Self-Shrinkers With Second Fundamental Form of Constant Length

TL;DR

This paper provides a new proof that smooth complete self-shrinkers in three-dimensional space with constant second fundamental form are generalized cylinders, and establishes a gap theorem for self-shrinkers across all dimensions.

Contribution

It introduces a simplified proof of a classification result for self-shrinkers and extends the understanding through a new gap theorem applicable in all dimensions.

Findings

Self-shrinkers with constant second fundamental form are generalized cylinders.

A new gap theorem for smooth self-shrinkers in all dimensions.

Simplified proof technique for classification results.

Abstract

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k \times \mathbb{R}^{2-k}$ for some $k\leq2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.