Berger’s formulas and their applications in symplectic mean curvature flow

Abstract

In this paper, we recall some well known Berger’s formulas. As their applications, we prove that if the local holomorphic pinching constant is $\lambda <2$, then there exists a positive constant $\delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$ such that $\cos \alpha \ge \delta$ is preserved along the mean curvature flow, improving Li–Yang’s main theorem in Li and Yang (Geom. Dedicata 170 (2014) 63–69). We also prove that when $\cos \alpha$ is close enough to 1, then the symplectic mean curvature flow exists globally and converges to a holomorphic curve.

1 Introduction

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

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

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

It is important to find the conditions to assure that the symplectic property is preserved along the mean curvature flow. In the case that M is a Kähler–Einstein surface, the symplectic property is preserved. Han and Li [7] proved that the symplectic property is also preserved if the ambient Kähler surface evolves along the Kähler–Ricci flow. In [10], Li and Yang found another condition to assure that the symplectic property is preserved along the mean curvature flow. In this paper, we improve Li and Yang’s result in [10].

Here, we only consider the ambient Kähler surface with positive holomorphic sectional curvature. We denote the minimum and maximum of holomorphic sectional curvatures at $p\in M$ by $k_{1}(p)$ and $k_{2}(p)$, respectively, and $\lambda (p)=\frac{k_{2}(p)}{k_{1}(p)}$. We define

$k_{1}:=\min _{p\in M}k_{1}(p)$   and    $k_{2}:=\max _{p\in M}k_{2}(p)$.

We also define the local holomorphic pinching constant by

$$\begin{aligned} \lambda :=\max _{p\in M}\lambda (p). \end{aligned}$$

Then we have the first main theorem.

Theorem 1.1

Suppose M is a Kähler surface with positive holomorphic sectional curvature. If $1\le \lambda <2$ and $\cos \alpha (\cdot ,0)\ge \delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$, then along the flow

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

(1)

where ${\bar{\nabla }}$ is the Levi–Civita connection with metric ${\bar{g}}$ on M and $|{\bar{\nabla }}J_{\Sigma _{t}}|^{2}$ is defined by (9) in subsection 2.1, and 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.

Remark 1.1

Li–Yang’s main theorem in [10], i.e., the lower bound of $\delta$ is $\frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^{2}+(48-24\lambda )^{2}}}$ for $\lambda \in [1, \frac{11}{7})$ and $\frac{8\lambda -5}{\sqrt{(8\lambda -5)^{2}+(12-6\lambda )^{2}}}$ for $\lambda \in [\frac{11}{7}, 2)$. It is easy to check that for each $\lambda \in [1, 2)$,

$$\begin{aligned} \begin{aligned}&\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}\\&\quad \le \min \left\{ \frac{53(\lambda -1)}{\sqrt{(53\lambda -53)^{2}+(48-24\lambda )^{2}}}, \frac{8\lambda -5}{\sqrt{(8\lambda -5)^{2}+(12-6\lambda )^{2}}}\right\} . \end{aligned} \end{aligned}$$

Hence we improve Li–Yang’s main result in [10].

Similar to Han–Li’s main theorem in [6], we also prove the following theorem for a Kähler surface with positive holomorphic sectional curvature and $1\le \lambda <2$.

Theorem 1.2

Suppose that M is a Kähler surface with positive holomorphic sectional curvature and $1\le \lambda <2$. Let $\alpha$ be the Kähler angle of the surface $\Sigma _{t}$ which evolves by the mean curvature flow. Suppose that $\cos \alpha (\cdot ,0)>\frac{58(\lambda -1)}{\sqrt{(48-24)^{2}+(58\lambda -58)^{2}}}$. $r_{0}$ is defined in Remark 4.2 and $\epsilon _{0}$ is the constant in Theorem 4.1, and define $\epsilon _{1}$ as

$$\begin{aligned} \epsilon _{1}= \frac{\pi ^{2}\epsilon _{0}^{2}r_{0}^{6}(1-\mathrm{e}^{-\frac{3}{8}(2-\lambda )k_{1}})^{2}}{4 \mathrm{Area}(\Sigma _{0})}. \end{aligned}$$

Then if $\int _{\Sigma _{0}}\frac{\sin ^{2}\alpha (x,0)}{\cos \alpha (x,0)}{ \mathrm{d}}\mu _{0}\le \epsilon _{1}$, the mean curvature flow with initial surface $\Sigma _{0}$ exists globally and it converges to a holomorphic curve.

By Theorem 1.2, it is easy to get the following corollary.

