Paper · Jun 29, 2021

Metrics with and flexibility in the Riemannian Penrose Inequality

Chao Li, Christos Mantoulidis

math.DGarXiv:2106.15709v3
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Abstract

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally for different values of k in the study of scalar curvature via minimal hypersurfaces, the Yamabe problem, and Perelman's Ricci flow with surgery. When k=1/2, the space models apparent horizons in time-symmetric initial data to the Einstein equations. We study these spaces in unison and generalize Codá Marques's path-connectedness theorem. Applying this with k=1/2, we compute the Bartnik mass of 3-dimensional apparent horizons and the Bartnik–Bray mass of their outer-minimizing generalizations in all dimensions. Our methods also yield efficient constructions for the scalar-nonnegative fill-in problem.

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Submitted
Jun 29, 2021
Last revised
Feb 19, 2023 (v3)
Subjects
math.DG, math-ph
DOI
10.1007/s00220-023-04679-9

Comments: To appear in Comm. Math. Phys. There are several refinements and expository improvements relative to previously uploaded version thanks to comments from the anonymous referees

Ricci flowSurgery3-manifoldsScalar curvatureGeometric inequalitiesEigenvalues & spectrumMinimal & CMC surfaces

Its neighbourhood

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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 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)5Bessières 2017 Constrained deformations of positive scalar curvature metrics (Alessandro Carlotto and Chao Li, 2019)1Carlotto 2019 Area, Scalar Curvature, and Hyperbolic 3-Manifolds (Ben Lowe, 2021)3Lowe 2021 Minimal Surface Entropy and Average Area Ratio (Ben Lowe and Andre Neves, 2021)4Lowe 2021 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)2Benatti 2022 A Minkowski type inequality for manifolds with positive spectrum (Ovidiu Munteanu and Jiaping Wang, 2023)8Munteanu 2023 The Penrose inequality with charge for 2-convex initial data sets (Tuan Dolmen, 2026)7Dolmen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Constrained deformations of positive scalar curvature metrics

    Alessandro Carlotto and Chao Li

    Topics Surgery, 3-manifolds, Scalar curvature · Same author Chao Li · Shared terms “Codá Marques”, “Perelman Ricci”, “Riemannian metrics”

  2. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

    Topics Geometric inequalities, Scalar curvature · Shared terms “Riemannian Penrose”, “Penrose inequality”

  3. Area, Scalar Curvature, and Hyperbolic 3-Manifolds

    Ben Lowe

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

  4. Minimal Surface Entropy and Average Area Ratio

    Ben Lowe and Andre Neves

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

  5. 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 · Shared terms “Riemannian metrics”

  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 “path connectedness”, “Marques”

  7. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows

    Topics Geometric inequalities · Shared terms “Penrose inequality”, “apparent horizon”, “Riemannian Penrose”

  8. A Minkowski type inequality for manifolds with positive spectrum

    Ovidiu Munteanu and Jiaping Wang

    Topics Minimal & CMC surfaces, Eigenvalues & spectrum, Geometric inequalities · Shared terms “minimal hypersurfaces”

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.