Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds

TL;DR

This paper investigates eigenvalues of the Witten-Laplacian on compact Riemannian manifolds, providing bounds and generalizations that enhance understanding of spectral properties in geometric analysis.

Contribution

It offers new eigenvalue estimates, sharp bounds for specific eigenvalues, and generalizes Reilly's result on the first eigenvalue for the Witten-Laplacian.

Findings

Sharp upper bounds for the k-th eigenvalue.

Explicit upper bound for the (n+3)-th eigenvalue.

Generalization of Reilly's result on the first eigenvalue.

Abstract

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the $k^{\text{th}}$ eigenvalue and for isoparametric minimal hypersurfaces in the unit sphere, an explicit upper bound of the $(n+3)^{\text{th}}$ eigenvalue of the Laplacian is obtained. Furthermore, we generalize the Reilly's result on the first eigenvalue of the Laplacian.