Area, Scalar Curvature, and Hyperbolic 3-Manifolds

TL;DR

This paper establishes lower bounds for the areas of stable minimal surfaces in certain hyperbolic 3-manifolds with scalar curvature constraints, and shows the hyperbolic metric uniquely maximizes a minimal surface counting functional.

Contribution

It provides new area bounds for minimal surfaces and proves the hyperbolic metric uniquely maximizes a minimal surface functional among metrics with scalar curvature ≥ -6.

Findings

Lower bounds for areas of stable minimal surfaces in hyperbolic 3-manifolds.

The hyperbolic metric uniquely maximizes the Calegari-Marques-Neves functional.

Proofs utilize Ricci flow with surgery.

Abstract

Let $M$ be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if $g$ is a Riemannian metric on $M$ with scalar curvature greater than or equal to $-6$, we find lower bounds for the areas of stable immersed minimal surfaces $\Sigma$ in $M$. Our bounds improve the closer $\Sigma$ is to being homotopic to a totally geodesic surface in the hyperbolic metric. We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in $(M,g)$. We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to $-6$, by the hyperbolic metric. Our proofs use the Ricci flow with surgery.