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arxiv: 2606.27155 · v1 · pith:IWVUSP44new · submitted 2026-06-25 · 🧮 math.DG

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

Virginia Agostiniani , Christina Karapiperi , Lorenzo Mazzieri This is my paper

Pith reviewed 2026-06-26 02:33 UTC · model grok-4.3

classification 🧮 math.DG
keywords Riemannian Penrose inequalitypositive mass theoremmanifolds with cornersasymptotically flat manifoldsapproximate monotonicitynonlinear potential theoryC^{2,α} regularity
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The pith

A direct proof establishes both the positive mass theorem and Riemannian Penrose inequality for three-dimensional asymptotically flat manifolds whose metrics fail to be C1 across a hypersurface.

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

The paper gives a unified argument that proves the positive mass theorem and the Riemannian Penrose inequality on three-dimensional asymptotically flat manifolds that possess a corner hypersurface across which the metric is not C1. It works by showing an approximate monotonicity property for a previously introduced quantity, using an approximation scheme that preserves C^{2,α} regularity up to the corner. The argument recovers the two results without first reducing the problem to the already-known smooth-metric case. A reader would care because these statements constrain the total mass and the area of apparent horizons in initial data sets that arise in general relativity, and corners appear in many physically motivated constructions.

Core claim

By establishing an approximate monotonicity for the quantity introduced by Agostiniani-Mantegazza-Mazzieri-Oronzio and applying Miao's approximation scheme to metrics of class C^{2,α} up to the hypersurface Σ, both the positive mass theorem and the Riemannian Penrose inequality are recovered directly for three-dimensional asymptotically flat manifolds with corners.

What carries the argument

Approximate monotonicity of the AMMO quantity under Miao's approximation scheme applied to C^{2,α} metrics up to the corner hypersurface.

If this is right

  • The positive mass theorem holds for asymptotically flat 3-manifolds whose metric is C^{2,α} up to a corner hypersurface.
  • The Riemannian Penrose inequality holds for the same class of manifolds.
  • Both results follow from a single monotonicity argument rather than separate reductions to the smooth case.
  • The method applies whenever the metric satisfies the stated regularity up to Σ.

Where Pith is reading between the lines

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

  • The same approximation technique could be tested on other monotonicity quantities that appear in general-relativity inequalities with low regularity.
  • Numerical codes that evolve initial data containing corners could monitor the approximate monotonicity as a diagnostic for code accuracy.
  • If the monotonicity survives further weakening of regularity, the results might extend to metrics that are only Lipschitz across Σ.

Load-bearing premise

The approximate monotonicity of the chosen quantity must continue to hold when the metric is merely C^{2,α} across the hypersurface rather than smoother.

What would settle it

Construct or numerically approximate a three-dimensional asymptotically flat manifold with a C^{2,α} corner hypersurface whose total mass is negative or whose horizon area violates the Penrose bound while satisfying the dominant energy condition away from the corner.

Figures

Figures reproduced from arXiv: 2606.27155 by Christina Karapiperi, Lorenzo Mazzieri, Virginia Agostiniani.

Figure 1
Figure 1. Figure 1: Enclosing smoothing region between two level sets. second homology group. Finally, note also that the proof in [22] holds for any dimension for which the classical Positive Mass Theorem holds, while the proof of the Riemannian Penrose Inequaity with corners in [23],[20] holds for dimension 3 ≤ n ≤ 7. Instead, our proof applies only to the case n = 3 due to the need of the Gauss-Bonnet Theorem. We recall th… view at source ↗
read the original abstract

We present a new proof of the Positive Mass Theorem and the Riemannian Penrose Inequality for three-dimensional asymptotically flat Riemannian manifolds whose metrics fail to be $C^1$ across a hypersurface $\Sigma$, first proven by Miao and McCormick-Miao, respectively. Unlike these approaches, ours recovers these results directly, without relying on their original formulations for smooth metrics. The proofs are based on a unified argument which applies to both theorems. We achieve this by establishing an approximate monotonicity for the quantity introduced by Agostiniani-Mantegazza-Mazzieri-Oronzio, employing the approximation scheme of Miao, for metrics with $C^{2,\alpha}$ regularity up to $\Sigma$.