COROLLARY 1.1

Under the same conditions and same notations as in Theorem 1.2, except $\int _{\Sigma _{0}}\frac{\sin ^{2}\alpha (x,0)}{\cos \alpha (x,0)}\mathrm{d}\mu _{0}\le \epsilon _{1}$, there exists a constant $\epsilon _{2}$ depending only on $\epsilon _{1}$ and $\mathrm{Area}(\Sigma _{0}),$ such that if

$$\begin{aligned} 1-\cos \alpha (\cdot , 0)\le \epsilon _{2}, \end{aligned}$$

then the mean curvature flow with initial surface exists globally and it converges to a holomorphic curve.

Remark 1.2

By Corollary 2.3, if $\lambda <\frac{3}{2}$, then the sectional curvature of M is positive, which implies that the bisectional curvature is positive. By Frankel conjecture, which was proved by Siu and Yau [12] and by Mori [11] independently, the Kähler surface is biholomorphic to ${\mathbb {C}}{\mathbb {P}}^{2}$. Recently, Yang and Zheng (see Proposition 2.6 in [16]) proved that the Kähler manifold $M^{n}$ with $\lambda <2$ must be biholomorphic to ${\mathbb {C}}{\mathbb {P}}^{n}$.

2 Preliminaries

In this section, we recall some preliminaries about the curvature and the evolution equations of the mean curvature flow.

2.1 Evolution equations for the mean curvature flow

In this subsection, we recall some evolution equations for the mean curvature flow.

Suppose that $\Sigma$ is a sub manifold in a Riemannian manifold M. We choose an orthonormal basis $\{e_{i}\}$ for $T\Sigma$ and $\{e_{\alpha }\}$ for $N\Sigma$. 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 which satisfies the mean curvature flow equation:

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

(2)

Recall the evolution equation for the second fundamental form $h_{ij}^{\alpha }$ and $|A|^{2}$ along the mean curvature flow (see [4, 9, 13, 14]).

Lemma 2.1

$$\begin{aligned} \frac{\partial }{\partial t} h_{ij}^{\alpha }&=\Delta h_{ij}^{\alpha }+({\bar{\nabla }}_{k}Rm)_{\alpha ijk}+({\bar{\nabla }}_{j}Rm)_{\alpha kik}-2R_{lijk}h_{lk}^{\alpha }\nonumber \\&\quad +\,2R_{\alpha \beta jk}h_{ik}^{\beta }+2R_{\alpha \beta ik}h_{jk}^{\beta }-R_{lkik}h_{lj}^{\alpha }-R_{lkjk}h_{il}^{\alpha }\nonumber \\&\quad +\,R_{\alpha k\beta k}h_{ij}^{\beta }-H^{\beta }(h_{ik}^{\beta }h_{jk}^{\alpha }+h_{jk}^{\beta }h_{ik}^{\alpha })+h_{im}^{\alpha }h_{mk}^{\beta }h_{kj}^{\beta }\nonumber \\&\quad -\,2h_{im}^{\beta }h_{mk}^{\alpha }h_{kj}^{\beta }+h_{ik}^{\beta }h_{km}^{\beta }h_{mj}^{\alpha }+h_{km}^{\alpha }h_{mk}^{\beta }h_{ij}^{\beta }+h_{ij}^{\beta }\langle e_{\beta }, {\bar{\nabla }}_{H}e_{\alpha }\rangle , \end{aligned}$$

(3)

where $R_{ABCD}$ is the curvature tensor of M and ${\bar{\nabla }}$ is the covariant derivative of M. Therefore,

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial t}|A|^{2}&=\Delta |A|^{2}-2|\nabla A|^{2}\\&\quad +\,[({\bar{\nabla }}_{k}Rm)_{\alpha ijk}+({\bar{\nabla }}_{j}Rm)_{\alpha kik}]h_{ij}^{\alpha } -4R_{lijk}h_{lk}^{\alpha }h_{ij}^{\alpha }\\&\quad +\,8R_{\alpha \beta jk}h_{ik}^{\beta }h_{ij}^{\alpha }-4R_{lkik}h_{lj}^{\alpha }h_{ij}^{\alpha }+2R_{\alpha k\beta k}h_{ij}^{\beta }h_{ij}^{\alpha }+2P_{1}+2P_{2}, \end{aligned} \end{aligned}$$

(4)

where

$$\begin{aligned} P_{1}= & {} \Sigma _{\alpha ,\beta ,i,j}\left( \Sigma _{k}\left( h_{ik}^{\alpha }h_{jk}^{\beta }-h_{jk}^{\alpha }h_{ik}^{\beta }\right) \right) ^{2}, \nonumber \\ P_{2}= & {} \Sigma _{\alpha ,\beta }(\Sigma _{i,j}h_{ij}^{\alpha }h_{ij}^{\beta })^{2}. \nonumber \\ \frac{\partial }{\partial t}|H|^{2}= & {} \Delta |H|^{2}-2|\nabla H|^{2}+2R_{\alpha k\beta k}H^{\alpha }H^{\beta }+2P_{3}, \end{aligned}$$

(5)

where

$$\begin{aligned} P_{3}=\Sigma _{i,j}(\Sigma _{\alpha }H^{\alpha }h_{ij}^{\alpha })^{2}. \end{aligned}$$

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 frame of the tangent bundle $T \Sigma _{t}$ and $\{e_{3},e_{4}\}$ is the frame of the normal bundle $N\Sigma _{t}$. Then along the surface $\Sigma _{t}$, we can take the complex structure on M as the form (cf. [9])

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

(6)

or

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

(7)

Since Kähler form is self-dual, then J must be of the form (7).

