Minimal Surface Entropy and Average Area Ratio

TL;DR

This paper establishes a precise relationship between minimal surface entropy and average area ratio on hyperbolizable 3-manifolds, proving the hyperbolic metric maximizes entropy under scalar curvature constraints and solving Gromov's conjecture.

Contribution

It introduces a sharp relation between two geometric invariants and proves the hyperbolic metric maximizes minimal surface entropy among metrics with scalar curvature ≥ -6.

Findings

Maximizes minimal surface entropy with hyperbolic metric

Solves Gromov's conjecture on average area ratio

Uses Ricci flow with surgery and invariant laminar measures

Abstract

On any closed hyperbolizable 3-manifold, we find a sharp relation between the minimal surface entropy (introduced by Calegari-Marques-Neves) and the average area ratio (introduced by Gromov), and we show that, among metrics g with scalar curvature greater than or equal to -6, the former is maximized by the hyperbolic metric. One corollary is to solve a conjecture of Gromov regarding the average area ratio. Our proofs use Ricci flow with surgery and laminar measures invariant under a PSL(2,R)-action.