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

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

2001-12-28

arxiv.org/abs/math/0112297v1

Mathematics

Differential Geometry

to be published in Inventiones Mathematicae

Scientific paper

10.1007/s002220100201

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.

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