Estimates for the higher eigenvalues of the drifting Laplacian on Hadamard manifolds

Abstract

In this paper, we investigate the eigenvalues of the drifting Laplacian on the bounded domain in Hardamard manifolds. By using upper half-plane model, we establish a universal inequality for the drifting Laplacian with a specific class of potential functions on the hyperbolic space, which can be viewed as a rigidity result associated with such a class of functions. Furthermore, we consider general Hardamard manifolds with pinching condition of sectional curvature, and obtain an eigenvalue inequality without the condition of Bakry-Émery curvature. As a by-product, we obtain a universal bound for the case of radial drifting Laplacian satisfying certain growth condition along with the radial direction.