Remark 2.1

In fact, the above argument also shows that the Kähler form is self-dual. If J is of the form (6), then the Kähler form is anti-self-dual, i.e., $*\omega =-\omega$, and hence it is impossible to obtain Kähler form. Hence J must be of the form (7), then the Kähler form $\omega$ must be self-dual.

Recall the evolution equation of the Kähler angle $\cos \alpha$ (cf. [4, 8]).

Lemma 2.2

The evolution equation for $\cos \alpha$ along $\Sigma _{t}$ is

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

(8)

Here

$$\begin{aligned} |{\bar{\nabla }}J_{\Sigma _{t}}|^{2}=|h^{4}_{1k}+h^{3}_{2k}|^{2}+|h^{4}_{2k}-h^{3}_{1k}|^{2}. \end{aligned}$$

(9)

Then $|{\bar{\nabla }}J_{\Sigma _{t}}|^{2}$ is independent of the choice of the frame and depend only on the orientation of the frame. It is proved in [4, 7] that

$$\begin{aligned} |{\bar{\nabla }}J_{\Sigma _{t}}|^{2}\ge \frac{1}{2}|H|^{2} \end{aligned}$$

(10)

and

$$\begin{aligned} |\nabla \cos \alpha |^{2}\le \sin ^{2}\alpha |{\bar{\nabla }}J_{\Sigma _{t}}|^{2}. \end{aligned}$$

(11)

2.2 Berger’s formulas

In this subsection, we recall some well known identities, which are called Berger’s formulas. We first recall the definitions of the Riemannian curvature and the holomorphic sectional curvature; secondly, we recall some Berger’s formulas, which are the relations between Riemannian curvatures and the holomorphic sectional curvature.

The Riemann curvature tensor R of (M, g) is defined by

$$\begin{aligned} R(X, Y, Z, W)=-g(\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z, W) \end{aligned}$$

for any vector fields X, Y, Z, W.

Set $R(X,Y)=R(X,Y,X,Y)$ and $R(X)=R(X,JX)$. Fix a point $p\in M$ and a two-dimensional plane $\Pi \subset T_{p}M$. The sectional curvature of $\Pi$ is defined by

$$\begin{aligned} K(\Pi )=\frac{R(X,Y)}{g(X,X)g(Y,Y)-g(X,Y)^{2}}, \end{aligned}$$

where $\{X,Y\}$ is a basis of $\Pi$. We also denote it by K(X, Y). For a Kähler manifold (M, g, J), if the two-dimensional plane $\Pi$ is spanned by $\{X,JX\}$, i.e., $\Pi$ is a holomorphic plane, then the sectional curvature of $\Pi$ is called a holomorphic sectional curvature of $\Pi$. We denote it by H(X), where $\{X,JX\}$ is a basis of $\Pi$. Then

$$\begin{aligned} H(X)=\frac{R(X)}{g(X,X)^{2}}. \end{aligned}$$

For any orthogonal four-frames $\{e_1, e_2, e_3,e_4\}\subset T_{p}M$, we have for any index $A, B, C,D\in \{1, 2, 3, 4\}$ the below lemma.

Lemma 2.3

[1].

$$\begin{aligned} \begin{aligned}&24R_{ABCD}\\&\quad =R(e_{A}+e_{C},e_{B}+e_{D})-R(eS_{A}+e_{C},e_{B}-e_{D})\\&\qquad -R(e_{A}-e_{C},e_{B}+e_{D})\\&\qquad +R(e_{A}-e_{C},e_{B}-e_{D})-R(e_{A}+e_{D},e_{B}+e_{C})\\&\qquad +R(e_{A}+e_{D},e_{B}-e_{C})\\&\qquad +R(e_{A}-e_{D},e_{B}+e_{C})-R(e_{A}-e_{D},e_{B}-e_{C}) \end{aligned} \end{aligned}$$

(12)

For the proof, see the proof of Proposition 1.9 in [3]. Then we get the following property.

