Complete self-shrinkers of the mean curvature flow

TL;DR

This paper studies complete self-shrinkers in Euclidean space, providing new estimates and rigidity theorems without requiring polynomial volume growth, and classifies certain low-dimensional cases with constant second fundamental form.

Contribution

Introduces a generalized maximum principle for the -operator, removing the polynomial volume growth assumption in self-shrinker estimates and classifications.

Findings

Established estimates on the second fundamental form without polynomial volume growth.

Proved rigidity theorems for complete self-shrinkers.

Classified 2- and 3-dimensional proper self-shrinkers with constant second fundamental form.

Abstract

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is assumed in Cao and Li. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on \emph{polynomial volume growth}. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.