New inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space

TL;DR

This paper establishes new eigenvalue inequalities for the Dirichlet Laplacian on hyperbolic space, verifying Cheng's conjecture for specific bounded domains with some approximation.

Contribution

It introduces novel inequalities for eigenvalues in hyperbolic space and confirms Cheng's conjecture for particular domain types.

Findings

Proved new inequalities for Dirichlet Laplacian eigenvalues in hyperbolic space.

Verified Cheng's conjecture up to an epsilon loss for certain bounded domains.

Extended understanding of eigenvalue bounds in non-Euclidean geometries.

Abstract

In this paper, motivated by study on universal inequalities for eigenvalues of the Dirichlet Laplacian, we prove some new inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space. In particular, we verify Cheng's conjecture (Adv. Lect. Math. 37, 2017) up to loss of $\epsilon$ for two special kinds of bounded domains in the hyperbolic space.