COROLLARY 2.1

[1]

Let (M, g) be a Riemannian manifold, and let p be an arbitrary point in M. Suppose that ${\underline{\kappa }}\le K(\pi ) \le {\bar{\kappa }}$ for all two-dimensional planes $\pi \subset T_{p}M$. Then

$$\begin{aligned} R(e_{1},e_{2},e_{3},e_{4})\le \frac{2}{3}({\bar{\kappa }}-{\underline{\kappa }}) \end{aligned}$$

(13)

for all orthonormal four-frames $\{e_1, e_2, e_3,e_4\}\subset T_{p}M$.

If we take $e_{A}=e_{C}=X, e_{B}=Y, e_{D}=Z$ in the equality (12), then we obtain the following lemma.

Lemma 2.4

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

$$\begin{aligned} R(X,Y,X,Z)=\frac{1}{4}(R(X,Y+Z)-R(X,Y-Z)). \end{aligned}$$

(14)

It is well known that we can express the sectional curvatures by the holomorphic sectional curvatures (see Proposition 2.1 in [2]).

Lemma 2.5

Let $(M,\omega , J)$ be a Kähler manifold. Then

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

(15)

Then we have the following corollary, also see Corollary 2.1 in [2].

COROLLARY 2.2

For any two orthonormal vectors X and Y, if $\langle X,JY\rangle =x$, then

$$\begin{aligned} \begin{aligned} K(X,Y)&=\frac{1}{8}[3(1+x)^{2}H(X+JY)\\&\quad +3(1-x)^{2}H(X-JY)-H(X+Y)-H(X-Y)\\&\quad -H(X)-H(Y)]. \end{aligned} \end{aligned}$$

Thus we have the following Corollary.

COROLLARY 2.3

For any two orthonormal vectors X and Y, if $\langle X,JY\rangle =x$, then

$$\begin{aligned} \frac{1}{4}[(3(1+x^{2})k_{1}-2k_{2}]\le K(X,Y)\le \frac{1}{4}[(3(1+x^{2})k_{2}-2k_{1}] \end{aligned}$$

(16)

Remark 2.2

In fact, Bishop and Goldberg (see Proposition 3.1 in [2]) also obtained the following interesting formula.

Lemma 2.6

Let X, Y be the orthonormal vectors, if $\langle X, JY\rangle =\cos \alpha .$ Denote

$$\begin{aligned} H(X,Y)= & {} \frac{1}{\pi }\int _{0}^{\pi }H(X\cos \alpha +Y\sin \alpha )\mathrm{d}\alpha , \\ A(X,Y)= & {} \frac{1}{\pi }\int _{0}^{\pi }K(X\cos \alpha +Y\sin \alpha , -JX\sin \alpha +JY\cos \alpha )\mathrm{d}\alpha . \end{aligned}$$

Then

$$\begin{aligned} K(X,Y)=H(X,Y)-3A(X,Y)\sin ^{2}\alpha . \end{aligned}$$

If $\cos \alpha =0$, by the Corollary 2.3, we have

$$\begin{aligned} \frac{1}{4}(3k_{1}-2k_{2})\le K(X,Y)\le \frac{1}{4}(3k_{2}-2k_{1}). \end{aligned}$$

On the other hand, for any orthonormal vectors X, Y with $\langle X, JY\rangle \ne 0$ and $|\langle X, JY\rangle |\ne 1$, let ${\tilde{Y}}=\langle X, JY\rangle X- JY$. Then $\langle X, {\tilde{Y}}\rangle =\langle X, J{\tilde{Y}}\rangle =0$ and Span$\{X, JY\}\,=\,$Span$\{X, {\tilde{Y}}\}.$ Hence we obtain

$$\begin{aligned} \frac{3k_{1}-2k_{2}}{4}\le K(X\cos \theta +Y\sin \theta , -JX\sin \theta +JY\cos \theta )\le \frac{3k_{2}-2k_{1}}{4} \end{aligned}$$

for every $\theta \in [0,\pi ].$ Hence for any orthonormal vectors X, Y, we have

$$\begin{aligned} \frac{3k_{1}-2k_{2}}{4}\le A(X,Y) \le \frac{3k_{2}-2k_{1}}{4}. \end{aligned}$$

On the other hand, K(X, Y) also can be expressed as follows:

$$\begin{aligned} K(X,Y)= & {} \frac{1}{4}[(1+\cos \alpha )^{2}H(X+JY)+(1-\cos \alpha )^{2}H(X-JY)]\\&-A(X,Y)\sin ^{2}\alpha . \end{aligned}$$

Then Bishop and Goldberg (Proposition 4.2 in [2]) established the following estimate.