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 paper claims a new direct proof of the Positive Mass Theorem and the Riemannian Penrose Inequality for 3-dimensional asymptotically flat Riemannian manifolds whose metrics are not C¹ across a hypersurface Σ (first proven by Miao and McCormick-Miao). The unified argument establishes approximate monotonicity of the AMMO quantity (Agostiniani-Mantegazza-Mazzieri-Oronzio) via Miao's approximation scheme, for metrics with C^{2,α} regularity up to Σ, without reducing to the smooth-metric case.

Significance. If the error control in the approximate monotonicity is rigorous, the result offers a unified treatment of both theorems for manifolds with corners using non-linear potential theory; this is a methodological strength relative to prior reductions to smooth metrics. The approach is internally consistent with standard approximation techniques in geometric inequalities.

minor comments (2)
  1. The introduction could include a brief comparison table or paragraph contrasting the new direct argument with the original Miao/McCormick-Miao reductions, to clarify the novelty for readers.
  2. Notation for the approximation parameter (e.g., ε) and the precise statement of the C^{2,α} regularity up to Σ should be fixed consistently between the abstract and §2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external approximation and prior independent quantity

full rationale

The paper's central argument establishes approximate monotonicity of the AMMO quantity (introduced in prior work by overlapping authors) via Miao's external approximation scheme for C^{2,α} metrics. No equations or steps in the provided abstract or description reduce the target theorems to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The cited quantity and scheme are treated as independent inputs, with the new contribution being their application to the corner case without relying on smooth-metric formulations. This is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

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Works this paper leans on

27 extracted references · 19 canonical work pages

  1. [1]

    A new proof of the Riemannian Penrose inequality

    V. Agostiniani, C. Mantegazza, L. Mazzieri, and F. Oronzio. “A new proof of the Riemannian Penrose inequality”. In:Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.34.3 (2023), pp. 715– 726.issn: 1120-6330,1720-0768.doi:10.4171/rlm/1024.url:https://doi.org/10.4171/ rlm/1024

    work page doi:10.4171/rlm/1024.url:https://doi.org/10.4171/ 2023

  2. [2]

    Agostiniani, L

    V. Agostiniani, L. Mazzieri, and F. Oronzio.A Green’s Function Proof of the Positive Mass Theorem. 2024.doi:10.48550/arXiv.2108.08402. arXiv:2108.08402

    work page doi:10.48550/arxiv.2108.08402 2024

  3. [3]

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

    L. Benatti, M. Fogagnolo, and L. Mazzieri. “Nonlinear isocapacitary concepts of mass in 3- manifolds with nonnegative scalar curvature”. In:SIGMA Symmetry Integrability Geom. Meth- ods Appl.19 (2023), Paper No. 091, 29.issn: 1815-0659.doi:10.3842/SIGMA.2023.091.url: https://doi.org/10.3842/SIGMA.2023.091

    work page doi:10.3842/sigma.2023.091.url: 2023

  4. [4]

    Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature

    L. Benatti, M. Fogagnolo, and L. Mazzieri. “Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature”. In:Anal. PDE17.9 (2024), pp. 3039–3077.issn: 2157-5045,1948-206X.doi:10.2140/apde.2024.17.3039.url:https://doi.org/10.2140/ apde.2024.17.3039

    work page doi:10.2140/apde.2024.17.3039.url:https://doi.org/10.2140/ 2024

  5. [5]

    On the isoperimetric Riemannian Penrose inequal- ity

    L. Benatti, M. Fogagnolo, and L. Mazzieri. “On the isoperimetric Riemannian Penrose inequal- ity”. In:Comm. Pure Appl. Math.78.5 (2025), pp. 1042–1085.issn: 0010-3640,1097-0312.doi: 10.1002/cpa.22239.url:https://doi.org/10.1002/cpa.22239

    work page doi:10.1002/cpa.22239.url:https://doi.org/10.1002/cpa.22239 2025

  6. [6]

    Stability of a shell around a black hole

    P. R. Brady, J. Louko, and E. Poisson. “Stability of a shell around a black hole”. In:Phys. Rev. D (3)44.6 (1991), pp. 1891–1894.issn: 0556-2821.doi:10.1103/PhysRevD.44.1891. url:https://doi.org/10.1103/PhysRevD.44.1891

    work page doi:10.1103/physrevd.44.1891 1991

  7. [7]

    Orientation and symmetries of Alexandrov spaces with applications in positive curvature

