About Bounds for Eigenvalues of the Laplacian with Density

Abstract

Let M denote a compact, connected Riemannian manifold of dimension n ∈ N. We assume that M has a smooth and connected boundary. Denote by g and dv g respectively, the Riemannian metric on M and the associated volume element. Let ∆ be the Laplace operator on M equipped with the weighted volume form dm := e −h dv g. We are interested in the operator L h • := e −h(α−1) (∆ • +αg(∇h, ∇•)), where α > 1 and h ∈ C 2 (M) are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian L h with the Neumann boundary condition if the boundary is non-empty.