Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Abstract

Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin 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 Kähler forms ω' and ω'' that determine different orientations andΣis symplectic with respect to both ω' and ω'', we prove the mean curvature flow of Σexists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.