PROPOSITION 2.1

[2]

Let X, Y be the orthonormal vectors with $\langle X, JY\rangle =\cos \alpha$. Then

$$\begin{aligned} k_1-\frac{3k_2}{4}\sin ^{2}\alpha \le K(X,Y)\le k_{2}-\frac{3k_{1}}{4}\sin ^{2}\alpha . \end{aligned}$$

It is easy to check that

$$\begin{aligned} \frac{1}{4}[(3+3\cos ^{2}\alpha )k_{2}-2k_{1}]\ge (or \le )k_{2}-\frac{3k_{1}}{4}\sin ^{2}\alpha , \quad if \ \cos ^{2}\alpha \ge (or \le )1/3. \end{aligned}$$

Lemma 2.7

For the orthonormal basis $\{e_{1},e_{2},e_{3},e_{4}\}$ on (M, g) along $\Sigma _{t}$, it takes the form J as (7). Hence $\cos \alpha , y, z$ are defined by (7). Then we have the following estimates:

• (1) $\frac{1}{4}(3+3\cos ^{2}\alpha )k_{1}-\frac{1}{2}k_{2}\le R_{1212} \le \frac{1}{4}(3+3\cos ^{2}\alpha )k_{2}-\frac{1}{2}k_{1}$;

• (2) $\frac{1}{4}(3+3y^{2})k_{1}-\frac{1}{2}k_{2}\le R_{2424}\le \frac{1}{4}(3+3y^{2})k_{2}-\frac{1}{2}k_{1}$;

• (3) $\frac{1}{4}(3+3z^{2})k_{1}-\frac{1}{2}k_{2}\le R_{2323} \le \frac{1}{4}(3+3z^{2})k_{2}-\frac{1}{2}k_{1}$;

• (4) $\frac{1}{32}[(23+6(\cos \alpha +y)^{2})k_{1}-(23+6(\cos \alpha -y)^{2})k_{2}]\le R_{2131} \le \frac{1}{32}[(23+6(\cos \alpha +y)^{2})k_{2}-(23+6(\cos \alpha -y)^{2})k_{1}]$;

• (5) $\frac{1}{32}[(23+6(\cos \alpha -y)^{2})k_{1}-(23+6(\cos \alpha +y)^{2})k_{2}]\le R_{2434} \le \frac{1}{32}[(23+6(\cos \alpha -y)^{2})k_{2}-(23+6(\cos \alpha +y)^{2})k_{1}]$.

Proof

By (7), we have

• $Je_{1}=\cos \alpha e_{2}+y e_{3}+z e_{4},$

• $Je_{2}=-\cos \alpha e_{1}+z e_{3}-y e_{4},$

• $Je_{3}=-y e_{1}-z e_{2}+\cos \alpha e_{4},$

• $Je_{4}=-ze_{1}+y e_{2}-\cos \alpha e_{3}.$

Hence $\langle Je_{1},e_{2}\rangle =\cos \alpha$, $\langle Je_{4},e_{2}\rangle =y$, $\langle Je_{2},e_{3}\rangle =z$. Then by Corollary 2.3, we get (1)–(3).

By Lemma 2.4,

$$\begin{aligned} R_{1213}=\frac{1}{4}(R(e_{1}, e_{2}+e_{3})-R(e_{1}, e_{2}-e_{3})). \end{aligned}$$

(17)

Hence $Je_{1}=\cos \alpha e_{2}+y e_{3}+z e_{4}$, $\langle Je_{1}, e_{2}+e_{3}\rangle =\cos \alpha +y$ and $\langle Je_{1}, e_{2}-e_{3}\rangle =\cos \alpha -y$. Then by Corollary 2.3,

$$\begin{aligned}&\frac{1}{16}[(27+12(\cos \alpha +y)^{2})k_{1}-19k_{2}]\nonumber \\&\quad \le R(e_{1}, e_{2}+e_{3})\le \frac{1}{16}[(27+12(\cos \alpha +y)^{2})k_{2}-19k_{1}] \end{aligned}$$

(18)

and

$$\begin{aligned}&\frac{1}{16}[(27+12(\cos \alpha -y)^{2})k_{1}-19k_{2}]\le R(e_{1}, e_{2}-e_{3})\nonumber \\&\quad \le \frac{1}{16}[(27+12(\cos \alpha -y)^{2})k_{2}-19k_{1}]. \end{aligned}$$

(19)

Hence

$$\begin{aligned} \begin{aligned} R_{1213}&\le \frac{1}{64}[(46+12(\cos \alpha +y)^{2})k_{2}-(46+12(\cos \alpha -y)^{2})k_{1}]\\&=\frac{1}{32}[(23+6(\cos \alpha +y)^{2})k_{2}-(23+6(\cos \alpha -y)^{2})k_{1}] \end{aligned} \end{aligned}$$

(20)

and

$$\begin{aligned} R_{1213}\ge \frac{1}{32}[(23+6(\cos \alpha +y)^{2})k_{1}-(23+6(\cos \alpha -y)^{2})k_{2}]. \end{aligned}$$

(21)

Hence we obtain (4).

