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Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets

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Pith's one-line read Free boundary MOTS in charged initial data sets satisfy area-charge inequalities, with local rigidity at equality.

desk verdict This paper extends area-charge inequalities to the free-boundary MOTS case in charged initial data with vanishing magnetic fields and adds a local rigidity statement. read the letter →

arxiv 2606.22184 v1 pith:HFCCH4MT submitted 2026-06-20 math.DG

Edilson Ferreira , Ivaldo Nunes This is my paper
classification math.DG
keywords area-chargeinequalityfreeboundaryMOTSEinstein-Maxwellequationslocalrigidityinitialdatasetsmarginallyoutertrappedsurfacesdominantenergycondition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that free boundary marginally outer trapped surfaces obey area-charge inequalities inside initial data sets for the Einstein-Maxwell equations when the magnetic field vanishes. It further shows that equality in the inequality forces local rigidity of the data near the surface. A reader would care because the result supplies a concrete geometric constraint linking surface area directly to electric charge under the Einstein-Maxwell constraints, thereby restricting admissible configurations of trapped surfaces in charged spacetimes.

What carries the argument

The area-charge inequality obtained by integrating the Einstein-Maxwell constraints along the surface and applying the energy condition, which directly bounds area from below by a multiple of the squared charge.

What would settle it

An explicit free boundary MOTS in an Einstein-Maxwell initial data set with zero magnetic field whose area lies strictly below the charge bound given by the inequality.

Watch

Extended reading notes

Core claim

In initial data sets for the Einstein-Maxwell equations with vanishing magnetic fields that satisfy the dominant energy condition, every free boundary marginally outer trapped surface satisfies an inequality relating its area to its electric charge. Equality holds only when the initial data is locally rigid in a neighborhood of the surface.

Load-bearing premise

The initial data must satisfy the Einstein-Maxwell equations with vanishing magnetic field together with the dominant energy condition.

Editorial extensions

If this is right

  • The inequality supplies a lower bound on area in terms of charge for every such surface.
  • Equality forces the initial data to be locally isometric to a model solution near the surface.
  • The result applies to any free-boundary problem whose boundary data meet the Einstein-Maxwell constraints.
  • It recovers the corresponding inequality for closed MOTS when the boundary is empty.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same technique may adapt to initial data with small but nonzero magnetic fields if suitable decay is assumed.
  • The rigidity statement could be strengthened to global uniqueness if the data set is asymptotically flat and the surface is outermost.
  • The inequality offers a test for numerical initial-data constructions that include electric charge and free boundaries.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The manuscript proves area-charge inequalities for free boundary marginally outer trapped surfaces (MOTS) in initial data sets for the Einstein-Maxwell equations with vanishing magnetic fields. It additionally establishes a local rigidity result when equality is attained in these inequalities.

Significance. If the proofs hold, the results would extend area-charge type inequalities to the setting of free-boundary MOTS in charged initial data, providing new tools for analyzing the Einstein-Maxwell system with boundaries. The local rigidity statement would further characterize the equality cases, which is of independent interest in mathematical relativity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our manuscript, which accurately describes the area-charge inequalities and local rigidity results for free boundary MOTS in charged initial data sets for the Einstein-Maxwell system with vanishing magnetic fields. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The provided abstract states the main results (area-charge inequalities and local rigidity for free boundary MOTS under Einstein-Maxwell with vanishing magnetic fields and energy conditions) but contains no equations, fitted parameters, self-citations, or derivation steps that could be inspected for reduction to inputs by construction. Without load-bearing steps, ansatzes, or uniqueness claims visible in the text, the derivation chain cannot be shown to collapse; the paper is treated as self-contained pending explicit equations.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The work relies on the Einstein-Maxwell equations and standard assumptions in mathematical relativity for initial data sets.

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Cite this review

Pith. "Pith review of Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets." pith.science (2026). https://pith.science/paper/HFCCH4MT

@misc{pith2026260622184,
  author       = {Pith},
  title        = {Pith review of: Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFCCH4MT}},
  note         = {Machine review of arXiv:2606.22184}
}
read the original abstract

In this work, we prove area-charge inequalities for free boundary MOTS in initial data sets for the Einstein-Maxwell equations with vanishing magnetic fields. In addition, we prove a local rigidity result under the assumption that equality holds.

