pith. sign in

arxiv: 2606.23529 · v1 · pith:EQC6V7J6new · submitted 2026-06-22 · 🧮 math.DG · gr-qc· math.AP

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

Marcus Khuri , Jian Wang , Jinmin Wang This is my paper

Pith reviewed 2026-06-26 07:10 UTC · model grok-4.3

classification 🧮 math.DG gr-qcmath.AP
keywords positive mass theoremasymptotically flat manifoldssingular metricsMinkowski dimensionADM massscalar curvaturerigidity
0
0 comments X

The pith

ADM mass is nonnegative for each asymptotically flat end when singularities have Minkowski dimension below n-3 + 2/n.

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

The paper proves that the Riemannian positive mass theorem continues to hold when the metric is allowed to have a compact singular set of sufficiently low Minkowski dimension. Under the stated dimension bound and with scalar curvature nonnegative on the smooth part, the ADM mass of every end is nonnegative, and vanishes in an end only when that end is Euclidean space. The result supplies an asymptotically flat counterpart to Schoen's codimension-three conjecture for positive scalar curvature. The extension is obtained by first approximating the singular metric by smooth ones, then using capacity estimates that cross the singular set, followed by a conformal blow-up and a dimension-descent argument with μ-bubbles.

Core claim

We prove that for complete asymptotically flat manifolds whose metrics are smooth away from a compact singular set of Minkowski dimension less than n-3 + 2/n and whose scalar curvature is nonnegative on the regular set, the ADM mass of each asymptotically flat end is nonnegative and vanishes only in the Euclidean case. For the rigidity statement the Minkowski dimension is required to be at most n-3 + 1/(n-1). The proof proceeds via a density theorem for singular asymptotically flat metrics, capacity estimates across the singular set, a conformal blow-up, and a μ-bubble dimension-descent argument.

What carries the argument

Capacity estimates across the singular set combined with the μ-bubble dimension-descent argument.

If this is right

  • The ADM mass remains nonnegative for every asymptotically flat end under the given singularity bound.
  • Rigidity to Euclidean space holds when the singular-set dimension satisfies the stricter upper bound n-3 + 1/(n-1).
  • The theorem supplies an asymptotically flat analogue of Schoen's codimension-three conjecture for positive scalar curvature.
  • The result applies in every dimension n.

Where Pith is reading between the lines

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

  • The same capacity and descent machinery may be adaptable to other curvature-positivity statements that tolerate controlled singularities.
  • The explicit dimension threshold indicates a possible transition point at which negative-mass examples could first appear.
  • The approximation-plus-descent strategy suggests a route for studying mass-type invariants on manifolds whose singular sets are only marginally larger than the present bound.

Load-bearing premise

The Minkowski dimension of the singular set is strictly less than n-3 + 2/n so that capacity estimates and the dimension-descent argument remain valid.

What would settle it

An asymptotically flat manifold with nonnegative scalar curvature on the regular set, a singular set whose Minkowski dimension meets or exceeds n-3 + 2/n, and negative ADM mass on some end would falsify the claim.

read the original abstract

We prove the Riemannian positive mass theorem in all dimensions for asymptotically flat $L^\infty$-metrics with subcritical singular sets. More precisely, we consider complete asymptotically flat manifolds whose metrics are smooth away from a compact singular set of Minkowski dimension less than $n-3+\frac{2}{n}$, and whose scalar curvature is nonnegative on the regular set. We show that the ADM mass of each asymptotically flat end is nonnegative, and that the mass vanishes in some end only in the Euclidean case. For the rigidity statement, we require additionally that the Minkowski dimension of the singular set is not larger than $n-3+\frac{1}{n-1}$. This gives an asymptotically flat analogue of Schoen's codimension-three conjecture for positive scalar curvature. The proof combines a density theorem for singular asymptotically flat metrics, capacity estimates across the singular set, conformal blow-up inspired by Bi-Hao-He-Shi-Zhu [3], and a $\mu$-bubble dimension-descent argument adapted from Brendle-Wang [6].

