Instantaneously complete Chern–Ricci flow and Kähler–Einstein metrics

Abstract

In this work, we obtain some existence results of Chern–Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These results can be viewed as a generalization of an existence result of Ricci flow by Giesen and Topping for surfaces of hyperbolic type to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete non-compact Hermitian manifolds, which generalizes the work of Lott–Zhang and Tosatti–Weinkove to complete non-compact Hermitian manifolds with possibly unbounded curvature.

Appendices

Appendix A: Some basic relations

Let g(t) be a solution to the Chern–Ricci flow,

$$\begin{aligned} \partial _tg=-\text {Ric}(g) \end{aligned}$$

and h is another Hermitian metric. Let $\omega (t)$ be the Kähler form of g(t), $\theta _0$ be the Kähler form of h. Let

$$\begin{aligned} \phi (t)=\int _0^t\log \frac{\omega ^n(s)}{\theta _0^n}ds. \nonumber \\ \omega (t)=\omega (0)-t\text {Ric}(\theta _0)+\sqrt{-1}\partial \bar{\partial }\phi . \end{aligned}$$

(A.1)

Let ${\dot{\phi }}=\frac{\partial }{\partial t}\phi$. Then

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) {\dot{\phi }}=-{\text {tr}}_g(\text {Ric}(\theta _0)), \end{aligned}$$

(A.2)

where $\Delta$ is the Chern Laplacian with respect to g.

On the other hand, if g is as above, the solution $\widetilde{g}$ of the corresponding normalized Chern–Ricci flow with the same initial data

$$\begin{aligned} \partial _t\widetilde{g}=-\text {Ric}(\widetilde{g})-\widetilde{g} \end{aligned}$$

is given by

$$\begin{aligned} \widetilde{g}(x,t)=e^{-t}g(x,e^{t}-1). \end{aligned}$$

The corresponding potential u is given by

$$\begin{aligned} u(t)=e^{-t}\int _0^te^s\log \frac{\widetilde{\omega }^n(s)}{\theta _0^n}ds \end{aligned}$$

where $\widetilde{\omega }(s)$ is the Kähler form of $\widetilde{g}(s)$. Also,

$$\begin{aligned} \widetilde{\omega }(t)= & {} -\text {Ric}(\theta _0)+e^{-t}(\text {Ric}(\theta _0)+\omega (0))+\sqrt{-1}\partial \bar{\partial }u. \end{aligned}$$

(A.3)

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\widetilde{\Delta }\right) (\dot{u}+u)= & {} -{\text {tr}}_{\widetilde{g}}\text {Ric}(\theta _0)-n, \end{aligned}$$

(A.4)

where $\widetilde{\Delta }$ is the Chern Laplacian with respect to $\widetilde{g}$.

Lemma A.1

(See [17, 30]) Let g(t) be a solution to the Chern–Ricci flow and let $\Upsilon ={\text {tr}}_{ h}g$, and $\Theta ={\text {tr}}_g h$.

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) \log \Upsilon =\mathrm {I+II+III} \end{aligned}$$

where

$$\begin{aligned} \mathrm {I}\le & {} 2\Upsilon ^{-2}\mathbf{Re }\left( h^{i{{\bar{l}}}} g^{k\bar{q}}( T_0)_{ki{{\bar{l}}}} {\hat{\nabla }}_{{{\bar{q}}}}\Upsilon \right) . \\ \mathrm {II}= & {} \Upsilon ^{-1} g^{i\bar{j}}{{\hat{h}}}^{k{{\bar{l}}}}g_{k{{\bar{q}}}} \left( {\hat{\nabla }}_i \overline{({{\hat{T}}})_{jl}^q}- {{\hat{h}}}^{p{{\bar{q}}}}{{\hat{R}}}_{i{{\bar{l}}}p{{\bar{j}}}}\right) \\ \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \mathrm {III}=&-\Upsilon ^{-1} g^{{i\bar{j}}} h^{k{{\bar{l}}}}\left( \hat{\nabla }_i\left( \overline{( T_0)_{jl{{\bar{k}}}}} \right) +{\hat{\nabla }}_{{{\bar{l}}}}\left( {( T_0)_{ik{{\bar{j}}}} }\right) -\overline{ ({{\hat{T}}})_{jl}^q}( T_0)_{ik{{\bar{q}}}}^p \right) . \end{aligned} \end{aligned}$$

where $T_0$ is the torsion of $g_0=g(0)$, ${{\hat{T}}}$ is the torsion of h and ${\hat{\nabla }}$ is the derivative with respect to the Chern connection of h.

Appendix B: A maximum principle

We have the following maximum principle, see [16] for example.

Lemma B.1

Let $(M^n,h)$ be a complete non-compact Hermitian manifold satisfying condition: There exists a smooth positive real exhaustion function $\rho$ such that $|\partial \rho |^2_h+|\sqrt{-1}\partial {\bar{\partial }} \rho |_h\le C_1$. Suppose g(t) is a solution to the Chern–Ricci flow on $M\times [0,S)$. Assume for any $0<S_1<S$, there is $C_2>0$ such that

$$\begin{aligned} C_2^{-1}h\le g(t) \end{aligned}$$

for $0\le t\le S_1$. Let f be a smooth function on $M\times [0,S)$ which is bounded from above such that

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) f\le 0 \end{aligned}$$

on $\{f>0\}$ in the sense of barrier. Suppose $f\le 0$ at $t=0$, then $f\le 0$ on $M\times [0,S)$.

We say that

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) f\le \phi \end{aligned}$$

in the sense of barrier means that for fixed $t_1>0$ and $x_1$, for any $\epsilon >0$, there is a smooth function $\sigma (x)$ near x such that $\sigma (x_1)=f(x_1,t_1)$, $\sigma (x)\le f(x,t_1)$ near $x_1$, such that $\sigma$ is $C^2$ and at $(x_1,t_1)$

$$\begin{aligned} \frac{\partial _-}{\partial t}f(x,t)-\Delta \sigma (x)\le \phi (x)+\epsilon . \end{aligned}$$

Here

$$\begin{aligned} \frac{\partial _-}{\partial t}f(x,t)=\liminf _{h\rightarrow 0^+}\frac{f(x,t)-f(x,t-h)}{h}. \end{aligned}$$

for a function f(x, t).