Symplectic mean curvature flows in Kähler surfaces with positive holomorphic sectional curvatures

Abstract

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than $2$, then there exists a positive constant $\delta$ depending on the ratio such that $\cos \alpha \ge \delta$ is preserved along the flow.

1 Introduction

Mean curvature flows were studied by many authors, for example Huisken [14, 15], Ecker and Huisken [6], Huisken and Sinestrari [16], Carlo Ilmanen [17], Neves [18], Smoczyk [19], Wang [21], White [22], etc.

In this paper we mainly concentrated on the symplectic mean curvature flows, which were studied by Chen and Tian [4], Chen and Li [2], Chen et al. [3], Wang [21], Han and Li [8–10], Han and Sun [13], and Han et al. [11, 12]. The basic fact is that the symplectic property is preserved by the mean curvature flow if the ambient space $M$ is Kähler–Einstein, or if the ambient Kähler surface evolves along the Kähler–Ricci flow [10].

Let $(M,J,\overline{\omega },\bar{g})$ be a Kähler surface. For a compact oriented real surface $\Sigma$ which is smoothly immersed in $M$, the Kähler angle [5] $\alpha$ of $\Sigma$ in $M$ was defined by

$$\begin{aligned} \omega |_\Sigma =\cos \alpha d\mu _\Sigma \end{aligned}$$

where $d\mu _\Sigma$ is the area element of $\Sigma$ in the induced metric from $\overline{g}$. We say that $\Sigma$ is a symplectic surface if $\cos \alpha > 0$; $\Sigma$ is a holomorphic curve if $\cos \alpha \equiv 1$.

Given an immersed $F_0: \Sigma \rightarrow M$, we consider a one-parameter family of smooth maps $F_t=F(\cdot , t): \Sigma \rightarrow M$ with corresponding images $\Sigma _t=F_t(\Sigma )$ immersed in $M$ and $F$ satisfies the mean curvature flow equation:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial }{\partial t} F(x,t) =H(x,t) \\ F(x,0)=F_{0}(x), \end{array}\right. \end{aligned}$$

(1.1)

where $H(x,t)$ is the mean curvature vector of $\Sigma _t$ at $F(x,t)$ in $M$.

Choose an orthonormal basis $\{e_1,e_2,e_3,e_4\}$ on ($M,\bar{g}$) along $\Sigma _t$ such that $\{e_1,e_2\}$ is the basis of $\Sigma _t$ and the symplectic form $\omega _t$ takes the form

$$\begin{aligned} \omega _t=\cos \alpha u_1\wedge u_2+\cos \alpha u_3\wedge u_4+\sin \alpha u_1\wedge u_3-\sin \alpha u_2\wedge u_4, \end{aligned}$$

(1.2)

where $\{u_1,u_2,u_3,u_4\}$ is the dual basis of $\{e_1,e_2,e_3,e_4\}$. Then along the surface $\Sigma _t$ the complex structure on $M$ takes the form ([2])

$$\begin{aligned} J= \left( \begin{array}{cccc} 0 &{} \cos \alpha &{} \sin \alpha &{} 0\\ -\cos \alpha &{} 0 &{} 0 &{} -\sin \alpha \\ -\sin \alpha &{} 0 &{} 0 &{} \cos \alpha \\ 0 &{} \sin \alpha &{} -\cos \alpha &{} 0 \end{array}\right) . \end{aligned}$$

(1.3)

Recall the evolution equation of the Kähler angle along the mean curvature flow deduced in [10],

Theorem 1.1

The evolution equation for $\cos \alpha$ along $\Sigma _t$ is

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) \cos \alpha&= \left| \overline{\nabla }J_{\Sigma _t}\right| ^2\cos \alpha +\sin ^2\alpha Ric(Je_1,e_2). \end{aligned}$$

(1.4)

Here

$$\begin{aligned} \left| \overline{\nabla }J_{\Sigma _t}\right| ^2=\left| h^4_{1k}+h^3_{2k}\right| ^2+\left| h^4_{2k}-h^3_{1k}\right| ^2\ge \frac{1}{2}|H|^2. \end{aligned}$$

(1.5)

