Remarks on symplectic mean curvature flows in K\"ahler surfaces with positive holomorphic sectional curvatures

Abstract

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $\lambda$ of the maximum and the minimum of the holomorphic sectional curvatures $< 2$, then there exists a positive constant $\delta>\frac{29(\lambda-1)}{\sqrt{(48-24\lambda)^{2}+(29\lambda-29)^{2}}}$ such that $\cos\alpha\geq\delta$ is preserved along the flow, improving the main theorem in [LY]; Secondly, as similar as the main theorem in [HL0], we prove that when $\cos\alpha$ is close to $1$ enough, then the symplectic mean curvature flow exists for long time and converges to a holomorphic curve; Finally, we prove that the symplectic mean curvature flow on K\"ahler surfaces with $\lambda\leq 1+\frac{1}{200}$ exists for long time and converges to a holomorphic curve if the initial surface satisfies a pinching condition, which generalize one of the main theorems in [HLY].