Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds

TL;DR

This paper establishes universal inequalities for the eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds, with specific results for hyperbolic space, advancing understanding of spectral geometry in these settings.

Contribution

It introduces new universal inequalities for Dirichlet Laplacian eigenvalues on conformally flat manifolds, including hyperbolic space, which were previously unknown.

Findings

Universal eigenvalue inequalities for hyperbolic space.

Extension of inequalities to conformally flat Riemannian manifolds.

Advancement in spectral geometry understanding.

Abstract

In this paper we study eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds. In particular we establish some universal inequality for eigenvalues of the Dirichlet Laplacian on the hyperbolic space $\mathbb{H}^n(-1)$.