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 / 1 minor

Summary. The paper proves the Riemannian positive mass theorem for complete asymptotically flat L^∞-metrics that are smooth away from a compact singular set of Minkowski dimension less than n-3 + 2/n (with scalar curvature nonnegative on the regular set). It establishes nonnegativity of the ADM mass in each end, with vanishing mass in some end only in the Euclidean case; rigidity requires the stricter bound of Minkowski dimension ≤ n-3 + 1/(n-1). The argument uses a density theorem for singular AF metrics, capacity estimates across the singular set, conformal blow-up, and a μ-bubble dimension-descent argument.

Significance. If the technical steps hold, the result extends the positive mass theorem to low-codimension singularities in all dimensions, providing an AF analogue of Schoen's codimension-three conjecture for positive scalar curvature. It builds on prior work via density theorems, capacity estimates, Bi-Hao-He-Shi-Zhu conformal blow-up, and Brendle-Wang μ-bubbles, and the dimension bounds are explicitly tied to the success of the capacity and descent arguments.

minor comments (1)
  1. The abstract references specific prior works [3] and [6] for the blow-up and μ-bubble steps; the manuscript should include a brief comparison paragraph clarifying the precise adaptations made to the singular setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, which accurately reflects the stated results, dimension bounds, and proof strategy. No specific major comments or technical objections are raised in the report. The recommendation of 'uncertain' appears to stem from a general concern about the validity of the technical steps (density theorem, capacity estimates, conformal blow-up, and μ-bubble descent), but without identified gaps we have no concrete points to rebut or revise. We stand by the completeness of the arguments as written and remain available to supply expanded details on any step if requested.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation combines a density theorem for singular AF metrics, capacity estimates across the singular set, conformal blow-up from Bi-Hao-He-Shi-Zhu [3], and a μ-bubble dimension-descent argument adapted from Brendle-Wang [6]. These are external results by non-overlapping authors, with the Minkowski dimension bound stated explicitly as a hypothesis needed for the estimates to succeed rather than derived from the conclusion. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the proof outline or abstract. The argument is self-contained against external benchmarks and does not reduce the mass nonnegativity statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard definitions and background results from Riemannian geometry and analysis; no free parameters are fitted and no new entities are postulated.

axioms (3)
  • domain assumption Definition of asymptotically flat L^∞ manifold with ends and ADM mass
    Standard background assumption invoked to state the setting and the mass functional.
  • domain assumption Scalar curvature nonnegative on the regular part of the manifold
    Central hypothesis of the theorem, stated explicitly in the abstract.
  • domain assumption Minkowski dimension of the singular set satisfies the stated upper bound
    Load-bearing condition that enables the capacity estimates and dimension-descent argument.

pith-pipeline@v0.9.1-grok · 5716 in / 1642 out tokens · 40230 ms · 2026-06-26T07:10:29.031677+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

54 extracted references · 9 linked inside Pith

  1. [1]

    The mass of an asymptotically flat manifold.Comm

    Robert Bartnik. The mass of an asymptotically flat manifold.Comm. Pure Appl. Math., 39(5):661–693, 1986

    1986

  2. [2]

    Boundary value problems for Dirac-type equations

    Robert Bartnik and Piotr Chru´ sciel. Boundary value problems for Dirac-type equations. Journal f¨ ur die reine und angewandte Mathematik, 579:13–73, 2005

    2005

  3. [3]

    A proof for the riemannian positive mass theorem up to dimension 19

    Yuchen Bi, Tianze Hao, Shihang He, Yuguang Shi, and Jintian Zhu. A proof for the riemannian positive mass theorem up to dimension 19. arXiv:2603.02769, 2026

    arXiv 2026

  4. [4]

    Positive scalar curvature obstructions via singular dimension descent

    Yuchen Bi and Jintian Zhu. Positive scalar curvature obstructions via singular dimension descent. arXiv:2606.20528, 2026

    Pith/arXiv arXiv 2026

  5. [5]

    Proof of the Riemannian Penrose inequality using the positive mass theorem

    Hubert Bray. Proof of the Riemannian Penrose inequality using the positive mass theorem. Journal of Differential Geometry, 59(2):177–267, 2001

    2001

  6. [6]

    A dimension descent scheme for the positive mass the- orem in arbitrary dimension

    Simon Brendle and Yipeng Wang. A dimension descent scheme for the positive mass the- orem in arbitrary dimension. arXiv:2604.08473, 2026

    Pith/arXiv arXiv 2026

  7. [7]