    H. L. Bray, D. P. Kazaras, M. A. Khuri, and D. L. Stern. “Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds”. In:J. Geom. Anal.32.6 (2022), Paper No. 184, 29.issn: 1050-6926,1559-002X.doi:10.1007/s12220- 022- 00924- 0.url: https://doi.org/10.1007/s12220-022-00924-0

    work page doi:10.1007/s12220- 2022

  8. [8]

    C1+α local regularity of weak solutions of degenerate elliptic equations

    E. DiBenedetto. “C1+α local regularity of weak solutions of degenerate elliptic equations”. In: Nonlinear Anal.7.8 (1983), pp. 827–850.issn: 0362-546X,1873-5215.doi:10 . 1016 / 0362 - 546X(83)90061-5.url:https://doi.org/10.1016/0362-546X(83)90061-5

    work page doi:10.1016/0362-546x(83)90061-5 1983

  9. [9]

    Strings and other distributional sources in general relativity

    R. Geroch and J. Traschen. “Strings and other distributional sources in general relativity”. In: Phys. Rev. D (3)36.4 (1987), pp. 1017–1031.issn: 0556-2821.doi:10.1103/PhysRevD.36. 1017.url:https://doi.org/10.1103/PhysRevD.36.1017

    work page doi:10.1103/physrevd.36 1987

  10. [10]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Sec- ond. Vol. 224. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1983, pp. xiii+513.isbn: 3-540-13025-X.doi: 10.1007/978-3-642-61798-0.url:https://doi.org/10.1007/978-3-642-61798-0

    work page doi:10.1007/978-3-642-61798-0.url:https://doi.org/10.1007/978-3-642-61798-0 1983

  11. [11]

    Heinonen, T

    J. Heinonen, T. Kilpeläinen, and O. Martio.Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993, pp. vi+363.isbn: 0-19-853669-0

    1993

  12. [12]

    The inverse mean curvature flow and the Riemannian Penrose inequality

    G. Huisken and T. Ilmanen. “The inverse mean curvature flow and the Riemannian Penrose inequality”. In:J. Differential Geom.59.3 (2001), pp. 353–437.issn: 0022-040X,1945-743X. url:http://projecteuclid.org/euclid.jdg/1090349447

    arXiv 2001

  13. [13]

    Singular hypersurfaces and thin shells in general relativity

    W. Israel. “Singular hypersurfaces and thin shells in general relativity”. In:Il Nuovo Cimento B (1965-1970)44.1 (1966), pp. 1–14

    1965

  14. [14]

    The positive mass theorem for manifolds with distributional curvature

    D. A. Lee and P. G. LeFloch. “The positive mass theorem for manifolds with distributional curvature”. In:Comm. Math. Phys.339.1 (2015), pp. 99–120.issn: 0010-3616,1432-0916.doi: 10.1007/s00220-015-2414-9.url:https://doi.org/10.1007/s00220-015-2414-9. 22 REFERENCES

    work page doi:10.1007/s00220-015-2414-9.url:https://doi.org/10.1007/s00220-015-2414-9 2015

  15. [15]

    Symmetric Green’s functions on complete manifolds

    P. Li and L.-F. Tam. “Symmetric Green’s functions on complete manifolds”. In:Amer. J. Math.109.6 (1987), pp. 1129–1154.issn: 0002-9327,1080-6377.doi:10.2307/2374588.url: https://doi.org/10.2307/2374588

    work page doi:10.2307/2374588.url: 1987

  16. [16]

    The positive mass theorem for non-spin manifolds with distributional curvature

    Y. Li. “The positive mass theorem for non-spin manifolds with distributional curvature”. In: Ann. Henri Poincaré21.6 (2020), pp. 2093–2114.issn: 1424-0637,1424-0661.doi:10.1007/ s00023-020-00915-3.url:https://doi.org/10.1007/s00023-020-00915-3

    work page doi:10.1007/s00023-020-00915-3 2020

  17. [17]

    Lindqvist.Notes on thep-Laplace Equation

    P. Lindqvist.Notes on thep-Laplace Equation. 161. University of Jyväskylä, 2017

    2017

  18. [18]

