pith. sign in

arxiv: 2605.18730 · v1 · pith:HET2BBBAnew · submitted 2026-05-18 · 🌀 gr-qc · math.AP· math.DG

The spacetime Penrose inequality under a quasi final state hypothesis

Ahmed Ellithy This is my paper

Pith reviewed 2026-05-20 09:15 UTC · model grok-4.3

classification 🌀 gr-qc math.APmath.DG
keywords spacetime Penrose inequalityquasi final state hypothesisapparent horizon tubemarginally outer trapped surfacetangentially maximal hypersurfaceHawking massdominant energy condition
0
0 comments X

The pith

Under the quasi final state hypothesis, every asymptotically flat initial data set with a MOTS boundary on a black-hole apparent horizon tube satisfies the spacetime Penrose inequality.

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

The paper proves the spacetime Penrose inequality for asymptotically flat spacetimes that contain a black-hole-type apparent horizon tube and obey the dominant energy condition. It replaces the strong black hole final state conjecture with a weaker quasi final state hypothesis consisting of late-time decay on the normal shift component, decay of the timelike-to-spacelike mean curvature ratio, and convergence of cross-sectional areas to a finite limit. The proof introduces tangentially maximal hypersurfaces foliated by spheres of vanishing timelike mean curvature; these surfaces are shown to exist globally as solutions to a quasilinear inward-parabolic PDE. On such surfaces the spacetime Hawking mass coincides with the Riemannian Hawking mass, the dominant energy condition implies nonnegative scalar curvature, and the Riemannian Penrose inequality together with horizon area laws then yields the desired mass lower bound.

Core claim

For an asymptotically flat globally hyperbolic spacetime containing a black-hole-type apparent horizon tube H_app that satisfies the dominant energy condition and the quasi final state hypothesis, every asymptotically flat initial data set whose boundary is a MOTS cross-section of H_app obeys the spacetime Penrose inequality. The argument proceeds by constructing tangentially maximal hypersurfaces on which the spacetime Hawking mass reduces to the Riemannian Hawking mass, applying the known Riemannian Penrose inequality, and invoking the area laws for dynamical and isolated horizons.

What carries the argument

Tangentially maximal hypersurface: a spacelike hypersurface carrying a foliation by spheres of vanishing timelike mean curvature, shown to exist globally as the solution of a quasilinear inward-parabolic PDE.

If this is right

  • The ADM mass is bounded from below by the square root of the horizon area divided by 16 pi.
  • The bound holds for any initial data set whose boundary is a MOTS cross-section of the apparent horizon tube.
  • Area monotonicity along the tube from dynamical to isolated horizons connects the initial area to the final area used in the inequality.
  • Nonnegative scalar curvature on the tangentially maximal hypersurface follows directly from the dominant energy condition.

Where Pith is reading between the lines

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

  • The parabolic PDE construction may extend to other horizon inequalities or to spacetimes with different matter content where the dominant energy condition is relaxed.
  • Numerical evolution codes could test whether the quasi final state decay conditions are realized in concrete gravitational collapse simulations.
  • If the hypothesis can be verified for a given family of spacetimes, the inequality becomes a theorem rather than a conjecture for those families.

Load-bearing premise

Global existence of tangentially maximal hypersurfaces on which the spacetime Hawking mass reduces exactly to the Riemannian Hawking mass.

What would settle it

An asymptotically flat spacetime obeying the dominant energy condition and quasi final state hypothesis, yet containing an initial data set with MOTS boundary whose ADM mass lies below the square root of its horizon area over 16 pi, would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.18730 by Ahmed Ellithy.

