Lower Bounds For The First Eigenvalue Of The Laplacian With Dirichlet Boundary Conditions In A Hyperbolic Space Of A Negative Constant Curvature

TL;DR

This paper develops a new technique to improve lower bounds for the first eigenvalue of the Laplacian with Dirichlet boundary conditions in hyperbolic spaces of negative constant curvature.

Contribution

It introduces a novel approach that enhances existing bounds for eigenvalues in negatively curved spaces, building on prior work by Savo (2009).

Findings

Improved lower bounds for the first eigenvalue in hyperbolic spaces.

Development of a new technique for eigenvalue estimation.

Extension of previous results to broader classes of domains.

Abstract

In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.