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

TL;DR

This paper establishes an optimal upper bound for the second eigenvalue of the Jacobi operator on compact hypersurfaces with constant scalar curvature in a sphere, characterizing cases of equality with specific geometric conditions.

Contribution

It derives the first optimal upper bound for the second eigenvalue of the Jacobi operator on such hypersurfaces and characterizes the equality cases.

Findings

Optimal upper bound for the second eigenvalue of the Jacobi operator.

Equality cases characterized by totally umbilical hypersurfaces or specific Riemannian products.

Abstract

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ 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 $\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.