Paper · Feb 6, 2021

Area, Scalar Curvature, and Hyperbolic 3-Manifolds

Ben Lowe

math.DGarXiv:2102.03660v3
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Abstract

Let be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if is a Riemannian metric on with scalar curvature greater than or equal to , we find lower bounds for the areas of stable immersed minimal surfaces in . Our bounds improve the closer 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 . We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to , by the hyperbolic metric. Our proofs use the Ricci flow with surgery.

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Submitted
Feb 6, 2021
Last revised
Oct 19, 2021 (v3)
Subjects
math.DG

Comments: This preprint is superseded by arXiv:2110.09451

Ricci flowSurgery3-manifoldsScalar curvatureMinimal & CMC surfaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)7Kleiner 2006 The moduli space of two-convex embedded spheres (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2016)6Buzano 2016 Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature (Laurent Bessières, Gérard Besson, Sylvain Maillot et al., 2017)4Bessières 2017 Metrics with λ₁(−Δ+ k R) ≥ 0 and flexibility in the Riemannian Penrose Inequality (Chao Li and Christos Mantoulidis, 2021)3Li 2021 Minimal Surface Entropy and Average Area Ratio (Ben Lowe and Andre Neves, 2021)1Lowe 2021 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)8Sun 2025 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)2Jiang 2025 On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume (Ruojing Jiang and Franco Vargas Pallete, 2025)5Jiang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Minimal Surface Entropy and Average Area Ratio

    Ben Lowe and Andre Neves

    Topics Surgery, Minimal & CMC surfaces, 3-manifolds · Same author Ben Lowe · Shared terms “curvature greater”, “Calegari Marques”, “Marques Neves”

  2. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete

    Topics Minimal & CMC surfaces, 3-manifolds, Scalar curvature · Shared terms “hyperbolic metric”, “uniquely maximized”

  3. Metrics with and flexibility in the Riemannian Penrose Inequality

    Chao Li and Christos Mantoulidis

    Topics Surgery, Minimal & CMC surfaces, 3-manifolds · Shared terms “Marques”

  4. Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

    Laurent Bessières, Gérard Besson, Sylvain Maillot et al.

    Topics Surgery, 3-manifolds, Scalar curvature

  5. On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

    Ruojing Jiang and Franco Vargas Pallete

    Topics Minimal & CMC surfaces, 3-manifolds · Shared terms “hyperbolic metric”

  6. The moduli space of two-convex embedded spheres

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery, 3-manifolds, Scalar curvature · Shared terms “Marques”

  7. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon

    Topics Surgery, 3-manifolds

  8. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu

    Topics Minimal & CMC surfaces, Scalar curvature

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  3. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.