Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature

Abstract

Let $x:M\to {\mathbb{S}}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,r\ge 1$, in a unit sphere ${\mathbb{S}}^{n+1}(1),n\ge 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral ${\int }_{M}Hdv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riemannian product ${\mathbb{S}}^m(c)\times {\mathbb{S}}^{n-m}(\sqrt{1-c^2}),1\le m\le n-2,c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.

arXiv commentCorrected typos