Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

TL;DR

This paper characterizes hyperbolic space among compact Riemannian manifolds with Ricci curvature bounds by establishing a sharp spectral and entropy estimate, showing equality iff the manifold is hyperbolic.

Contribution

It provides a new characterization of hyperbolic space using spectral bounds and entropy estimates, extending previous results.

Findings

Equality of the bottom of the spectrum and a specific value characterizes hyperbolic space.

A sharp estimate for the Kaimanovich entropy is established.

The result links geometric, spectral, and entropy properties of manifolds.

Abstract

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is well known that the bottom of spectrum $\lambda_{0}$ of its unverversal covering satisfies $\lambda_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.