A gap theorem of self-shrinkers

TL;DR

This paper classifies complete self-shrinkers in Euclidean space with constant second fundamental form norm below a certain threshold, showing they are isometric to standard models like Euclidean space or spheres.

Contribution

It establishes a gap theorem for self-shrinkers with constant second fundamental form norm, identifying specific geometric models under a new curvature bound.

Findings

Self-shrinkers with constant S and S<(10/7) are isometric to standard models.

Classifies self-shrinkers with polynomial volume growth in Euclidean space.

Provides a new curvature threshold for geometric classification.

Abstract

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker with polynomial volume growth in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$, or $\mathbb{R}^{n-m}\times S^m (\sqrt{m})$, $1\leq m\leq n-1$, if the squared norm $S$ of the second fundamental form is constant and satisfies $S<(10/7)$.