    On the spacetime positive energy theorem in arbitrary dimension

    Simon Brendle and Yipeng Wang. On the spacetime positive energy theorem in arbitrary dimension. arXiv:2604.18561, 2026

    Pith/arXiv arXiv 2026

  8. [8]

    SmoothingL ∞ Riemannian metrics with nonnegative scalar cur- vature outside of a singular set

    Paula Burkhardt-Guim. SmoothingL ∞ Riemannian metrics with nonnegative scalar cur- vature outside of a singular set. arXiv:2406.04564, 2024. RIEMANNIAN PMT WITH LOW-CODIMENSION SINGULARITIES 47

    arXiv 2024

  9. [9]

    Positive scalar curvature with point singularities

    Simone Cecchini, Georg Frenck, and Rudolf Zeidler. Positive scalar curvature with point singularities. arXiv:2407.20163, 2024

    arXiv 2024

  10. [10]

    Quantitative stratification and the regularity of har- monic maps and minimal currents.Communications on Pure and Applied Mathematics, 66(6):965–990, 2013

    Jeff Cheeger and Aaron Naber. Quantitative stratification and the regularity of har- monic maps and minimal currents.Communications on Pure and Applied Mathematics, 66(6):965–990, 2013

    2013

  11. [11]

    Singular metrics with negative scalar curvature.International Journal of Mathematics, 33(7):2250047, 2022

    Man-Chuen Cheng, Man-Chun Lee, and Luen-Fai Tam. Singular metrics with negative scalar curvature.International Journal of Mathematics, 33(7):2250047, 2022

    2022

  12. [12]

    Generalized soap bubbles and the topology of manifolds with positive scalar curvature.Annals of Mathematics, 199(2):707–740, 2024

    Otis Chodosh and Chao Li. Generalized soap bubbles and the topology of manifolds with positive scalar curvature.Annals of Mathematics, 199(2):707–740, 2024

    2024

  13. [13]

    Generic regularity for minimizing hypersurfaces in dimensions 9 and 10

    Otis Chodosh, Christos Mantoulidis, and Felix Schulze. Generic regularity for minimizing hypersurfaces in dimensions 9 and 10. arXiv:2302.02253, 2023

    arXiv 2023

  14. [14]

    Generic regularity for minimizing hypersurfaces in dimension 11

    Otis Chodosh, Christos Mantoulidis, Felix Schulze, and Zhihan Wang. Generic regularity for minimizing hypersurfaces in dimension 11. arXiv:2506.12852, 2025

    arXiv 2025

  15. [15]

    Boundary conditions at spatial infinity from a Hamiltonian point of view

    Piotr Chru´ sciel. Boundary conditions at spatial infinity from a Hamiltonian point of view. NATO Adv. Sci. Inst. Ser. B: Phys.,, 138:49–59, 1986

    1986

  16. [16]

    Positive scalar curvature and isolated conical singularity

    Xianzhe Dai, Yukai Sun, and Changliang Wang. Positive scalar curvature and isolated conical singularity. arXiv:2412.02941, 2024

    arXiv 2024

  17. [17]

    The positive mass theorem for asymp- totically flat manifolds with isolated conical singularities.Science China Mathematics, 68:1671–1686, 2025

    Xianzhe Dai, Yukai Sun, and Changliang Wang. The positive mass theorem for asymp- totically flat manifolds with isolated conical singularities.Science China Mathematics, 68:1671–1686, 2025

    2025

  18. [18]

    Singular metrics with nonnegative scalar curvature and rcd

    Xianzhe Dai, Changliang Wang, Lihe Wang, and Guofang Wei. Singular metrics with nonnegative scalar curvature and rcd. arXiv:2412.09185, 2024

    arXiv 2024

  19. [19]

    The plateau problem for marginally outer trapped surfaces.Journal of Differential Geometry, 83(3):551–583, 2009

    Michael Eichmair. The plateau problem for marginally outer trapped surfaces.Journal of Differential Geometry, 83(3):551–583, 2009

    2009

  20. [20]

    North-Holland, Amsterdam, 1978

    Ryszard Engelking.Dimension Theory. North-Holland, Amsterdam, 1978

    1978

  21. [21]

    Springer, 2015

    David Gilbarg and Neil Trudinger.Elliptic partial differential equations of second order. Springer, 2015

