A new characterization of the Clifford torus via scalar curvature pinching

TL;DR

This paper characterizes the Clifford torus among hypersurfaces in spheres by establishing a scalar curvature pinching condition that leads to a precise geometric classification.

Contribution

It introduces a new curvature pinching condition involving mean curvature and second fundamental form that uniquely characterizes the Clifford torus and certain product spheres.

Findings

Under the pinching condition, the hypersurface is isometric to a product of spheres.

The scalar curvature bounds imply the hypersurface is a Clifford torus or a similar product.

The result provides a new geometric characterization of the Clifford torus.

Abstract

Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $\mathbb{S}^{n+1}$. Denote by $S$ the squared norm of the second fundamental form of $M$. We prove that there exists a positive constant $\gamma(n)$ depending only on $n$ such that if $|H|\leq\gamma(n)$ and $\beta(n,H)\leq S\leq\beta(n,H)+\frac{n}{23}$, then $S\equiv\beta(n,H)$ and $M$ is one of the following cases: (i) $\mathbb{S}^{k}(\sqrt{\frac{k}{n}})\times \mathbb{S}^{n-k}(\sqrt{\frac{n-k}{n}})$, $\,1\le k\le n-1$; (ii) $\mathbb{S}^{1}(\frac{1}{\sqrt{1+\mu^2}})\times \mathbb{S}^{n-1}(\frac{\mu}{\sqrt{1+\mu^2}})$. Here $\beta(n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$ and $\mu=\frac{n|H|+\sqrt{n^2H^2+4(n-1)}}{2}$. This provides a new characterization of the Clifford torus.