    Rigidity of Riemannian Penrose inequality with corners and its impli- cations

    S. Lu and P. Miao. “Rigidity of Riemannian Penrose inequality with corners and its impli- cations”. In:J. Funct. Anal.281.10 (2021), Paper No. 109231, 11.issn: 0022-1236,1096-0783. doi:10.1016/j.jfa.2021.109231.url:https://doi.org/10.1016/j.jfa.2021.109231

    work page doi:10.1016/j.jfa.2021.109231.url:https://doi.org/10.1016/j.jfa.2021.109231 2021

  19. [19]

    Mazurowski and X

    L. Mazurowski and X. Yao.A Positive Mass Theorem for Continuous Metrics. 2026. arXiv: 2606.19123 [math.DG].url:https://arxiv.org/abs/2606.19123

    Pith/arXiv arXiv 2026

  20. [20]

    On a Penrose-like inequality in dimensions less than eight

    S. McCormick and P. Miao. “On a Penrose-like inequality in dimensions less than eight”. In:Int. Math. Res. Not. IMRN7 (2019), pp. 2069–2084.issn: 1073-7928,1687-0247.doi: 10.1093/imrn/rnx181.url:https://doi.org/10.1093/imrn/rnx181

    work page doi:10.1093/imrn/rnx181.url:https://doi.org/10.1093/imrn/rnx181 2019

  21. [21]

    On the positive mass theorem for manifolds with corners

    D. McFeron and G. Székelyhidi. “On the positive mass theorem for manifolds with corners”. In:Comm. Math. Phys.313.2 (2012), pp. 425–443.issn: 0010-3616,1432-0916.doi:10.1007/ s00220-012-1498-8.url:https://doi.org/10.1007/s00220-012-1498-8

    work page doi:10.1007/s00220-012-1498-8 2012

  22. [22]

    Positive mass theorem on manifolds admitting corners along a hypersurface

    P. Miao. “Positive mass theorem on manifolds admitting corners along a hypersurface”. In: Adv. Theor. Math. Phys.6.6 (2002), 1163–1182 (2003).issn: 1095-0761,1095-0753.doi:10. 4310/ATMP.2002.v6.n6.a4.url:https://doi.org/10.4310/ATMP.2002.v6.n6.a4

    work page doi:10.4310/atmp.2002.v6.n6.a4 2002

  23. [23]

    Truncated Affine Rozansky–Witten Models as Extended Defect TQFTs

    P. Miao. “On a localized Riemannian Penrose inequality”. In:Comm. Math. Phys.292.1 (2009), pp. 271–284.issn: 0010-3616,1432-0916.doi:10.1007/s00220- 009- 0834- 0.url:https: //doi.org/10.1007/s00220-009-0834-0

    work page doi:10.1007/s00220- 2009

  24. [24]

    On the proof of the positive mass conjecture in general relativity

    R. Schoen and S. T. Yau. “On the proof of the positive mass conjecture in general relativity”. In:Comm. Math. Phys.65.1 (1979), pp. 45–76.issn: 0010-3616,1432-0916.url:http : / / projecteuclid.org/euclid.cmp/1103904790

    arXiv 1979

  25. [25]

    On the rigidity of Riemannian-Penrose inequality for asymp- totically flat 3-manifolds with corners

    Y. Shi, W. Wang, and H. Yu. “On the rigidity of Riemannian-Penrose inequality for asymp- totically flat 3-manifolds with corners”. In:Math. Z.291.1-2 (2019), pp. 569–589.issn: 0025- 5874,1432-1823.doi:10 . 1007 / s00209 - 018 - 2095 - 0.url:https : / / doi . org / 10 . 1007 / s00209-018-2095-0

    2019

  26. [26]

    Space-times with distribution-valued curvature tensors

    A. H. Taub. “Space-times with distribution-valued curvature tensors”. In:J. Math. Phys.21.6 (1980), pp. 1423–1431.issn: 0022-2488,1089-7658.doi:10 . 1063 / 1 . 524568.url:https : //doi.org/10.1063/1.524568

    work page doi:10.1063/1.524568 1980

  27. [27]

    A new proof of the positive energy theorem

    E. Witten. “A new proof of the positive energy theorem”. In:Comm. Math. Phys.80.3 (1981), pp. 381–402.issn: 0010-3616,1432-0916.url:http : / / projecteuclid . org / euclid . cmp / 1103919981. Università degli Studi di Trento Email address:virginia.agostiniani@unitn.it Università degli Studi di Trento Email address:christina.karapiperi@unitn.it Università ...

    1981