A new pinching theorem for complete self-shrinkers and its generalization

TL;DR

This paper establishes a pinching theorem for complete self-shrinkers and extends it to $ ext{lambda}$-hypersurfaces, showing under certain conditions they must be spheres or cylinders, thus characterizing their geometric structure.

Contribution

It introduces a new pinching theorem for complete self-shrinkers and generalizes it to $ ext{lambda}$-hypersurfaces with polynomial volume growth, identifying conditions for them to be spheres or cylinders.

Findings

If $|A|^2$ is close to 1, the self-shrinker is a sphere or cylinder.

For $ ext{lambda}$-hypersurfaces with small $ ext{lambda}$, $|A|^2$ is constant and the hypersurface is a cylinder.

The results provide sharp conditions for the geometric classification of self-shrinkers and $ ext{lambda}$-hypersurfaces.

Abstract

In this paper, we firstly verify that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$, and if the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{18}$, then $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. More generally, let $M$ be a complete $\lambda$-hypersurface with polynomial volume growth in $\mathbb{R}^{n+1}$ with $\lambda\neq0$. Then we prove that there exists an positive constant $\gamma$, such that if $|\lambda|\leq\gamma$ and the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|^2-\beta_\lambda\leq\frac{1}{18}$, then $|A|^2\equiv \beta_\lambda$, $\lambda>0$ and $M$ is a cylinder. Here $\beta_\lambda=\frac{1}{2}(2+\lambda^2+|\lambda|\sqrt{\lambda^2+4})$.