pith. sign in

arxiv: 2606.19123 · v1 · pith:WCOY7E7Nnew · submitted 2026-06-17 · 🧮 math.DG

A Positive Mass Theorem for Continuous Metrics

Liam Mazurowski , Xuan Yao This is my paper

Pith reviewed 2026-06-26 19:57 UTC · model grok-4.3

classification 🧮 math.DG
keywords positive mass theoremcontinuous metricsasymptotically flatharmonic massADM massscalar curvaturerigidity theorem
0
0 comments X

The pith

A continuous asymptotically flat metric on R^3 approximable by smooth metrics with almost non-negative scalar curvature has non-negative harmonic mass, vanishing only if the metric is flat.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a positive mass theorem for continuous metrics on Euclidean space that are asymptotically flat and can be approximated by smooth metrics with nearly non-negative scalar curvature. It introduces a harmonic mass defined using harmonic functions, which reduces to the standard ADM mass for smooth metrics. The theorem states that this mass is always at least zero, and equals zero only when the metric is the flat Euclidean metric. This extends the classical result to a larger class of metrics while keeping the rigidity statement intact.

Core claim

Let g be a continuous metric on R^3 which is asymptotically flat with |g_ij(x) - delta_ij| = O(|x|^{-tau}) for tau > 1/2, and which can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. The harmonic mass m(g) defined using harmonic functions satisfies m(g) >= 0, and if m(g) = 0 then g is flat. This m(g) coincides with the ADM mass for smooth g and equals the limit of the C^0 local mass.

What carries the argument

The harmonic mass m(g), computed from the asymptotic behavior of harmonic functions with respect to the continuous metric g.

If this is right

  • The positive mass theorem applies to continuous metrics under the approximation condition.
  • Zero harmonic mass implies the metric is flat, providing rigidity.
  • The harmonic mass agrees with the ADM mass in the smooth case.
  • The mass equals the limit of Burkhardt-Guim's C^0 local mass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may enable the study of mass in settings where metrics are only continuous, such as in some numerical or weak solutions in general relativity.
  • Extensions could involve other notions of mass or different approximation conditions for non-smooth metrics.
  • The result suggests that positivity of mass is robust under C^0 limits when curvature conditions are preserved in approximation.

Load-bearing premise

The continuous metric can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature.

What would settle it

Construct a continuous asymptotically flat metric on R^3 that is approximable by smooth metrics with non-negative scalar curvature but has negative harmonic mass, or has zero harmonic mass without being flat.

read the original abstract

Let $g$ be a continuous metric on $\mathbb R^3$ which is asymptotically flat in the sense that $\vert g_{ij}(x) - \delta_{ij}\vert = O(\vert x\vert^{-\tau})$ for some $\tau > \frac{1}{2}$. Further assume that $g$ can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. For such a metric $g$, we define a synthetic ADM mass $m(g)$ using harmonic functions. The harmonic mass $m(g)$ coincides with the usual ADM mass whenever $g$ is smooth and decays rapidly enough that the latter is defined. The harmonic mass can also be computed as a limit of the $C^0$ local mass introduced by Burkhardt-Guim. Our main result is a positive mass theorem: the harmonic mass satisfies $m(g)\geq 0$ and if $m(g) = 0$ then $g$ is flat.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a positive mass theorem for continuous asymptotically flat metrics g on R^3 satisfying |g_ij(x) - δ_ij| = O(|x|^{-τ}) for τ > 1/2. Under the additional hypothesis that g admits uniform approximation on compact sets by smooth metrics with almost non-negative scalar curvature, the authors define a harmonic mass m(g) via solutions to the Laplace-Beltrami equation. They prove that m(g) coincides with the classical ADM mass for sufficiently regular smooth metrics, equals the limit of the Burkhardt-Guim C^0 local mass, and satisfies m(g) ≥ 0 with equality if and only if g is flat.

