Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface is a graph in 4 if there is a unit constant 2-form on 4 such that >e 1e 2, v 0>0 where {e 1, e 2} is an orthonormal frame on . We prove that, if 012 on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface is a graph in M 1M 2 where M 1 and M 2 are Riemann surfaces, if >e 1e 2, 1>v 0>0 where 1 is a Khler form on M 1. We prove that, if M is a Khler-Einstein surface with scalar curvature R,012 on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R0 it converges to a totally geodesic surface which is holomorphic.