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

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: $W(M)\geq n^{\frac{n}{2}}Vol(M)$.