Significance. If the result holds, the work extends the positive mass theorem to continuous metrics under an explicit approximation hypothesis, which is relevant for weak regularity settings in geometric analysis and general relativity. The synthetic definition of mass via harmonic functions, together with the explicit verification that it recovers the ADM mass on smooth metrics and the limit of the Burkhardt-Guim local mass, constitutes a clear strength of the approach.

minor comments (2)
  1. [Main theorem statement] The precise quantitative meaning of 'almost non-negative scalar curvature' in the approximation hypothesis (e.g., the rate at which the negative part tends to zero) should be stated explicitly in the main theorem, as this controls the limit passage.
  2. Notation for the harmonic functions and the surface integrals defining m(g) would benefit from a dedicated preliminary section to improve readability before the limit arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report, so there are no specific points requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the harmonic mass m(g) directly via solutions to the Laplace-Beltrami equation on the given continuous asymptotically flat metric. It then invokes an explicit external hypothesis (uniform approximation on compact sets by smooth metrics with almost non-negative scalar curvature) to pass the known smooth-case positive mass theorem to the limit. The coincidence with classical ADM mass on smooth metrics and the representation as a limit of Burkhardt-Guim C^0 local mass are shown as consistency checks rather than as the source of the inequality. No load-bearing step reduces the claimed non-negativity or rigidity statement to a definition, a fitted parameter, or a self-citation chain; the approximation hypothesis is stated up front and the argument remains conditional on it.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and properties of harmonic functions on asymptotically flat manifolds, the classical positive mass theorem for smooth metrics, and the uniform approximation hypothesis; no free parameters or new postulated entities are introduced.

axioms (2)
  • domain assumption Existence and regularity properties of harmonic functions on asymptotically flat manifolds with continuous metrics
    Invoked to define the synthetic mass m(g)
  • standard math The classical positive mass theorem holds for smooth asymptotically flat metrics with non-negative scalar curvature
    Used as the base case from which the continuous result is obtained by approximation

pith-pipeline@v0.9.1-grok · 5690 in / 1308 out tokens · 23521 ms · 2026-06-26T19:57:35.271945+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Riemannian Positive Mass Theorem in All Dimensions in the Presence of Low-Codimension Singularities

    math.DG 2026-06 unverdicted novelty 7.0

    Proves Riemannian positive mass theorem for asymptotically flat L^∞ metrics with subcritical singular sets of Minkowski dimension less than n-3 + 2/n (rigidity for ≤ n-3 + 1/(n-1)), using density theorem, capacity est...

  2. Riemannian Penrose Inequality for Manifolds with Corners via Non-Linear Potential Theory

    math.DG 2026-06 unverdicted novelty 6.0

    New unified proof of the Positive Mass Theorem and Riemannian Penrose Inequality for 3D asymptotically flat manifolds with C^{2,α} metrics up to a hypersurface, via approximate monotonicity of a potential-theoretic quantity.

Reference graph

Works this paper leans on

31 extracted references · 5 linked inside Pith · cited by 2 Pith papers

  1. [1]

    A Green’s function proof of the positive mass theorem.Communications in Mathematical Physics, 405(2):54, 2024

    Virginia Agostiniani, Lorenzo Mazzieri, and Francesca Oronzio. A Green’s function proof of the positive mass theorem.Communications in Mathematical Physics, 405(2):54, 2024

    2024

  2. [2]

    A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature.Mathematical Research Letters, 23(2):325–337, 2016

    Richard Bamler. A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature.Mathematical Research Letters, 23(2):325–337, 2016

    2016

  3. [3]

    The mass of an asymptotically flat manifold.Communications on pure and applied mathematics, 39(5):661–693, 1986

    Robert Bartnik. The mass of an asymptotically flat manifold.Communications on pure and applied mathematics, 39(5):661–693, 1986

    1986

  4. [4]

    On the isoperimetric Riemannian Penrose inequality

    Luca Benatti, Mattia Fogagnolo, and Lorenzo Mazzieri. On the isoperimetric Riemannian Penrose inequality. Communications on Pure and Applied Mathematics, 78(5):1042–1085, 2025

    2025

  5. [5]

    Nonlinear isocapacitary concepts of mass in 3-manifolds with nonnegative scalar curvature.SIGMA

    Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri, et al. Nonlinear isocapacitary concepts of mass in 3-manifolds with nonnegative scalar curvature.SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 19:091, 2023. A POSITIVE MASS THEOREM FOR CONTINUOUS METRICS 35

    2023

  6. [6]

    Scalar curvature rigidity of convex polytopes.Inventiones mathematicae, 235(2):669–708, 2024

    Simon Brendle. Scalar curvature rigidity of convex polytopes.Inventiones mathematicae, 235(2):669–708, 2024

    2024

  7. [7]

    Pointwise lower scalar curvature bounds forC 0 metrics via regularizing Ricci flow.Geo- metric and Functional Analysis, 29(6):1703–1772, 2019

    Paula Burkhardt-Guim. Pointwise lower scalar curvature bounds forC 0 metrics via regularizing Ricci flow.Geo- metric and Functional Analysis, 29(6):1703–1772, 2019

    2019

  8. [8]

    ADM mass forC 0 metrics and distortion under Ricci-DeTurck flow.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2024(806):187–245, 2024

    Paula Burkhardt-Guim. ADM mass forC 0 metrics and distortion under Ricci-DeTurck flow.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2024(806):187–245, 2024

    2024

  9. [9]

    On the smoothness of isometries.Duke Math

    Eugenio Calabi and Philip Hartman. On the smoothness of isometries.Duke Math. J., 37(1):741–750, 1970

    1970

  10. [10]

    WeightedL q-theory for the Stokes resolvent in exterior domains.Journal of the Mathematical Society of Japan, 49(2):251–288, 1997

    Reinhard Farwig and Hermann Sohr. WeightedL q-theory for the Stokes resolvent in exterior domains.Journal of the Mathematical Society of Japan, 49(2):251–288, 1997

    1997

  11. [11]

    Scalar curvature bounds for 3D continuous metrics through the inverse mean curvature flow.arXiv preprint arXiv:2605.20114, 2026

    Mattia Fogagnolo, Giorgio Gatti, and Alessandra Pluda. Scalar curvature bounds for 3D continuous metrics through the inverse mean curvature flow.arXiv preprint arXiv:2605.20114, 2026

    Pith/arXiv arXiv 2026

  12. [12]

    Springer, 1998

    David Gilbarg and Neil S Trudinger.Elliptic partial differential equations of second order. Springer, 1998

    1998

  13. [13]

    Dirac and Plateau billiards in domains with corners.Central European Journal of Mathematics, 12(8):1109–1156, 2014

    Misha Gromov. Dirac and Plateau billiards in domains with corners.Central European Journal of Mathematics, 12(8):1109–1156, 2014

    2014

  14. [14]

    Four lectures on scalar curvature.arXiv preprint arXiv:1908.10612, 2019

    Misha Gromov. Four lectures on scalar curvature.arXiv preprint arXiv:1908.10612, 2019

    arXiv 1908

  15. [15]

    The Green function for uniformly elliptic equations.Manuscripta mathe- matica, 37(3):303–342, 1982

    Michael Gr¨ uter and Kjell-Ove Widman. The Green function for uniformly elliptic equations.Manuscripta mathe- matica, 37(3):303–342, 1982

    1982

  16. [16]

    Jacobians of Sobolev homeomorphisms.Calculus of Variations and Partial Differ- ential Equations, 38(1):233–242, 2010

    Stanislav Hencl and Jan Mal` y. Jacobians of Sobolev homeomorphisms.Calculus of Variations and Partial Differ- ential Equations, 38(1):233–242, 2010

    2010

  17. [17]

    An isoperimetric concept for the mass in general relativity

    Gerhard Huisken. An isoperimetric concept for the mass in general relativity. InOberwolfach Reports. European Mathematical Society, Z¨ urich, 2006

    2006

  18. [18]

    ADM mass and the capacity-volume deficit at infinity.Communications in Analysis and Ge- ometry, 31(6):1565–1610, 2024

    Jeffrey L Jauregui. ADM mass and the capacity-volume deficit at infinity.Communications in Analysis and Ge- ometry, 31(6):1565–1610, 2024