We want to see whether the symplectic property is preserved along the mean curvature flow. In the case that $M$ is a Kähler–Einstein surface, we have $Ric(Je_1,e_2)=\bar{\rho }\cos \alpha$, where $\bar{\rho }$ is the scalar curvature of $M$, so the symplectic property is preserved. If the ambient Kähler surface evolves along the Kähler–Ricci flow, Han and Li [10] derived the evolution equation for $\cos \alpha$ and consequently they showed that the symplectic property is also preserved. In this paper, we find another condition to assure that along the flow, at each time the surface is symplectic. Note that we don’t require $M$ to be Einstein. Denote the minimum and maximum of holomorphic sectional curvatures of $M$ by $k_1$ and $k_2$. We state our main theorem as follows:

Main Theorem Suppose $M$ is a Kähler surface with positive holomorphic sectional curvatures. Set $\lambda =\frac{k_2}{k_1}$. If the flow satisfies either

I. :

$1\le \lambda <\frac{11}{7}$ and $\cos \alpha (\cdot ,0)\ge \delta >\frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^2+(48-24\lambda )^2}}$,

or

II. :

$\frac{11}{7}\le \lambda <2$ and $\cos \alpha (\cdot ,0)\ge \delta >\frac{8\lambda -5}{\sqrt{(8\lambda -5)^2+(12-6\lambda )^2}}$,

then along the flow

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) \cos \alpha \ge \left| \overline{\nabla }J_{\Sigma _t}\right| ^2\cos \alpha +C\sin ^2\alpha , \end{aligned}$$

(1.6)

where $C$ is a positive constant depending only on $k_1, k_2$ and $\delta$. As a corollary, $\min _{\Sigma _t}\cos \alpha$ is increasing with respect to $t$. In particular, at each time $t, \Sigma _t$ is symplectic. Therefore, we call this flow the symplectic mean curvature flow.

Since we obtain (1.6), many theorems in “symplectic mean curvature flows in Kähler–Einstein surfaces” still hold in our case. For example,

Arguing as in [5] by strong maximum principle, we have

Corollary 1.2

I. Suppose $M$ is a Kähler surface with positive holomorphic sectional curvatures and $1\le \lambda <\frac{11}{7}$, then every symplectic minimal surface satisfying

$$\begin{aligned} \cos \alpha >\frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^2+(48-24\lambda )^2}} \end{aligned}$$

in $M$ is a holomorphic curve.

II. Suppose $M$ is a Kähler surface with positive holomorphic sectional curvatures and $\frac{11}{7}\le \lambda <2$, then every symplectic minimal surface satisfying

$$\begin{aligned} \cos \alpha >\frac{8\lambda -5}{\sqrt{(8\lambda -5)^2+(12-6\lambda )^2}} \end{aligned}$$

in $M$ is a holomorphic curve.

Arguing exactly in the same way as in [2] or [21], we have

Theorem 1.3

Under the same condition of the Main Theorem, the symplectic mean curvature flow has no type I singularity at any $T>0$.

2 Curvature tensor, sectional curvature and holomorphic sectional curvature

Denote the curvature tensor of $M$ by $K$. Set $K(X)=K(X,JX,X,JX)$ and $K(X,Y)=K(X,Y,X,Y)$, where $X,Y$ are arbitrary vector fields on $M$. It is known that (c.f. [1, 20]) we can express the sectional curvatures by holomorphic sectional curvatures.

Theorem 2.1

The sectional curvatures of $M$ can be determined by the holomorphic sectional curvatures by

$$\begin{aligned} K(X,Y)&= \frac{1}{32}[3K(X+JY)+3K(X-JY)-K(X+Y)-K(X-Y)\nonumber \\&\quad -\,4K(X)-4K(Y)]. \end{aligned}$$

(2.1)

Using (2.1), it is easy to check that,

Theorem 2.2

For any vector fields $X, Y$ and $Z$ on $M$,

$$\begin{aligned} K(X,Y,X,Z)&= \frac{1}{2}[K(Y+Z,X)-K(X,Y)-K(X,Z)]\nonumber \\&= \frac{1}{64}[3K(Y+Z+JX)+3K(Y+Z-JX)-K(Y+Z+X)\nonumber \\&-\,K(Y+Z-X)-3K(Y+JX)-3K(Y-JX)-3K(Z+JX)\nonumber \\&-\,3K(Z-JX)-4K(Y+Z)+K(Y+X)+K(Y-X)\nonumber \\&+\,K(Z+X)+K(Z-X)+4K(X)+4K(Y)+4K(Z)]. \end{aligned}$$

(2.2)

Denote the minimum and the maximum of sectional curvatures by $K_{min}$ and $K_{max}$, respectively, we have the following estimates.