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Works this paper leans on

42 extracted references · 1 canonical work pages

  1. [1]

    Alaee, M

    A. Alaee, M. Lesourd, and S.-T. Yau,Stable surfaces and free boundary marginally outer trapped surfaces, Calc. Var. Partial Differential Equations60(2021), no. 5, 186

    2021

  2. [2]

    Almaraz, L

    S. Almaraz, L. L. de Lima, and L. Mari,Spacetime positive mass theorems for initial data sets with non-compact boundary, Int. Math. Res. Not. IMRN2021(2021), no. 4, 2783-2841

    2021

  3. [3]

    L. C. Ambrozio,Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds, J. Geom. Anal.25(2015), no. 2, 1001-1017. MR 3319958

    2015

  4. [4]

    Andersson, M

    L. Andersson, M. Mars, and W. Simon,Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys.12(2008), no. 4, 853-888. MR 2420905

    2008

  5. [5]

    Barbosa and F

    E. Barbosa and F. Conrado,Discs area-minimizing in mean convex Riemannian n- manifolds, Proc. Roy. Soc. Edinburgh Sect. A152(2022), no. 6, 1361-1382. MR 4514025

    2022

  6. [6]

    Barros, R

    A. Barros, R. Batista, and T. Cruz,Hawking mass and local rigidity of minimal surfaces in three-manifolds, Comm. Anal. Geom.25(2017), no. 1, 1-23. MR 3663311

    2017

  7. [7]

    Barros and C

    A. Barros and C. Cruz,Free boundary hypersurfaces with non-positive Yamabe in- variant in mean convex manifolds, J. Geom. Anal.30(2020), no. 4, 3542-3562. MR 4167257

    2020

  8. [8]

    Barros, C

    A. Barros, C. Cruz, R. Batista, and P. Sousa,Rigidity in dimension four of area- minimising Einstein manifolds, Math. Proc. Cambridge Philos. Soc.158(2015), no. 2, 355-363. MR 3310250

    2015

Show all 42 references
  1. [9]

    Batista, B

    R. Batista, B. Lima, and J. Silva,Rigidity of free boundary minimal disks in mean convex three-manifolds, J. Geom. Anal.34(2024), no. 9, Paper No. 279, 18. MR 4766901

    2024

  2. [10]

    H. Bray, S. Brendle, and A. Neves,Rigidity of area-minimizing two-spheres in three- manifolds, Comm. Anal. Geom.18(2010), no. 4, 821-830. MR 2765731

    2010

  3. [11]

    Cai,Volume minimizing hypersurfaces in manifolds of nonnegative scalar curva- ture, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv

    M. Cai,Volume minimizing hypersurfaces in manifolds of nonnegative scalar curva- ture, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1-7. MR 1925731

    1999

  4. [12]

    Cai and G

    M. Cai and G. J. Galloway,Rigidity of area minimizing tori in 3-manifolds of non- negative scalar curvature, Comm. Anal. Geom.8(2000), no. 3, 565-573. MR 1775139 16 FERREIRA AND NUNES

    2000

  5. [13]

    Castro and C

    K. Castro and C. Rosales,Free boundary stable hypersurfaces in manifolds with den- sity and rigidity results, J. Geom. Phys.79(2014), 14-28. MR 3176286

    2014

  6. [14]

    Chodosh, M

    O. Chodosh, M. Eichmair, and V. Moraru,A splitting theorem for scalar curvature, Comm. Pure Appl. Math.72(2019), no. 6, 1231-1242. MR 3948556

    2019

  7. [15]

    Cruz and A

    T. Cruz and A. Mendes,Area-charge inequalities and rigidity of time-symmetric ini- tial data sets, arXiv preprint arXiv:2507.13040 (2025)

    2025

  8. [16]

    S. Dain, J. L. Jaramillo, and M. Reiris,Area-charge inequality for black holes, Classical Quantum Gravity29(2012), no. 3, 035013

    2012

  9. [17]

    de Almeida and A

    D. de Almeida and A. Mendes,Rigidity results for free boundary hypersurfaces in initial data sets with boundary: d. de almeida, a. mendes, Lett. Math. Phys.116 (2026), no. 1, 21

    2026

  10. [18]

    J. M. Espinar and H. Rosenberg,Area estimates and rigidity of capillary H-surfaces in three-manifolds with boundary, Math. Z.289(2018), no. 3-4, 1261-1279. MR 3830248

    2018

  11. [19]

    Fischer-Colbrie and R

    D. Fischer-Colbrie and R. Schoen,The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math.33(1980), no. 2, 199-211. MR 562550

    1980

  12. [20]

    G. J. Galloway,Rigidity of outermost MOTS: the initial data version, Gen. Relativity Gravitation50(2018), no. 3, Paper No. 32, 7. MR 3768955

    2018

  13. [21]

    G. J. Galloway and A. Mendes,Rigidity of marginally outer trapped 2-spheres, Comm. Anal. Geom.26(2018), no. 1, 63-83. MR 3761653

    2018

  14. [22]

    G. J. Galloway and A. Mendes,Some rigidity results for charged initial data sets, Nonlinear Anal.256(2025), Paper No. 113780, 9. MR 4871647

    2025

  15. [23]

    G. W. Gibbons,Some comments on gravitational entropy and the inverse mean cur- vature flow, Classical Quantum Gravity16(1999), no. 6, 1677-1687. MR 1697098

    1999

  16. [24]

