The Cayley hyperbolic space and volume entropy rigidity

TL;DR

This paper addresses a gap in the proof of the volume entropy rigidity theorem for manifolds locally isometric to the Cayley hyperbolic space, ensuring the theorem's validity in this specific case.

Contribution

It repairs a gap in the existing proof of the volume entropy rigidity theorem specifically for the Cayley hyperbolic space case.

Findings

Confirmed the validity of the volume entropy rigidity theorem for Cayley hyperbolic spaces.

Provided a corrected proof addressing the previously identified gap.

Abstract

Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem asserts that $g_0$ minimizes a normalized volume growth entropy among all complete, finite-volume, Riemannian metric on $M$. We will repair a gap in the proof when $g_0$ is locally isometric to the Cayley hyperbolic space.