Using Lemma 2.4, Corollary 2.3 and the same argument as in the proof of (4) , we can obtain (5). $\square$

3 Lower bound along the mean curvature flow

In this section, we follow the argument in [10] to prove the first main theorem of this paper, which improves the main theorem in [10].

Theorem 3.1

Suppose M is a Kähler surface with positive holomorphic sectional curvatures. If $1\le \lambda <2$ and $\cos \alpha (\cdot ,0)\ge \delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$, then along the flow

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

(22)

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.

Proof

For simplicity, we can take $y=\sin \alpha , z=0$ in the form of J. Due to the evolution of $\cos \alpha$ (see Lemma 2.2),

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta \right) \cos \alpha =|{\bar{\nabla }}J_{\Sigma _{t}}|^{2}\cos \alpha +\mathrm{Ric}(Je_{1},e_{2})\sin ^{2}\alpha . \end{aligned}$$

(23)

In order to prove this theorem, we need to estimate $\mathrm{Ric}(Je_{1}, e_{2})$. Then

$$\begin{aligned} \begin{aligned} \mathrm{Ric}(Je_{1}, e_{2})&=\sum _{i=1}^{4}R(Je_{1}, e_{i}, e_{2}, e_{i})\\&=\sum _{i=1}^{4}R(\cos \alpha e_{2}+\sin \alpha e_{3}, e_{i}, e_{2}, e_{i})\\&=\cos \alpha R_{22}+\sin \alpha (R_{1213}+R_{4243})\\&=\cos \alpha R_{22}+\sin \alpha R_{23}. \end{aligned} \end{aligned}$$

(24)

By Lemma 2.7, we have

$$\begin{aligned} R_{22}=R_{1212}+R_{3232}+R_{4242} \ge 3k_{1}-\frac{3}{2}k_{2} \end{aligned}$$

(25)

and

$$\begin{aligned} |R_{23}|=|R_{1213}+R_{4243}|\le \frac{29}{16}(k_{2}-k_{1}). \end{aligned}$$

(26)

Hence we have

$$\begin{aligned} \begin{aligned}&\mathrm{Ric}(Je_{1}, e_{2})\\&\quad \ge \cos \alpha \left( 3k_{1}-\frac{3}{2}k_{2}\right) -\sqrt{1-\cos ^{2}\alpha }\frac{29}{16}(k_{2}-k_{1})\\&\quad =\left( 3\cos \alpha +\frac{29}{16}\sqrt{1-\cos ^{2}\alpha }\right) k_{1}-\left( \frac{3}{2}\cos \alpha +\frac{29}{16}\sqrt{1-\cos ^{2}\alpha }\right) k_{2}\\&\quad =k_{1}\left\{ \frac{3}{2}\cos \alpha (2-\lambda )+\frac{29}{16}\sqrt{1-\cos ^{2}}\alpha (1-\lambda )\right\} . \end{aligned} \end{aligned}$$

(27)

We set

$$\begin{aligned} f_{\lambda }(x)=\frac{3}{2}(2-\lambda )x+\frac{29}{16}(1-\lambda )\sqrt{1-x^{2}}. \end{aligned}$$

(28)

When $1\le \lambda <2$, $f_{\lambda }(x)>0$ is equivalent to

$$\begin{aligned} \frac{3}{2}(2-\lambda )x>\frac{29}{16}(\lambda -1)\sqrt{1-x^{2}}. \end{aligned}$$

Furthermore, if $x>0$, $f_{\lambda }(x)>0$ is equivalent to

$$\begin{aligned} \left( \frac{3}{2}(2-\lambda )x\right) ^{2}>\left( \frac{29}{16}(\lambda -1)(1-x^{2})\right) , \end{aligned}$$

it is equivalent to

$$\begin{aligned} \left\{ \left( \frac{3}{2}(2-\lambda )\right) ^{2}+\left( \frac{29}{16}(\lambda -1)\right) ^{2}\right\} x^{2}>\left( \frac{29}{16}(\lambda -1)\right) ^{2}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} x^{2}>\frac{29^{2}(\lambda -1)^{2}}{24^{2}(2-\lambda )^{2}+29^{2}(\lambda -1)^{2}}. \end{aligned}$$

Hence if $1\le \lambda <2$ and $\cos \alpha >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$, we have $f_{\lambda }(\cos \alpha )>0$, that is, $\mathrm{Ric}(Je_{1},e_{2})>0$. Furthermore, if $\cos \alpha \ge \delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$, then $f_{\lambda }(\cos \alpha )\ge f_{\lambda }(\delta )>0$. Then by the maximum principle, the condition $\cos \alpha \ge \delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}$ is preserved by the mean curvature flow. Hence we obtain the theorem. $\square$

Remark 3.1

For the estimate of the term $R_{1213}+R_{4243}$, we use more better estimate of $R_{1213}$ and $R_{4243}$ than the Li–Yang’s estimate [10], which is the key point to improve the Li–Yang’s main result.

