Paper · Oct 18, 2021

Minimal Surface Entropy and Average Area Ratio

Ben Lowe, Andre Neves

math.DGarXiv:2110.09451v3
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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.

Comments: Published version

Ricci flowSurgery3-manifoldsScalar curvatureMinimal & CMC surfaces

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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 2021, 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)3Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)7Kleiner 2014 Free boundary flow with surgery (Robert Haslhofer, 2023)6Haslhofer 2023 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)4Mramor 2024 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)5Sun 2025 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)1Jiang 2025 On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume (Ruojing Jiang and Franco Vargas Pallete, 2025)2Jiang 2025 Small Normal Curvature and Three-Manifold Topology (Tsz-Kiu Aaron Chow and Jingbo Wan, 2026)8Chow 2026
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 applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete PDF

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

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

    Ruojing Jiang and Franco Vargas Pallete PDF

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

  3. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  4. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces, 3-manifolds

  5. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu PDF

    Topics Minimal & CMC surfaces, Scalar curvature

  6. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Surgery, Minimal & CMC surfaces

  7. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  8. Small Normal Curvature and Three-Manifold Topology

    Tsz-Kiu Aaron Chow and Jingbo Wan PDF

    Topics Surgery, 3-manifolds · Shared terms “Neves”

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.