Paper · Sep 22, 2026

A continuous Positive Mass Theorem for perturbations of Euclidean space

Paula Burkhardt-Guim

math.DGarXiv:2609.25550
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Abstract

We prove a Positive Mass Theorem for -asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a -asymptotically flat Riemannian metric that is a perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a analog of the classical ADM mass previously introduced by the author.

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Submitted
Sep 22, 2026
Subjects
math.DG

Comments: 40 pages

Ricci flowScalar curvatureBoundaries & non-compactPhysics

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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. ADM mass for metrics and distortion under Ricci-DeTurck flow

    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 “positive mass”, “mass theorem”, “ADM mass”

  3. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Physics, Scalar curvature, Boundaries & non-compact · Shared terms “mass theorem”, “positive mass”, “Euclidean”

  4. Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

    Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al.Wider flows

    Topics Physics, Scalar curvature · Shared terms “positive mass”, “nonnegative scalar”, “mass theorem”

  5. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch

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

  6. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick

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

  7. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz

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

  8. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

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

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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.