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

Let f:\Sigma_1 --> \Sigma_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 \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable conditions on the curvature of \Sigma_1 and \Sigma_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.Comment: to be published in Inventiones Mathematica

Mean curvature flows and isotopy problems

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.Comment: 10 pages, contribution to "Survey in Differential Geometry

Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.Comment: submitte