We also have the following corollary and theorem as Corollary 1.2 and Theorem 1.3 in [10]. Using the same argument as in [5], we have the following.

COROLLARY 3.1

Suppose M is a Kähler surface with positive holomorphic sectional curvatures and $1\le \lambda <2$. Then every symplectic minimal surface satisfying

$$\begin{aligned} \cos \alpha >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}} \end{aligned}$$

in M is a holomorphic curve.

Using the same argument as in [4] or [14], we have as follows.

Theorem 3.2

Under the same condition of Theorem 3.1, the symplectic mean curvature flow has no type I singularity at any $T>0$.

4 When $\cos \alpha$ is close to 1

In this section, we use the same argument of Han and Li [6]. We prove Kähler manifold M with positive holomorphic sectional curvature and $1\le \lambda <2$, when $\cos \alpha$ is close enough to 1. Then the mean curvature flow exists globally and converges to a holomorphic curve.

PROPOSITION 4.1

Suppose that M is a Kähler surface with positive holomorphic sectional curvature and $1\le \lambda <2$. Let $\alpha$ be the Kähler angle of the surface $\Sigma _{t}$ which evolves by the mean curvature flow. Suppose that $\cos \alpha (\cdot ,0)>\frac{58(\lambda -1)}{\sqrt{(48-24)^{2}+(58\lambda -58)^{2}}}$. Then

$$\begin{aligned} \begin{aligned}&\int _{\Sigma _{t}}\frac{\sin ^{2}\alpha }{\cos \alpha }\mathrm{d}\mu _{t}\le C_{0}\mathrm{e}^{-\frac{3}{4}(2-\lambda )k_{1}t},\\&\int _{t}^{t+1}\int _{\Sigma _{t}}|H|^{2}\mathrm{d}\mu _{t}\mathrm{d}t\le C_{0}\mathrm{e}^{-\frac{3}{4}(2-\lambda )k_{1}t}, \end{aligned} \end{aligned}$$

(29)

where $C_{0}$ is defined by $C_{0}=\int _{\Sigma _{0}}\frac{\sin ^{2}\alpha (x,0)}{\cos \alpha (x,0)}\mathrm{d}\mu _{0}$.

Proof

By Theorem 3.1, we know $\cos \alpha (\cdot ,t)>\frac{58(\lambda -1)}{\sqrt{(48-24)^{2}+(58\lambda -58)^{2}}}$ is preserved along the mean curvature flow. Since $\cos \alpha >\frac{58(\lambda -1)}{\sqrt{(48-24)^{2}+(58\lambda -58)^{2}}}$, then by (27), we have

$$\begin{aligned} \mathrm{Ric}(Je_{1},e_{2})>\frac{3}{4}(2-\lambda )k_{1}\cos \alpha . \end{aligned}$$

Hence

$$\begin{aligned}&\left( \frac{\partial }{\partial t}-\Delta \right) \cos \alpha >|{\bar{\nabla }}J_{\Sigma _{t}}|^{2}\cos \alpha +\frac{3}{4}(2-\lambda )k_{1}\cos \alpha \sin ^{2}\alpha \\&\quad \ge \frac{3}{4}(2-\lambda )k_{1}\cos \alpha \sin ^{2}\alpha . \end{aligned}$$

Then using the same argument as in the proof of Proposition 2.1 in [6], we get the proposition. $\square$

Using the same argument as in the proof of Proposition 2.2 in [6], we also get the following.

PROPOSITION 4.2

Suppose that M is a Kähler surface with positive holomorphic sectional curvature and $1\le \lambda <2$. Let $\alpha$ be the Kähler angle of the surface $\Sigma _{t}$ which evolves by the mean curvature flow. Suppose that $\cos \alpha (\cdot ,0)>\frac{29(\lambda -1)}{\sqrt{(48-24)^{2}+(29\lambda -29)^{2}}}$. Then

$$\begin{aligned} \int _{0}^{T}\int _{\Sigma _{t}}|H|\mathrm{d}\mu _{t}\mathrm{d}t\le (C_{0})^{1/2}\frac{\mathrm{Area}(\Sigma _{0})^{1/2}}{1-\mathrm{e}^{-\frac{3}{8}(2-\lambda )k_{1}}}, \end{aligned}$$

(30)

where the constant $C_{0}$ is defined in Proposition 4.1.

