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arXiv · 2405.05896

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

Hiroyasu Satoh
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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.

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Hiroyasu Satoh. 2024-05-09. A note on the volume entropy of harmonic manifolds of hypergeometric type. https://doi.org/10.1007/s00025-025-02521-3

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