Asymptotic behavior of the first Dirichlet eigenvalue of AHE manifolds

TL;DR

This paper studies how the first Dirichlet eigenvalue of geodesic balls in AHE manifolds behaves asymptotically, showing it matches hyperbolic space under certain conditions on the conformal infinity.

Contribution

It establishes the two-term asymptotic of the eigenvalues for AHE manifolds with nonnegative Yamabe type conformal infinity, extending hyperbolic space results.

Findings

Eigenvalue decay rate matches hyperbolic space asymptotics

Asymptotic behavior depends on conformal infinity properties

Results apply to AHE manifolds with nonnegative Yamabe type

Abstract

In this article, we investigate the rate at which the first Dirichlet eigenvalue of geodesic balls decreases as the radius approaches infinity. We prove that if the conformal infinity of an asymptotically hyperbolic Einstein manifold is of nonnegative Yamabe type, then the two-term asymptotic of the eigenvalues is the same as that in hyperbolic space.