New result on Chern conjecture for minimal hypersurfaces and its application

TL;DR

This paper proves new rigidity results for minimal hypersurfaces in spheres and self-shrinkers in Euclidean space, showing under certain curvature bounds they must be specific well-known geometric objects, thus advancing the understanding of the Chern conjecture.

Contribution

It establishes sharper curvature bounds that guarantee minimal hypersurfaces are Clifford tori and self-shrinkers are spheres or cylinders, improving previous rigidity theorems.

Findings

Minimal hypersurfaces with curvature bounds are Clifford tori.

Self-shrinkers with curvature bounds are spheres or cylinders.

Results improve existing rigidity theorems by Ding and Xin.

Abstract

We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$ whose equation is given by (\ref{selfshr}), and if the squared length of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{21}$, then $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. Our results improve the rigidity theorems due to Q. Ding and Y. L. Xin \cite{DX1,DX2}.