Characterization of hypersurfaces via the second eigenvalue of the Jacobi operator

TL;DR

This paper characterizes specific hypersurfaces in various ambient manifolds by analyzing the second eigenvalue of the Jacobi operator, identifying the Clifford torus and slices of warped products as extremal cases.

Contribution

It provides new characterizations of hypersurfaces using the second eigenvalue of the Jacobi operator, including the maximality of the Clifford torus and the uniqueness of slices in warped products.

Findings

Clifford torus maximizes the second eigenvalue among certain surfaces in S^3.

Slices of warped products are uniquely characterized by an integral inequality involving the second eigenvalue.

In R x S^n, the second eigenvalue of hypersurfaces is bounded above by n, with slices achieving equality.

Abstract

In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $\mathbb S^3$ with genus bigger than zero. After, we characterize the slices of the warped product $I\times_h\mathbb S^n$, under a suitable hypothesis on the warping function $h:I\subset\mathbb R\to\mathbb R$, as the only hypersurfaces which saturate a certain integral inequality involving the second eigenvalue of the Jacobi operator. As a consequence, we obtain that if $\Sigma$ is a closed immersed hypersurface of $\mathbb R\times\mathbb S^n$, then the second eigenvalue of the Jacobi operator of $\Sigma$ satisfies $\lambda_2\le n$ and the slices are the only hypersurfaces which satisfy $\lambda_2=n$.