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

TL;DR

This paper investigates the behavior of symplectic mean curvature flows in K"ahler surfaces with positive holomorphic sectional curvatures, establishing conditions for long-term existence and convergence to holomorphic curves.

Contribution

It improves existing results by providing new bounds for the preservation of the angle and extends long-time existence and convergence results under broader conditions.

Findings

Preservation of a lower bound for oslpha along the flow.

Long-time existence and convergence to holomorphic curves when oslpha is close to 1.

Long-time existence and convergence under a pinching condition for ased on \u03bb  + 1/200.

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].