Abstract A surface is a graph in 4 if there is a unit constant 2-form on 4 such that v00 where {e1, e2} is an orthonormal frame on . We prove that, if v0[MATHEMATICAL FORMULA] 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 M1M2 where M1 and M2 are Riemann surfaces, if v00 where 1 is a Khler form on M1. We prove that, if M is a Khler-Einstein surface with scalar curvature R, v0[MATHEMATICAL FORMULA] 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.