Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds

Abstract

In this paper, we investigate eigenvalues of the eigenvalue problem with Dirichlet boundary condition of the drifting Laplacian on an n-dimensional, complete noncompact Riemannian manifold. Some estimates for eigenvalues are obtained. By utilizing Cheng and Yang recursion formula, we give a sharp upper bound of the kth eigenvalue. As we know, product Riemannian manifolds, Ricci solitons and self-shrinkers are some important Riemannian manifolds. Therefore, we investigate the eigenvalues of the drifting Laplacian on those Riemannian manifolds. In particular, by some theorems of classification for Ricci solitons, we can obtain some eigenvalue inequalities of drifting Laplacian on the Ricci solitons with certain conditions.