On embedded minimal hypersurfaces in $S^{n+1}$ with symmetries

TL;DR

This paper extends the characterization of the Clifford torus among minimal hypersurfaces in spheres with symmetries, providing new inequalities involving the second fundamental form and implications for the Willmore energy.

Contribution

It generalizes a known characterization of the Clifford torus to hypersurfaces with specific symmetries and establishes new inequalities related to the second fundamental form and Willmore energy.

Findings

The average of the squared second fundamental form satisfies $ar S \,\geq\, n$ with equality only for Clifford tori.

A Simons'-type theorem states that if the integral of $(n - S)$ is non-negative, then $S$ is either identically zero or equal to $n$.

The Willmore energy of such hypersurfaces satisfies $W(M) \geq n^{\frac{n}{2}} Vol(M)$.

Abstract

In this note, we generalize a characterization of the Clifford torus due to Ros. Let $f:M\rightarrow S^{n+1}$ be an embedded closed minimal hypersurface. Assume there are $(n+2)$ great hyperspheres of $S^{n+1}$ perpendicular to each other, such that $M$ is symmetric with respect to them. Let $S$ denote the square of the length of the second fundamental form of $f$ and let $\bar S=\frac{1}{Vol(M)}\int_{M} Sd M$ be the average of $S$. Then $\bar S\geq n$ with equality holding if and only if $f$ is the Clifford torus $C_{m,n-m}$. It can be rewritten as a Simons' type theorem: If $0\leq \int_M (n-S)d M$, then either $S\equiv0$ or $S\equiv n$. This answers partially a conjecture by Perdomo. Moreover, the estimate of the Willmore energy of $f$ is built: $W(M)\geq n^{\frac{n}{2}}Vol(M)$.