Long-time Existence and Convergence of Graphic Mean Curvature Flow in Arbitrary Codimension

Abstract

Let f:Σ_1 --> Σ_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of Σ_1 and Σ_2 by the mean curvature flow. Under suitable conditions on the curvature of Σ_1 and Σ_2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f_t and f_t converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschtz initial data.