Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds

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.

arXiv commentFrom the remark [arXiv:1304.3813] of Professor S. Ilias, we will withdraw this paper [arXiv:1304.3212] from arXiv. In fact, we did not know the relation of Witten-Laplacian and Schrödinger operator before his remark. Hence, we would like to thank Professor Ilias for his information

References in corpus (2)

• Eigenvalue estimate and compactness for closed $f$-minimal surfaces
• Stability and compactness for complete $f$-minimal surfaces