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

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 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.