Long time existence of the symplectic mean curvature flow

Abstract

Let $(M,\bar{g})$ be a Kähler 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 $|A{|}^2\le 2/3|H{|}^2+1/2k$ and $\cos \alpha \ge \frac{\sqrt{30}}{6}$ or $|A{|}^2\le 2/3|H{|}^2+4/5k\cos \alpha$ and $\cos \alpha \ge 251/265$.

17pages

References in corpus (1)

• Translating solutions to Lagrangian mean curvature flow