Area, Scalar Curvature, and Hyperbolic 3Manifolds
Abstract
Let $M$ be a closed hyperbolic 3manifold that admits no infinitesimal conformallyflat 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 CalegariMarquesNeves 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.03660
 Bibcode:
 2021arXiv210203660L
 Keywords:

 Mathematics  Differential Geometry;
 53A10 (Primary) 53E20 (Secondary)
 EPrint:
 This preprint is superseded by arXiv:2110.09451