    2024

  19. [19]

    A note on Huisken’s isoperimetric mass.Letters in Mathematical Physics, 114(6):134, 2024

    Jeffrey L Jauregui, Dan A Lee, and Ryan Unger. A note on Huisken’s isoperimetric mass.Letters in Mathematical Physics, 114(6):134, 2024

    2024

  20. [20]

    The positive mass theorem for manifolds with distributional curvature

    Dan A Lee and Philippe G LeFloch. The positive mass theorem for manifolds with distributional curvature. Communications in Mathematical Physics, 339(1):99–120, 2015

    2015

  21. [21]

    Quantification of scalar curvature underC 0 convergence using smoothing.arXiv preprint arXiv:2604.17759, 2026

    Man-Chun Lee. Quantification of scalar curvature underC 0 convergence using smoothing.arXiv preprint arXiv:2604.17759, 2026

    Pith/arXiv arXiv 2026

  22. [22]

    A polyhedron comparison theorem for 3-manifolds with positive scalar curvature.Inventiones mathemat- icae, 219(1):1–37, 2020

    Chao Li. A polyhedron comparison theorem for 3-manifolds with positive scalar curvature.Inventiones mathemat- icae, 219(1):1–37, 2020

    2020

  23. [23]

    Elliptic equations and maps of bounded length distortion.Mathematische Annalen, 282(3):423–443, 1988

    Olli Martio and Jussi V¨ ais¨ al¨ a. Elliptic equations and maps of bounded length distortion.Mathematische Annalen, 282(3):423–443, 1988

    1988

  24. [24]

    Quantification ofC 0 convergence in dimension three.arXiv preprint arXiv:2604.14087, 2026

    Liam Mazurowski and Xuan Yao. Quantification ofC 0 convergence in dimension three.arXiv preprint arXiv:2604.14087, 2026

    Pith/arXiv arXiv 2026

  25. [25]

    Scalar curvature under weak limits of manifolds.arXiv preprint arXiv:2605.03136, 2026

    Liam Mazurowski and Xuan Yao. Scalar curvature under weak limits of manifolds.arXiv preprint arXiv:2605.03136, 2026

    Pith/arXiv arXiv 2026

  26. [26]

    AnL p-estimate for the gradient of solutions of second order elliptic divergence equations

    Norman G Meyers. AnL p-estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche, 17(3):189–206, 1963

    1963

  27. [27]

    On the proof of the positive mass conjecture in general relativity.Commu- nications in Mathematical Physics, 65(1):45–76, 1979

    Richard Schoen and Shing-Tung Yau. On the proof of the positive mass conjecture in general relativity.Commu- nications in Mathematical Physics, 65(1):45–76, 1979

    1979

  28. [28]

    Proof of the positive mass theorem

    Richard Schoen and Shing-Tung Yau. Proof of the positive mass theorem. II.Communications in Mathematical Physics, 79(2):231–260, 1981

    1981

  29. [29]

    On gromov’s dihedral extremality and rigidity conjectures.arXiv preprint arXiv:2112.01510, 2021

    Jinmin Wang, Zhizhang Xie, and Guoliang Yu. On gromov’s dihedral extremality and rigidity conjectures.arXiv preprint arXiv:2112.01510, 2021

    arXiv 2021

  30. [30]

    A new proof of the positive energy theorem.Communications in Mathematical Physics, 80(3):381– 402, 1981

    Edward Witten. A new proof of the positive energy theorem.Communications in Mathematical Physics, 80(3):381– 402, 1981

    1981

  31. [31]

    Gromov’s Euclidean endpointC 0 rigidity for the positive mass theorem.arXiv preprint arXiv:2606.15372, 2026

    Jiangcheng You and Heng Zhang. Gromov’s Euclidean endpointC 0 rigidity for the positive mass theorem.arXiv preprint arXiv:2606.15372, 2026. Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, 18015, United States Email address:lim624@lehigh.edu Department of Mathematics, Princeton University, Princeton, NJ 08540 Email address:xy1216@pri...

    Pith/arXiv arXiv 2026