Theorem 2.3

$K_{min}$ and $K_{max}$ satisfy

$$\begin{aligned} K_{max}\le \frac{3}{2}k_2-\frac{1}{2}k_1 \end{aligned}$$

(2.3)

and

$$\begin{aligned} K_{min}\ge \frac{3}{4}k_1-\frac{1}{2}k_2 \end{aligned}$$

(2.4)

Proof

Given any point $p\in M$ and any two unit orthogonal vectors $X$ and $Y$ at $p$, we can find two vectors $Z$ and $W$ such that $\{X,Y,Z,W\}$ form an orthonormal basis of $T_pM$. Suppose $JX=yY+zZ+wW$, then

$$\begin{aligned} {} \langle X+JY,X+JY\rangle =2-2y, \end{aligned}$$

(2.5)

and

$$\begin{aligned} {} \langle X-JY,X-JY\rangle =2+2y. \end{aligned}$$

(2.6)

Assume the Kähler form is anti-self-dual, it was shown in [12] that, $y^2+z^2+w^2=1$ and $J$ has the form

$$\begin{aligned} J=\left( \begin{array}{c@{\quad }l@{\quad }c@{\quad }r} 0 &{}y &{}z &{}w \\ -y &{}0 &{}w &{}-z\\ -z &{}-w &{}0 &{}y\\ -w &{}z &{}-y &{}0 \end{array}\right) . \end{aligned}$$

(2.7)

Combining (2.1) with (2.6) and (2.6), we get

$$\begin{aligned} K(X,Y)&\le \frac{1}{32}\Bigg [3(2-2y)^2k_2+3(2+2y)^2k_2-2^2k_1-2^2k_1-4k_1-4k_1\Bigg ]\\&= \frac{1}{4}\Bigg [(3+3y^2)k_2-2k_1\Bigg ]\\&\le \frac{1}{4}(6k_2-2k_1)\\&= \frac{3}{2}k_2-\frac{1}{2}k_1, \end{aligned}$$

and similarly

$$\begin{aligned} K(X,Y)&\ge \frac{1}{4}[(3+3y^2)k_1-2k_2]\\&\ge \frac{1}{4}(3k_1-2k_2)\\&= \frac{3}{4}k_1-\frac{1}{2}k_2. \end{aligned}$$

This proves the theorem. $\square$

3 Proof of the Main Theorem

In this section, we will prove the Main Theorem of this paper.

Proof of the Main Theorem

In order to prove this theorem, we need to estimate $Ric(Je_1,e_2)$. Using two different methods, we get two available estimates. We now deduce the first one.

$$\begin{aligned} Ric(Je_1,e_2)&= K(Je_1,e_1,e_2,e_1)+K(Je_1,e_3,e_2,e_3)+K(Je_1,e_4,e_2,e_4)\nonumber \\&= K(\cos \alpha e_2+\sin \alpha e_3,e_1,e_2,e_1)+K(\cos \alpha e_2+\sin \alpha e_3,e_3,e_2,e_3)\nonumber \\&+K(\cos \alpha e_2+\sin \alpha e_3,e_4,e_2,e_4)\nonumber \\&= cos\alpha R_{22}+\sin \alpha (K_{3121}+K_{3424}), \end{aligned}$$

(3.1)

where

$$\begin{aligned} R_{22}=K_{2121}+K_{2323}+K_{2424}. \end{aligned}$$

(3.2)

By (2.1), we have

$$\begin{aligned} K_{2121}&= \frac{1}{32}[3K(e_1+Je_2)+3K(e_1-Je_2)-K(e_1+e_2)-K(e_1-e_2)\\&\quad -4K(e_1)-4K(e_2)]. \end{aligned}$$

By our choice of the complex structure (1.3), we get

$$\begin{aligned} \langle e_1+Je_2,e_1+Je_2\rangle =2-2\cos \alpha , \end{aligned}$$

and

$$\begin{aligned} \langle e_1-Je_2,e_1-Je_2\rangle =2+2\cos \alpha . \end{aligned}$$

