Paper · Aug 30, 2022

ADM mass for metrics and distortion under Ricci-DeTurck flow

Paula Burkhardt-Guim

math.DGarXiv:2208.14550v3
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As on arXiv

Abstract

We show that there exists a quantity, depending only on data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the mass at infinity is independent of choice of -asymptotically flat coordinate chart, and the local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

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Submitted
Aug 30, 2022
Last revised
Feb 28, 2025 (v3)
Subjects
math.DG
Journal
Journal für die reine und angewandte Mathematik (Crelle's Journal) (2023) 2024, 806, 187 - 245
DOI
10.1515/crelle-2023-0085

Comments: v3: 59 pages. Journal version

Ricci flowScalar curvatureBoundaries & non-compactHarmonic maps & DeTurck trickPhysics

Its neighbourhood

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  1. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim

    Topics Physics, Scalar curvature, Boundaries & non-compact · Same author Paula Burkhardt-Guim · Shared terms “ADM mass”, “asymptotically flat”, “nonnegative scalar”

  2. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon

    Topics Physics, Scalar curvature, Boundaries & non-compact · Shared terms “ADM mass”, “asymptotically flat”

  3. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “Riemannian metric”

  4. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Boundaries & non-compact · Shared terms “Riemannian metric”, “coordinates”

  5. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick

    Topics Physics, Boundaries & non-compact · Shared terms “ADM mass”

  6. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu

    Topics Harmonic maps & DeTurck trick, Scalar curvature

  7. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al.

    Topics Harmonic maps & DeTurck trick, Scalar curvature

  8. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch

    Topics Physics, Scalar curvature · Shared terms “ADM mass”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.