Classification and rigidity of self-shrinkers in the Mean curvature flow

TL;DR

This paper extends classification and rigidity results for self-shrinkers in mean curvature flow, demonstrating that existing theorems hold under weaker conditions and providing new rigidity results across various codimensions.

Contribution

It generalizes Smoczyk's classification theorem to weaker conditions and establishes new rigidity results for self-shrinkers in arbitrary codimension.

Findings

Classification theorem holds under weaker conditions

Rigidity results for self-shrinkers in arbitrary codimension

Extension of existing results to broader settings

Abstract

In this paper, we first use the method of Colding and Minicozzi [5] to show that K. Smoczyk's classification theorem [16] for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.