Figure 1
Figure 1. Figure 1: Schematic of the proof. The apparent-horizon evolution has an initial cross-section [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

Penrose's original heuristic for his eponymous spacetime inequality -- a conjectured lower bound on the ADM mass in terms of the area of a horizon cross-section -- relies on the black hole final state conjecture. In this paper we isolate a substantially weaker but precise late-time condition, which we call the quasi final state hypothesis and prove the spacetime Penrose inequality under this hypothesis. More precisely, for an asymptotically flat globally hyperbolic spacetime with a black-hole-type apparent horizon tube ${H}_{{app}}$ satisfying the dominant energy condition and the quasi final state hypothesis, we show that every asymptotically flat initial data set whose boundary is a MOTS cross-section of ${H}_{{app}}$ satisfies the spacetime Penrose inequality. The quasi final state hypothesis requires only a late-time decay condition on the normal component of the shift and the ratio of timelike to spacelike mean curvature, together with convergence of the cross-sectional areas of ${H}_{{app}}$ to a finite limit. Our approach is new and formulated directly in spacetime. The main geometric object is what we call a \emph{tangentially maximal} hypersurface, carrying a foliation by spacelike spheres whose timelike mean curvature vanishes. We show that these hypersurfaces are governed by a quasilinear inward-parabolic PDE, and we develop the corresponding a priori theory and prove global existence. On these hypersurfaces, the spacetime Hawking mass reduces to the Riemannian Hawking mass, and the dominant energy condition gives nonnegative scalar curvature. The Riemannian Penrose inequality, combined with the area laws for dynamical and isolated horizons, then yields the result.

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

1 major / 2 minor

Summary. The paper claims that for an asymptotically flat globally hyperbolic spacetime with a black-hole-type apparent horizon tube H_app satisfying the dominant energy condition and the quasi final state hypothesis (late-time decay of the normal shift component and timelike-to-spacelike mean curvature ratio, plus convergence of cross-sectional areas to a finite limit), every asymptotically flat initial data set whose boundary is a MOTS cross-section of H_app satisfies the spacetime Penrose inequality. The proof introduces tangentially maximal hypersurfaces (with a foliation by spheres of vanishing timelike mean curvature) governed by a quasilinear inward-parabolic PDE, establishes global existence via a priori estimates, reduces the spacetime Hawking mass to the Riemannian Hawking mass on these surfaces, obtains nonnegative scalar curvature from the DEC, and combines the Riemannian Penrose inequality with area laws for dynamical and isolated horizons.

Significance. If the a priori estimates and global existence for the tangentially maximal hypersurfaces hold under the stated hypotheses, this constitutes a notable advance: it proves the spacetime Penrose inequality under a precise, substantially weaker late-time condition than the full black hole final state conjecture, while working directly in spacetime rather than reducing immediately to initial data. The new geometric construction of tangentially maximal hypersurfaces and the mass reduction step are original contributions that could extend to other inequalities involving dynamical horizons. Credit is due for the clean reduction to the Riemannian Penrose inequality (whose proof is external) and for formulating falsifiable late-time conditions.

major comments (1)
  1. [Global existence and a priori estimates for tangentially maximal hypersurfaces] The global existence proof for the quasilinear inward-parabolic PDE governing tangentially maximal hypersurfaces (detailed in the section developing the a priori theory) relies on the area convergence and late-time decay supplied by the quasi final state hypothesis. However, these conditions may not furnish uniform bounds on the second fundamental form or mean curvature sufficient to exclude finite-time singularities or loss of spacelike character prior to reaching the MOTS cross-section. This is load-bearing for the subsequent reduction of the spacetime Hawking mass to the Riemannian Hawking mass; without such control the mass inequality step does not apply to all initial data sets.
minor comments (2)
  1. [Introduction and notation] The notation for the apparent horizon tube H_app and its cross-sections is introduced in the abstract but would benefit from an explicit definition and diagram in the introduction or §2 to aid readers unfamiliar with dynamical horizon terminology.
  2. [Main theorem statement] In the statement of the main theorem, clarify whether the initial data sets are required to be maximal or if the tangentially maximal condition is imposed only on the auxiliary hypersurfaces; the current wording leaves this slightly ambiguous.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their recognition of the significance of the quasi final state hypothesis, and for raising this important point about the a priori estimates. We address the major comment in detail below.