    2015

  22. [22]

    Metric inequalities with scalar curvature.Geometric and Functional Anal- ysis, 28(3):645–726, 2018

    Misha Gromov. Metric inequalities with scalar curvature.Geometric and Functional Anal- ysis, 28(3):645–726, 2018

    2018

  23. [23]

    The Green function for uniformly elliptic equa- tions.Manuscripta Math., 37(3):303–342, 1982

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

    1982

  24. [24]

    American Mathematical Soc., 2011

    Qing Han and Fanghua Lin.Elliptic partial differential equations, volume 1. American Mathematical Soc., 2011

    2011

  25. [25]

    The hyperboloidal and spacetime positive mass theorem in all dimensions

    Sven Hirsch, Marcus Khuri, Martin Lesourd, and Yiyue Zhang. The hyperboloidal and spacetime positive mass theorem in all dimensions. arXiv:2604.24746, 2026

    Pith/arXiv arXiv 2026

  26. [26]

    A positive mass theorem for manifolds with boundary

    Sven Hirsch and Pengzi Miao. A positive mass theorem for manifolds with boundary. Pacific Journal of Mathematics, 306(1):185–201, 2020

    2020

  27. [27]

    Princeton University Press, Princeton, 1941

    Witold Hurewicz and Henry Wallman.Dimension Theory, volume 4 ofPrinceton Mathe- matical Series. Princeton University Press, Princeton, 1941

    1941

  28. [28]

    Desingularizing positive scalar curvature 4-manifolds.Mathematische Annalen, 390:4951–4972, 2024

    Demetre Kazaras. Desingularizing positive scalar curvature 4-manifolds.Mathematische Annalen, 390:4951–4972, 2024

    2024

  29. [29]

    A positive mass theorem for lipschitz metrics with small singular sets.Proceedings of the American Mathematical Society, 141(11):3997–4004, 2013

    Dan Lee. A positive mass theorem for lipschitz metrics with small singular sets.Proceedings of the American Mathematical Society, 141(11):3997–4004, 2013

    2013

  30. [30]

    American Mathematical Society, 2021

    Dan Lee.Geometric Relativity, volume 201. American Mathematical Society, 2021

    2021

  31. [31]

    The positive mass theorem for manifolds with distributional curvature.Communications in Mathematical Physics, 339(1):99–120, 2015

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

    2015

  32. [32]

    Density and positive mass theorems for in- complete manifolds.Calculus of Variations and Partial Differential Equations, 62(194), 2023

    Dan Lee, Martin Lesourd, and Ryan Unger. Density and positive mass theorems for in- complete manifolds.Calculus of Variations and Partial Differential Equations, 62(194), 2023

    2023

  33. [33]

    The positive mass theorem with arbi- trary ends.Journal of Differential Geometry, 128(1):257–293, 2024

    Martin Lesourd, Ryan Unger, and Shing-Tung Yau. The positive mass theorem with arbi- trary ends.Journal of Differential Geometry, 128(1):257–293, 2024. 48 MARCUS KHURI, JIAN W ANG, AND JINMIN W ANG

    2024

  34. [34]

    Positive scalar curvature with skeleton singularities

    Chao Li and Christos Mantoulidis. Positive scalar curvature with skeleton singularities. Mathematische Annalen, 374(1–2):99–131, 2019

    2019

  35. [35]

    Littman, G

    W. Littman, G. Stampacchia, and H. F. Weinberger. Regular points for elliptic equations with discontinuous coefficients.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17:43–77, 1963

    1963

  36. [36]

    The higher dimensional positive mass theorem I

    Joachim Lohkamp. The higher dimensional positive mass theorem I. arXiv:math/0608795, 2006

    Pith/arXiv arXiv 2006

  37. [37]

    The higher dimensional positive mass theorem II

    Joachim Lohkamp. The higher dimensional positive mass theorem II. arXiv:1612.07505, 2016

    Pith/arXiv arXiv 2016

  38. [38]

    Capacity, quasi-local mass, and singular fill-ins.Journal f¨ ur die reine und angewandte Mathematik, 768:55–92, 2020

    Christos Mantoulidis, Pengzi Miao, and Luen-Fai Tam. Capacity, quasi-local mass, and singular fill-ins.Journal f¨ ur die reine und angewandte Mathematik, 768:55–92, 2020

