Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface ∑ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that (e₁ ∧ a₂, ω) ≥ v₀ > 0 where {e₁,a₂} is an orthonormal frame on ∑. We prove that, if v₀ ≥ 1/2√ 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 M₁ × M₂ where M₁ and M₂ are Riemann surfaces, if (e₁ ∧ e₂,ω1) ≥ V₀ > 0 where ω₁ is a Kähler form on M₁. We prove that, if M is a Kähler-Einstein surface with scalar curvature R, V₀ ≥ 1/√2 on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R ≥ 0 it converges to a totally geodesic surface which is holomorphic.