Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

Abstract

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature.

16 pages, comments are welcomed! The paper is revised according to the comments of the journal referees, and some references are added. Accepted by The Journal of Geometric Analysis