Long time existence of the symplectic mean curvature flow

Details Long time existence of the symplectic mean curvature flow Long time existence of the symplectic mean curvature flow

2011-07-05

arxiv.org/abs/1107.0829v1

Mathematics

Differential Geometry

17pages

Scientific paper

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 $\cos\alpha\ge251/265$.

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