New features of the first eigenvalue on negatively curved spaces

TL;DR

This paper investigates the properties of the first eigenvalue in negatively curved spaces, providing sharp asymptotic expansions, a synthetic proof of Cheng's comparison theorem, and examples contrasting Riemannian and Finsler geometries.

Contribution

It offers new asymptotic formulas for eigenvalues in hyperbolic spaces, a synthetic proof of Cheng's theorem in metric measure spaces, and constructs examples of Finsler manifolds with zero first eigenvalue.

Findings

Asymptotically sharp eigenvalue expansions in hyperbolic space

A synthetic proof of Cheng's eigenvalue comparison theorem

Existence of Finsler manifolds with zero first eigenvalue

Abstract

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model $n$-dimensional hyperbolic space, complementing the results of Borisov and Freitas (\textit{Comm. Anal. Geom.} 25: 507--544, 2017). We then give a synthetic proof of Cheng's sharp eigenvalue comparison theorem in metric measure spaces satisfying a 'negatively curved' Bishop-Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (\textit{J. Differential Geom.} 4: 359--366, 1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.