Isolation phenomenon for small positive volume entropy

Ascertain whether there exists a dimension-dependent constant Qn > 0 such that no harmonic manifold with metric normalized by Ric = −(n − 1) has volume entropy Q in the interval (0, Qn).

Background

The authors raise the possibility of an isolation phenomenon near zero for the volume entropy among harmonic manifolds under Ricci normalization. If such a constant exists, it would imply a gap in the spectrum of achievable volume entropy values, suggesting rigidity at small entropy levels.

This question complements the paper’s bounds on volume entropy and the discussion of examples achieving extremal values, by investigating whether the lower end of possible entropy values is quantitatively isolated from zero.

References

Specifically, one may ask whether there exists a constant Qnsatisfying the following property: There exists no harmonic manifold (M ,g) satisfying Ric = −(n − 1) with volume entropy Q such that 0 < Q < Q . At present, this remains an open question.

— A note on the volume entropy of harmonic manifolds of hypergeometric type (2405.05896 - Satoh, 9 May 2024) in Section 1 (Introduction and Main Results), final paragraph, p. 2–3