Remark 4.1

Han and Li [6] proved the above propositions in the case of Kähler–Einstein manifold M with positive scalar curvature R. They obtained

$$\begin{aligned} \begin{aligned}&\int _{\Sigma _{t}}\frac{\sin ^{2}\alpha }{\cos \alpha }\mathrm{d}\mu _{t}\le C_{0}\mathrm{e}^{-Rt},\\&\int _{t}^{t+1}\int _{\Sigma _{t}}|H|^{2}\mathrm{d}\mu _{t}\mathrm{d}t\le C_{0}\mathrm{e}^{-Rt},\\&\int _{0}^{T}\int _{\Sigma _{t}}|H|\mathrm{d}\mu _{t}\mathrm{d}t\le (C_{0})^{1/2}\frac{\mathrm{Area}(\Sigma _{0})^{1/2}}{1-\mathrm{e}^{-R/2}}. \end{aligned} \end{aligned}$$

We recall White’s local regularity theorem. Let $H(X,X_{0},t)$ be the backward heat kernel on ${\mathbb {R}}^{4}$. Define

$$\begin{aligned} \rho (X,t)=4\pi (t_{0}-t)H(X,X_{0},t)=\frac{1}{4\pi (t_{0}-t)} \exp \left( -\frac{|X-X_{0}|^{2}}{4(t_{0}-t)}\right) \end{aligned}$$

(31)

for $t<t_{0}$. Let $i_{M}$ be the injective radius of $M^{4}$. We choose a cutoff function $\phi \in C_{0}^{\infty }(B_{2r}(X_{0}))$ with $\phi \equiv 1$ in $B_{r}(X_{0})$, where $X_{0}\in M, 0<2r<i_{M}$. Choose normal coordinates in $B_{2r}(X_{0})$ and express F using the coordinates $(F^{1}, F^{2}, F^{3}, F^{4})$ as a surface in ${\mathbb {R}}^{4}$. The parabolic density of the mean curvature flow is defined by

$$\begin{aligned} \Phi (X_{0}, t_{0}, t)=\int _{\Sigma _{t}}\phi (F)\rho (F,t)\mathrm{d}\mu _{t}. \end{aligned}$$

(32)

The following local regularity theorem was proved by White (see Theorems 3.1 and  4.1 in [15]).

Theorem 4.1

There is a positive constant $\epsilon _{0}>0$ such that if

$$\begin{aligned} \Phi (X_{0},t_{0}, t_{0}-r^{2})\le 1+\epsilon _{0}, \end{aligned}$$

(33)

then the second fundamental form A(t) of $\Sigma _{t}$ in M is bounded in $B_{r/2}(X_{0})$, that is,

$$\begin{aligned} \sup _{B_{r/2}\times (t_{0}-r^{2}/4, t_{0}]}|A|\le C, \end{aligned}$$

(34)

where C is a positive constant depending only on M.

Remark 4.2

Since $\Sigma _{0}$ is smooth, it is well known that

$$\begin{aligned} \lim _{r\rightarrow 0}\int _{\Sigma _{0}}\phi (F)\frac{\mathrm{e}^{-(|F-X_{0}|^{2}/4r^{2})}}{4\pi r^{2}}\mathrm{d}\mu _{0}=1 \end{aligned}$$

for any $X_{0}\in \Sigma _{0}.$ So we can find a sufficiently small $r_{0}$ such that

$$\begin{aligned} \int _{\Sigma _{0}}\phi (F)\frac{\mathrm{e}^{-(|F-X_{0}|^{2}/4r_{0}^{2})}}{4\pi r_{0}^{2}}\mathrm{d}\mu _{0}\le 1+\frac{\epsilon _{0}}{2}, \end{aligned}$$

i.e.,

$$\begin{aligned} \Phi (X_{0}, r_{0}^{2}, 0)\le 1+\frac{\epsilon _{0}}{2} \end{aligned}$$

for all $X_{0}\in M$, where $\epsilon _{0}$ is the constant in White’s theorem.

Using the same argument as in the proof of Theorem 2.5 in [6], we get the following theorem.

Theorem 4.2

Suppose that M is a Kähler surface with positive holomorphic sectional curvature and $1\le \lambda <2$. Let $\alpha$ be the Kähler angle of the surface $\Sigma _{t}$ which evolves by the mean curvature flow. Suppose that $\cos \alpha (\cdot ,0)>\frac{29(\lambda -1)}{\sqrt{(48-24)^{2}+(29\lambda -29)^{2}}}$. $C_{0}, r_{0}, \epsilon _{0}$ are defined as in Proposition 4.1, Remark 4.2 and Theorem 4.1, respectively. We denote

$$\begin{aligned} \epsilon _{1} = \frac{\pi ^{2}\epsilon _{0}^{2}r_{0}^{6}(1-\mathrm{e}^{-\frac{3}{8}(2-\lambda )k_{1}})^{2}}{4 \,\mathrm{Area}(\Sigma _{0})}. \end{aligned}$$

Then if $C_{0}\le \epsilon _{1}$, the mean curvature flow with initial surface $\Sigma _{0}$ exists globally and it converges to a holomorphic curve.