Long time existence of the symplectic mean curvature flow

TL;DR

This paper proves long-time existence and convergence of the symplectic mean curvature flow on certain Kähler surfaces, under specific initial curvature and angle conditions, leading to holomorphic limit surfaces.

Contribution

It establishes conditions ensuring the long-term existence and convergence of the symplectic mean curvature flow on Kähler surfaces with positive holomorphic sectional curvature.

Findings

Flow exists for a long time under given curvature bounds.

Flow converges to a holomorphic curve.

Specific initial conditions guarantee convergence.

Abstract

Let $(M,\bar{g})$ be a K\"ahler surface with a constant holomorphic sectional curvature $k>0$, and $\Sigma$ an immersed symplectic surface in $M$. Suppose $\Sigma$ evolves along the mean curvature flow in $M$. In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies $|A|^2\leq 2/3|H|^2+1/2 k$ and $\cos\alpha\geq \frac{\sqrt{30}}{6}$ or $|A|^2\leq 2/3 |H|^2+4/5 k\cos\alpha$ and $\cos\alpha\ge251/265$.