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 Σ 1 × 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.

On graphic Bernstein type results in higher codimension

“…We first recall a formula whose parabolic version was derived in [13]. To apply the formula in [13] to the current situation, we note that a minimal submanifold corresponds to the stationary phase of mean curvature flow.…”

A Bernstein Type Result for Special Lagrangian Submanifolds

“…When Σ is the graph over a Lagrangian subspace L, we calculate the Laplacian of ln * Ω where * Ω is the Jacobian of the projection from Σ to L. This is a positive function and when the Gauss map of Σ satisfies the above conditions it is indeed superharmonic. The parabolic version of this equation was first derived in [10] in the study of higher co-dimension mean curvature flows.…”

“…The following formula was essentially derived in [10]. To apply to the current situation, we note that a minimal submanifold corresponds to a stationary phase of the mean curvature flow.…”

Entire mean curvature flows of graphs

“…see [Wang 2002;Chen et al 2002;Li and Li 2003]. This formula plays the important role in these papers.…”