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 $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb {S}^{n+1}(1),$ $n\geq 5$, and let $J_s$ be the Jacobi operator of $M$. In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of $J_s$ of the hypersurface with constant scalar curvature $n(n-1)$ in $\mathbb {S}^{n+1}(1),~n\geq 3$. In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator $J_s$ of the hypersurface with constant scalar curvature $n(n-1)r, r>1$, in $\mathbb {S}^{n+1}(1)$. In this paper, we study the second eigenvalue of the Jacobi operator $J_s$ of $M$ and give an optimal upper bound for the second eigenvalue of $J_s$.