A mean value inequality for the generalized self-expander type submanifolds and its application

TL;DR

This paper establishes a mean value inequality for generalized self-expander submanifolds in Euclidean space and applies it to show that certain mean curvature flows converge to cones at singularities.

Contribution

It introduces a new mean value inequality for generalized self-expander submanifolds and demonstrates its application in analyzing the singularity formation in mean curvature flow.

Findings

Mean value inequality for generalized self-expander submanifolds

Convergence of mean curvature flow to cones at singularities

Characterization of limit varifolds as cones

Abstract

In this paper we get a version of mean value inequality for generalized self-expander type submanifolds in Euclidean space. As the application, we prove that if mean curvature flow $M(t)$ on the self-expander in Euclidean space subconverges to an $n$-rectifiable varifold $T$ in weak sense for $t$ goes to the singular time, then $T$ must be the cone.