Estimates of the first Dirichlet eigenvalue from exit time moment spectra

TL;DR

This paper links the first Dirichlet eigenvalue of geodesic balls in rotationally symmetric spaces to exit time spectra of Brownian motion, providing precise estimates and bounds with applications to submanifold eigenvalues.

Contribution

It introduces a novel method to compute and estimate the first Dirichlet eigenvalue using exit time spectra, extending classical results to broader geometric contexts.

Findings

Exact formulas for eigenvalues in model spaces

New bounds for eigenvalues of submanifolds

Generalizations of McKean and Cheung-Leung results

Abstract

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.