Eigenvalues of the bi-drifting Laplacian on the complete noncompact Riemannian manifolds

In this paper, we study the eigenvalues of bi-drifting Laplacian in the bounded domain in the complete noncompact Riemannian manifolds. By establishing a theorem of Barta type, we prove a universal inequality for bi-drifting Laplacian with a specific class of potential functions on the hyperbolic space, which can be viewed as a rigidity result in term of variables. As an application, we investigate the asymptotic behavior of the eigenvalues with any order when the bounded domain tends to the hyperbolic space. In addition, we obtain some eigenvalue inequalities for bi-drifting Laplacian on the complete Riemannian manifold with pinching condition of sectional curvature and without the assumption for Bakry-emery curvature.

Automated summary

This paper studies the eigenvalues of bi-drifting Laplacian, establishes a universal inequality for bi-drifting Laplacian with a specific class of potential functions on the hyperbolic space, and investigates the asymptotic behavior of the eigenvalues as the bounded domain tends to the hyperbolic space.