Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

TL;DR

This paper investigates the average area ratio and normalized total scalar curvature of hyperbolic n-manifolds, proving a local minimality property of the hyperbolic metric and exploring their interrelations.

Contribution

It establishes that the hyperbolic metric minimizes the average area ratio locally and discusses its connection to scalar curvature and minimal surface entropy.

Findings

Average area ratio attains a local minimum at the hyperbolic metric.

Relation between average area ratio and normalized total scalar curvature.

Connection to minimal surface entropy in odd dimensions.

Abstract

On closed hyperbolic manifolds of dimension $n\geq 3$, we review the definition of the average area ratio of a metric $h$ with $R_h\geq -n(n-1)$ relative to the hyperbolic metric $h_0$, and we prove that it attains the local minimum of one at $h_0$, which solves a local version of Gromov's conjecture. Additionally, we discuss the relation between the average area ratio and normalized total scalar curvature for hyperbolic $n$-manifolds, as well as its relation to the minimal surface entropy if $n$ is odd.