{i}}(t)\to g(t)$ in $C^{\infty}_{loc}(M\times(0,T(n,c,K))$ where $g(t)$ solves (1.1). The fact that we may extend this solution to $M\times(0,T_{h})$ follows from the same argument as in Theorem 1.1. The fact that $g(t)\to g_{0}$ as $t\to 0$ follows by Lemma 5.1 exactly as in the proof of part (1).

6. Proof of Theorem 1.3

Proof of Theorem 1.3.

Let $g_{0}\in S(c_{1},c_{2},h)$, by Theorem 1.1 and By [16, Theorem 1.3] there is a longtime solution $g(t)$ to (1.1) on $M\times[0,+\infty)$ with $g(0)=g_{0}$ such that for any $[a,b]\subset[0,+\infty)$, $g(t)$ is uniformly equivalent to $h$. If $k<0$, the longtime convergence follows from [9, Theorem 1.1] and the uniqueness of complete Kähler-Einstein metric. It suffices to discuss the case of $k=0$.

For simplicity, we work on the universal cover $\mathbb{C}^{n}$. According to the parabolic Schwarz Lemma (2.2) with $h$ being the flat metric, we can apply maximum principle to deduce that for all $t>0$, $g(t)\geq nc_{1}h$. Also, since $\left(\frac{\partial}{\partial t}-\Delta\right)\log{\det g}=0$ and $\sup_{M\times[0,T)}\log\det g(x,t)<\infty$ for any $0<T<\infty$, we can apply the maximum principle to show $\log\det g$ is uniformly bounded above on $\mathbb{C}^{n}\times[0,+\infty)$ and hence further infer that $\exists C>1$ such that for all $(x,t)\in\mathbb{C}^{n}\times[0,+\infty)$,

[TABLE]

By Evans-Krylov estimates [14, 18] or Kähler-Ricci flow local estimates [27], $g(t_{k})$ converges to some Kähler metric $g_{\infty}$ on $\mathbb{C}^{n}$ for some subsequence $t_{k}\rightarrow+\infty$ in $C^{\infty}_{loc}$.

On the other hand, for $L$ sufficeintly large the function $F=t|\Psi|_{g(t)}^{2}+L\operatorname{tr}_{g}h$ satisfies

[TABLE]

on $\mathbb{C}^{n}\times[0,+\infty)$. Since $\sup_{M\times[0,T)}F(x,t)<\infty$ for any $0<T<\infty$, we can apply the maximum principle to show that $F$ is uniformly bounded above and hence $|\partial g(t)|\leq C^{\prime}t^{-1/2}$ on $\mathbb{C}^{n}\times[0,+\infty)$ for some $C^{\prime}>0$. In particular, $\partial g_{\infty}=0$ implying $\omega_{\infty}=\sqrt{-1}\partial\bar{\partial}f$ for some quadratic polynomial $f$ and hence $\omega_{\infty}$ is a flat metric. ∎

7. Proof of Theroem 1.4

Proof.

The proof here is similar to that in [10] except that here we have a Kähler approximation of the initial metric. For the sake of completeness, we present the proof here. For $\epsilon>0$, let $g_{\epsilon,0}=g_{0}+\epsilon h$ be a complete Kähler metric uniformly equivalent to $h$. By Theorem 1.1, there is a solution to the Kähler-Ricci flow $g_{\epsilon}(t)$ on $M\times[0,T_{\epsilon})$ starting from $g_{\epsilon,0}$ such that $g_{\epsilon}(t)$ is uniformly equivalent to $h$ for all $t$, and we assume $T_{\epsilon}$ is the maximal such time. Consider the corresponding potential flow $\varphi_{\epsilon}(t)=\int^{t}_{0}\log\frac{\det g_{\epsilon}(r)}{\det h}\;dr$ where we can rewrite the Kähler form of $g_{\epsilon}(t)$ to be $\omega_{\epsilon}(t)=\omega_{0}+\epsilon\omega_{h}-t\text{\rm Ric}(h)+\sqrt{-1}\partial\bar{\partial}\varphi_{\epsilon}$.

Claim 7.1**.**

Then there exists $C(n,s,K,{\beta},f)>0$ such that for all $\epsilon>0$ and $t\in(0,s\wedge T_{\epsilon})$ we have

1. (i)

$\varphi_{\epsilon}\leq Ct$, 2. (ii)

$\varphi_{\epsilon}\geq nt\log t-Ct$, 3. (iii)

$\dot{\varphi}_{\epsilon}\leq C$,

proof of claim 7.1.

