Eigenvalues on Spherically Symmetric Manifolds

TL;DR

This paper investigates Dirichlet Laplace eigenvalues on balls within spherically symmetric manifolds, comparing them to Euclidean cases and revealing how curvature influences eigenvalue sizes.

Contribution

It provides a comparison of eigenvalues on spherically symmetric manifolds with Euclidean spaces, highlighting curvature effects on small and hyperbolic spaces.

Findings

Eigenvalues on small spherical balls are smaller than Euclidean counterparts.

Eigenvalues on hyperbolic space balls are larger than Euclidean counterparts.

Curvature affects eigenvalues differently depending on the manifold type.

Abstract

In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.