A note on the volume entropy of harmonic manifolds of hypergeometric type

TL;DR

This paper investigates the volume entropy of harmonic manifolds of hypergeometric type, establishing bounds and characterizing cases of equality, including real hyperbolic spaces and specific Damek-Ricci spaces.

Contribution

It provides new inequalities for volume entropy in harmonic manifolds of hypergeometric type and characterizes the manifolds that attain these bounds.

Findings

Upper bound of volume entropy achieved only by real hyperbolic spaces.

Existence of Damek-Ricci spaces attaining the lower entropy bound in four cases.

Normalized Ricci curvature condition used to derive entropy inequalities.

Abstract

Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature $\mathrm{Ric} = -(n-1)$, we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature $-1$, while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases.