Area, Scalar Curvature, and Hyperbolic 3-Manifolds

2021/02/06 by Ben Lowe, Lowe, Ben
Mathematics · #53A10 (Primary) 53E20 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2102.03660

openalex publication_date 2021/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Σ in M. Our bounds improve the closer Σ 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.

Citations

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

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