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

TL;DR

This paper introduces a new entropy concept for submanifolds in Riemannian manifolds, demonstrating its monotonicity along mean curvature flow and applying it to establish partial regularity of flow limits in specific curved spaces.

Contribution

It defines a generalized mean curvature flow entropy in Riemannian manifolds and proves its monotonicity, leading to partial regularity results for surface flows.

Findings

Entropy is equivalent to area growth in certain manifolds.

Entropy is monotone along mean curvature flow.

Partial regularity of flow limits is established in specific curved manifolds.

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.