Some extensions of the mean curvature flow in Riemannian manifolds

TL;DR

This paper investigates the behavior of mean curvature flow in Riemannian manifolds, showing that specific geometric quantities become unbounded at the first singularity, extending previous Euclidean and symmetric manifold results.

Contribution

It generalizes existing results on mean curvature flow singularities to locally symmetric Riemannian manifolds with bounded geometry.

Findings

Subcritical quantities blow up at the first singular time.

Generalizes results from Euclidean space to Riemannian manifolds.

Extends recent work by Le, Xu-Ye-Zhao, and others.

Abstract

Given a family of smooth immersions $F_t: M^n\to N^{n+1}$ of closed hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded geometry, moving by the mean curvature flow, we show that at the first finite singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a recent result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of N. Le in the Euclidean case (arXiv: math.DG/1002.4669v2).