The second gap on complete self-shrinkers

TL;DR

This paper classifies complete self-shrinkers in Euclidean space under specific curvature conditions, showing they are isometric to standard models without assuming polynomial volume growth.

Contribution

It provides a new classification of complete self-shrinkers with constant second fundamental form norm and third fundamental form, under a curvature bound, without volume growth assumptions.

Findings

Self-shrinkers are isometric to Euclidean space, spheres, or products of spheres and Euclidean space.

The classification holds when the squared norm of the second fundamental form is less than 1.83379.

Polynomial volume growth condition is not required for the classification.

Abstract

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$, or $S^k (\sqrt{k})\times\mathbb{R}^{n-k}$, $1\leq k\leq n-1$, if the squared norm $S$ of the second fundamental form, $f_3$ are constant and $S$ satisfies $S<1.83379$. We should remark that the condition of polynomial volume growth is not assumed.