Characterization of hypergeometric-type harmonic manifolds achieving the lower bound of volume entropy

Characterize all non-compact harmonic manifolds of hypergeometric type for which the volume entropy equals the lower bound in inequality (1.4) under the normalization Ric = −(n − 1).

Background

The main theorem establishes that for harmonic manifolds of hypergeometric type with the metric normalized by Ric = −(n − 1), the volume entropy Q satisfies specific upper and lower bounds. While the upper bound is achieved only by real hyperbolic spaces, the lower bound arises from an inequality argument, and Section 5 shows that only four Damek-Ricci spaces attain this lower bound.

Despite these examples, the authors note that a general structural characterization of all hypergeometric-type harmonic manifolds attaining the lower bound is not known, highlighting a gap in the understanding of the exact geometric or algebraic conditions necessary for this equality.