Since $h$ has bounded curvature and $g_{\epsilon}(t)$ is uniformly equivalent to $h$, there is an exhaustion function $\rho>0$ on $M$ such that $|\sqrt{-1}\partial\bar{\partial}\rho|_{h}\leq C$. For any $\delta>0$, consider $\tilde{\varphi}=\varphi_{\epsilon}-\delta\rho$ on $M\times[0,s\wedge T_{\epsilon}-\delta]$. Then at a point $(x_{0},t_{0})\in M\times[0,s\wedge T_{\epsilon}-\delta]$ where the function $\tilde{\varphi}-\tilde{C}t$ attains a maximum, we may calculate the following provided $t_{0}>0$ and $\tilde{C}$ is chosen sufficently large

[TABLE]

which is impossible. Hence $t_{0}=0$. Here we have used the $|\text{\rm Rm}(h)|\leq K$ and $\epsilon,\delta<<1$. By letting $\delta\rightarrow 0$, we see that (i) holds.

For the lower bound, we consider $\hat{\varphi}=\varphi_{\epsilon}+\delta\rho-tf-nt(\log t-1)+\tilde{C}t$ on $[0,s\wedge T_{\epsilon}-\delta]$ for some constant $\tilde{C}$. Then at a point $(x_{0},t_{0})\in M\times[0,s\wedge T_{\epsilon}-\delta]$ where this function attains a minimum we calculate the following provided $t_{0}>0$ $\tilde{C}$ is sufficiently large, and $\delta$ is sufficiently small compared to $\epsilon$:

[TABLE]

which is impossible. Hence $t_{0}=0$. By letting $\delta\rightarrow 0$, we see that (ii) holds.

To derive the upper bound on $\dot{\varphi}_{\epsilon}$ we use the fact

[TABLE]

and thus

[TABLE]

and apply the maximum principle as above to show that the function $t\dot{\varphi}_{\epsilon}-\varphi_{\epsilon}-nt-\delta\rho$ attains its maximum at on $M\times[0,s\wedge T_{\epsilon}-\delta]$ when $t=0$ provided $\delta$ is sufficiently small and the bound in (iii) follows by letting $\delta\to 0$ this and the bound in (i).

∎

Claim 7.2**.**

Then exists $C(n,s,K,{\beta},f,t)>0$ such that for all $\epsilon>0$ and $t\in(0,s\wedge T_{\epsilon})$ we have

[TABLE]

Proof.

From the above we may calculate that the function $v=(s-t)\dot{\varphi}_{\epsilon}+\varphi_{\epsilon}-sf+nt$ satisfies

[TABLE]

Then for sufficiently large $L>>1$ and using (2.2) the function $F=\log\operatorname{tr}_{g_{\epsilon}}h-Lv+(ns+1)\log t$ will satisfy

[TABLE]

Now let $(x_{0},t_{0})$ be the point in $M\times[0,s\wedge T_{\epsilon}-\delta]$ where the function $F_{\delta}=F-\delta e^{At}\rho$ is maximal where $A$ is chosen large enough so that $\left(\frac{\partial}{\partial t}-\Delta\right)(e^{At}\rho)\geq 0$. Now if $t_{0}=0$ then the claim follows from the estimates in Claim 7.1 and the hypothesis on $g_{0}$. If $t_{0}>0$ then by the maximum principle we have $0\leq\left(\frac{\partial}{\partial t}-\Delta\right)F_{\delta}(x_{0},t_{0})\leq\left(\frac{\partial}{\partial t}-\Delta\right)F(x_{0},t_{0})$ and thus $t_{0}\operatorname{tr}_{g_{\epsilon}}h(x_{0},t_{0})\leq m$ from the estimates in Claim 7.1. We may assume $\operatorname{tr}_{g_{\epsilon}}h(x_{0},t_{0})\geq 1$ as otherwise the claim trivially holds. Then we have the following where the RHS is evaluated at t $(x_{0},t_{0})$

[TABLE]

and the claim follows by letting $\delta\rightarrow 0$. ∎

By the Claims above we conclude that for any $[a,b]\subset(0,s\wedge T_{\epsilon})$, there is $C_{4}(n,s,f,K,a,b)>1$ such that

[TABLE]

and it follows from local estimate [14, 18, 27] and diagonal subsequence argument that we in fact have $T_{\epsilon}\geq s$ for all $\epsilon$ and further that we can let $\epsilon\rightarrow 0$ to obtain a smooth solution $\varphi$ to (1.4) on $M\times(0,s)$. By Claim 7.1, $\varphi(t)\rightarrow 0$ uniformly on $M$. The smoothness on $U=\{x:g_{0}>0\}$ follows from [10, Theorem 1.2] since the argument there is purely local once we have the estimates from claim 7.1. One can also argue using the pseudolocality of Ricci flow [19, 12], see [10, Remark 3.1].

∎