Mean curvature flow of surfaces in einstein four-manifolds

Abstract

Let Σ be a compact oriented surface immersed in a four dimensional Kähler- Einstein manifold (M,ω). We consider the evolution of Σ 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 Kähler forms ω′ and ω″ that determine different orientations and Σ is symplectic with respect to both ω′ and ω″, we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity. © 2001 Applied Probability Trust.