    Gromov and H

    M. Gromov and H. B. Lawson, Jr.,Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2)111(1980), no. 2, 209-230. MR 569070

    1980

  17. [25]

    Gromov and H

    M. Gromov and H. B. Lawson, Jr.,Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes ´Etudes Sci. Publ. Math. (1983), no. 58, 83-196. MR 720933

    1983

  18. [26]

    S. Lee, S. Park, and J. Pyo,Capillary stable minimal hypersurfaces in a high dimen- sional Riemannian manifold, J. Geom. Anal.35(2025), no. 5, Paper No. 157, 20. MR 4886612

    2025

  19. [27]

    A. B. Lima, P. A. Sousa, and R. M. Batista,Rigidity of marginally outer trapped surfaces in charged initial data sets, Lett. Math. Phys.115(2025), no. 2, 41

    2025

  20. [28]

    Lima,Area estimates and rigidity of non-compact H-surfaces in 3-manifolds, Proc

    V. Lima,Area estimates and rigidity of non-compact H-surfaces in 3-manifolds, Proc. Amer. Math. Soc.147(2019), no. 10, 4499-4512. MR 4002559

    2019

  21. [29]

    M´ aximo and Ivaldo Nunes,Hawking mass and local rigidity of minimal two-spheres in three-manifolds, Comm

    D. M´ aximo and Ivaldo Nunes,Hawking mass and local rigidity of minimal two-spheres in three-manifolds, Comm. Anal. Geom.21(2013), no. 2, 409-432. MR 3043752

    2013

  22. [30]

    Mazet and H

    L. Mazet and H. Rosenberg,On minimal spheres of area 4 and rigidity, Comment. Math. Helv.89(2014), no. 4, 921-928. MR 3284299

    2014

  23. [31]

    Mendes,Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds, Math

    A. Mendes,Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds, Math. Proc. Cambridge Philos. Soc.167(2019), no. 2, 345-353. MR 3991376

    2019

  24. [32]

    Mendes,Area-charge inequality and local rigidity in charged initial data sets, Clas- sical Quantum Gravity42(2025), no

    A. Mendes,Area-charge inequality and local rigidity in charged initial data sets, Clas- sical Quantum Gravity42(2025), no. 22, Paper No. 225020, 18. MR 4998680

    2025

  25. [33]

    Mendes,Rigidity of marginally outer trapped (hyper) surfaces with negative lambda- constant, Trans

    A. Mendes,Rigidity of marginally outer trapped (hyper) surfaces with negative lambda- constant, Trans. Amer. Math. Soc.372(2019), no. 8, 5851-5868

    2019

  26. [34]

    Mendes,Rigidity of free boundary mots, Nonlinear Anal.220(2022), 112841

    A. Mendes,Rigidity of free boundary mots, Nonlinear Anal.220(2022), 112841

    2022

  27. [35]

    Moraru,On area comparison and rigidity involving the scalar curvature, J

    V. Moraru,On area comparison and rigidity involving the scalar curvature, J. Geom. Anal.26(2016), no. 1, 294-312. MR 3441515

    2016

  28. [36]

    Nunes,Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J

    I. Nunes,Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J. Geom. Anal.23(2013), no. 3, 1290-1302. MR 3078354 AREA-CHARGE INEQUALITIES FOR FREE BOUNDARY MOTS 17

    2013

  29. [37]

    L. F. Pessoa, E. V´ eras, and B. Vieira,Area estimates for capillary cme hypersurfaces with nonpositive Yamabe invariant, Bull. Lond. Math. Soc.57(2025), no. 9, 2708-

    2025

  30. [38]

    Schoen and S

    R. Schoen and S. T. Yau,On the structure of manifolds with positive scalar curvature, Manuscripta Math.28(1979), no. 1-3, 159-183. MR 535700

    1979

  31. [39]

    Schoen and S

    R. Schoen and S. T. Yau,Existence of incompressible minimal surfaces and the topol- ogy of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2)110(1979), no. 1, 127-142. MR 541332

    1979

  32. [40]

    Schoen and S

    R. Schoen and S. T. Yau,On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys.65(1979), no. 1, 45-76. MR 526976

    1979

  33. [41]

    Schoen and S.-T

    R. Schoen and S.-T. Yau,Positive scalar curvature and minimal hypersurface sin- gularities. Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, Surv. Differ. Geom., vol. 24, Int. Press, Boston, MA, [2022] 2022, pp. 4...

    2019

  34. [42]

    Witten,A new proof of the positive energy theorem, Comm

    E. Witten,A new proof of the positive energy theorem, Comm. Math. Phys.80(1981), no. 3, 381-402. MR 626707 Departamento de Matem ´atica, Universidade Federal do Maranh ˜ao, S ˜ao Lu´ıs - Brazil Email address:ivaldo.nunes@ufma.br Faculdade de Matem´atica, Universidade Federal d...

    1981

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