Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds

TL;DR

This paper establishes universal eigenvalue inequalities for the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds, using Bochner formulas and comparison theorems, with applications to rigidity and the fundamental gap conjecture.

Contribution

It introduces new eigenvalue bounds for the drifted Cheng-Yau operator in curved manifolds, extending known results and providing a unified approach for various cases.

Findings

Derived universal eigenvalue inequalities for the operator.

Established a Rauch comparison theorem for the Cheng-Yau operator.

Proved a rigidity result for bounded annular domains.

Abstract

We show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our universal inequalities. In particular, we obtain the corresponding inequalities for both Cheng-Yau operator and drifted Laplacian cases, and we recover the known inequalities for the Laplacian case. We also obtain a rigidity result for a Cheng-Yau operator on a class of bounded annular domains in a pinched Cartan-Hadamard manifold. In particular, we can use, e.g., the potential function of the Gaussian shrinking soliton to obtain such a rigidity for the Euclidean space case. The fundamental gap conjecture is also addressed in this paper.