Hence $K_{2121}$ can be estimated by $k_1$ and $k_2$,

$$\begin{aligned} K_{2121}&\ge \frac{1}{32}[3(2-2\cos \alpha )^2k_1+3(2+2\cos \alpha )^2k_1-2^2k_2-2^2k_2-4k_2-4k_2]\nonumber \\&= \frac{1}{4}[(3+3\cos ^2\alpha )k_1-2k_2]. \end{aligned}$$

(3.3)

Similarly, we get

$$\begin{aligned} K_{2323}\ge \frac{1}{4}(3k_1-2k_2), \end{aligned}$$

(3.4)

and

$$\begin{aligned} K_{2424}\ge \frac{1}{4}[(3+3\sin ^2\alpha )k_1-2k_2]. \end{aligned}$$

(3.5)

Putting (3.3), (3.4) and (3.5) into (3.2), we obtain that

$$\begin{aligned} R_{22}\ge 3k_1-\frac{3}{2}k_2. \end{aligned}$$

(3.6)

Using (2.2) and (1.3), we can also estimate $K_{3121}$ and $K_{3424}$. We have

$$\begin{aligned} K_{3121}\ge \frac{1}{32}[(53+48\sin \alpha \cos \alpha )k_1-53k_2], \end{aligned}$$

(3.7)

and

$$\begin{aligned} K_{3424}\ge \frac{1}{32}[(53-48\sin \alpha \cos \alpha )k_1-53k_2]. \end{aligned}$$

(3.8)

Adding (3.7) and (3.8) yields

$$\begin{aligned} K_{3121}+K_{3424}\ge -\frac{53}{16}(k_2-k_1). \end{aligned}$$

(3.9)

By a similar computation in the opposite direction, we get

$$\begin{aligned} |K_{3121}+K_{3424}|\le \frac{53}{16}(k_2-k_1). \end{aligned}$$

(3.10)

Therefore by (3.1), (3.6), (3.10) and short time existence of the mean curvature flow, we have

$$\begin{aligned} Ric(Je_1,e_2)&\ge \cos \alpha (3k_1-\frac{3}{2}k_2)-\sqrt{1-\cos ^2\alpha }\frac{53}{16}(k_2-k_1)\nonumber \\&= \left( 3\cos \alpha \!+\!\frac{53}{16}\sqrt{1-\cos ^2\alpha }\right) k_1\!-\!\left( \frac{3}{2}\cos \alpha +\frac{53}{16}\sqrt{1-\cos ^2\alpha }\right) k_2.\qquad \qquad \end{aligned}$$

(3.11)

If $1\le \lambda <2$ and $\cos \alpha >\frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^2+(48-24\lambda )^2}}$, then the RHS of (3.11) is positive.

Another estimate follows directly from Theorem 2.3 and Berger inequality (c.f. [7]) that

$$\begin{aligned} |K_{3121}+K_{3424}|\le |K_{3121}|+|K_{3424}|\le K_{max}-K_{min}\le 2k_2-\frac{5}{4}k_1. \end{aligned}$$

(3.12)

Putting the above estimate into (3.1) yields

$$\begin{aligned} Ric(Je_1,e_2)&\ge \cos \alpha \left( 3k_1-\frac{3}{2}k_2\right) -\sqrt{1-\cos ^2\alpha }\left( 2k_2-\frac{5}{4}k_1\right) \nonumber \\&= \left( 3\cos \alpha +\frac{5}{4}\sqrt{1-\cos ^2\alpha }\right) k_1-\left( \frac{3}{2}\cos \alpha +\frac{5}{4}\sqrt{1-\cos ^2\alpha }\right) k_2.\qquad \quad \end{aligned}$$

(3.13)

It follows that if $1\le \lambda <2$ and $\cos \alpha >\frac{8\lambda -5}{\sqrt{(8\lambda -5)^2+(12-6\lambda )^2}}$, then the RHS of (3.11) is positive. Note that

$$\begin{aligned} \frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^2+(48-24\lambda )^2}}\le \frac{8\lambda -5}{\sqrt{(8\lambda -5)^2+(12-6\lambda )^2}} \end{aligned}$$

for $1\le \lambda <\frac{11}{7}$, and

$$\begin{aligned} \frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^2+(48-24\lambda )^2}}\ge \frac{8\lambda -5}{\sqrt{(8\lambda -5)^2+(12-6\lambda )^2}} \end{aligned}$$

for $\frac{11}{7}\le \lambda <2$, we get the conclusion. $\square$