    2020

  39. [39]

    A positive mass theorem for continuous metrics

    Liam Mazurowski and Xuan Yao. A positive mass theorem for continuous metrics. arXiv:2606.19123, 2026

    Pith/arXiv arXiv 2026

  40. [40]

    On the positive mass theorem for manifolds with corners.Communications in Mathematical Physics, 313(2):425–443, 2012

    Donovan McFeron and G´ abor Sz´ ekelyhidi. On the positive mass theorem for manifolds with corners.Communications in Mathematical Physics, 313(2):425–443, 2012

    2012

  41. [41]

    Positive mass theorem on manifolds admitting corners along a hypersurface

    Pengzi Miao. Positive mass theorem on manifolds admitting corners along a hypersurface. Advances in Theoretical and Mathematical Physics, 6(6):1163–1182, 2002

    2002

  42. [42]

    The singular structure and regularity of stationary and minimizing varifolds.Annals of Mathematics, 190(2):413–565, 2019

    Aaron Naber and Daniele Valtorta. The singular structure and regularity of stationary and minimizing varifolds.Annals of Mathematics, 190(2):413–565, 2019

    2019

  43. [43]

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

    1979

  44. [44]

    Proof of the positive mass theorem

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

    1981

  45. [45]

    International Press, Boston, 1994

    Richard Schoen and Shing-Tung Yau.Lectures on Differential Geometry. International Press, Boston, 1994

    1994

  46. [46]

    Positive scalar curvature and minimal hypersurface singularities

    Richard Schoen and Shing-Tung Yau. Positive scalar curvature and minimal hypersurface singularities. InSurveys in Differential Geometry 2019. Differential Geometry, Calabi–Yau Theory, and General Relativity. Part 2, volume 24 ofSurveys in Differential Geometry, pages 441–480. International Press, Boston, MA, 2022

    2019

  47. [47]

    Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature.Journal of Differential Geometry, 62(1):79–125, 2002

    Yuguang Shi and Luen-Fai Tam. Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature.Journal of Differential Geometry, 62(1):79–125, 2002

    2002

  48. [48]

    On the fill-in of nonnegative scalar curvature metrics.Mathematische Annalen, 379(1–2):235–270, 2021

    Yuguang Shi, Wenlong Wang, Guodong Wei, and Jintian Zhu. On the fill-in of nonnegative scalar curvature metrics.Mathematische Annalen, 379(1–2):235–270, 2021

    2021

  49. [49]

    Generic regularity of homologically area minimizing hypersurfaces in eight- dimensional manifolds.Communications in Analysis and Geometry, 1(2):217–228, 1993

    Nathan Smale. Generic regularity of homologically area minimizing hypersurfaces in eight- dimensional manifolds.Communications in Analysis and Geometry, 1(2):217–228, 1993

    1993

  50. [50]

    Positive mass theorem for initial data sets with arbitrary ends

    Tin-Yau Tsang. Positive mass theorem for initial data sets with arbitrary ends. arXiv:2604.26978, 2026

    Pith/arXiv arXiv 2026

  51. [51]

    arXiv:2606.77320, 2026

    Jian Wang, Jinmin Wang, and Zhizhang Xie.L ∞-metrics on tori and schoen’s conjecture. arXiv:2606.77320, 2026

    arXiv 2026

  52. [52]

    Scalar curvature rigidity of spheres with subsets removed andL ∞ metrics

    Jinmin Wang and Zhizhang Xie. Scalar curvature rigidity of spheres with subsets removed andL ∞ metrics. arXiv:2407.21312, 2024

    Pith/arXiv arXiv 2024

  53. [53]

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

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

    1981

  54. [54]

    Width estimate and doubly warped product.Transactions of the American Mathematical Society, 374(2):1497–1511, 2021

    Jintian Zhu. Width estimate and doubly warped product.Transactions of the American Mathematical Society, 374(2):1497–1511, 2021. RIEMANNIAN PMT WITH LOW-CODIMENSION SINGULARITIES 49 (Marcus Khuri)Department of Mathematics, Stony Brook University Email address:marcus.khuri@stonybrook.edu (Jian Wang)State Key Laboratory of Mathematical Sciences, Academy of ...

    2021