Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Abstract

Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel K\"ahler forms \omega' and \omega'' that determine different orientations and\Sigma is symplectic with respect to both \omega' and \omega'', we prove the mean curvature flow of \Sigma exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.