Estimates for the first eigenvalue of Jacobi operator on hypersurfaces with constant mean curvature in spheres

TL;DR

This paper derives an optimal upper bound for the first eigenvalue of the Jacobi operator on constant mean curvature hypersurfaces in spheres, depending only on mean curvature and dimension.

Contribution

It provides the first explicit upper bound for the first eigenvalue of the Jacobi operator in this geometric setting, depending solely on mean curvature and dimension.

Findings

Established an optimal upper bound for the eigenvalue

Bound depends only on mean curvature and dimension

Results apply to non-totally umbilical hypersurfaces

Abstract

In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$.