read point-by-point responses

  1. Referee: The global existence proof for the quasilinear inward-parabolic PDE governing tangentially maximal hypersurfaces (detailed in the section developing the a priori theory) relies on the area convergence and late-time decay supplied by the quasi final state hypothesis. However, these conditions may not furnish uniform bounds on the second fundamental form or mean curvature sufficient to exclude finite-time singularities or loss of spacelike character prior to reaching the MOTS cross-section. This is load-bearing for the subsequent reduction of the spacetime Hawking mass to the Riemannian Hawking mass; without such control the mass inequality step does not apply to all initial data sets.

    Authors: We appreciate the referee's scrutiny of this foundational step. The a priori theory section derives uniform bounds on the second fundamental form and mean curvature directly from the area convergence to a finite limit combined with the late-time decay of the normal shift component and the timelike-to-spacelike mean curvature ratio. These hypotheses are used to apply parabolic maximum principles and regularity theory to the quasilinear inward-parabolic PDE, yielding L^infty control that rules out finite-time singularities and preserves the spacelike character of the evolving hypersurfaces up to the MOTS cross-section. The resulting bounds are then used to reduce the spacetime Hawking mass to its Riemannian counterpart. We will revise the manuscript to include an explicit corollary summarizing these curvature bounds and their dependence on the quasi final state hypothesis, thereby making the argument more transparent. revision: partial

Circularity Check

0 steps flagged

No circularity in the derivation chain

full rationale

The paper introduces the quasi final state hypothesis as a precise late-time condition on the normal shift and mean curvature ratio with area convergence, then constructs tangentially maximal hypersurfaces via a quasilinear inward-parabolic PDE for which it develops a priori estimates and proves global existence. On these surfaces the spacetime Hawking mass reduces to the Riemannian Hawking mass, nonnegative scalar curvature follows from the dominant energy condition, and the result follows by invoking the established Riemannian Penrose inequality together with area laws for dynamical and isolated horizons. No step equates the target inequality to its inputs by definition, renames a fitted quantity as a prediction, or reduces via a self-citation chain; the central argument is built from independent geometric constructions and external theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The proof rests on standard general-relativity background assumptions plus one new geometric object whose existence is established by PDE analysis.

axioms (3)
  • domain assumption Dominant energy condition
    Invoked to obtain nonnegative scalar curvature after reduction to the hypersurface.
  • domain assumption Asymptotically flat globally hyperbolic spacetime
    Standard setup required for the spacetime Penrose inequality statement.
  • domain assumption Existence of black-hole-type apparent horizon tube
    Assumed to provide the MOTS cross-sections and area monotonicity.
invented entities (1)
  • tangentially maximal hypersurface no independent evidence
    purpose: To carry a foliation by spheres with vanishing timelike mean curvature so that the spacetime problem reduces to the Riemannian case
    New object introduced and analyzed via parabolic PDE theory in the paper.

pith-pipeline@v0.9.0 · 5815 in / 1536 out tokens · 48822 ms · 2026-05-20T09:15:20.581648+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages · 1 internal anchor

  1. [1]

    Spyros Alexakis,The Penrose inequality on perturbations of the Schwarzschild exterior, 2015, Preprint, arXiv:1506.06400 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  2. [2]

    Brian Allen, Edward Bryden, Demetre Kazaras, and Marcus Khuri,Proof of the spacetime Penrose inequality with suboptimal constant in the asymptotically flat and asymptotically hy- perboloidal regimes, 2025, Preprint, arXiv:2504.10641 [math.DG]

    work page arXiv 2025

  3. [3]

    Xinliang An and Taoran He,On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality, 2025, Preprint, arXiv:2505.11399 [gr-qc]

    work page arXiv 2025

  4. [4]

    8, 085018

    Lars Andersson, Marc Mars, Jan Metzger, and Walter Simon,The time evolution of marginally trapped surfaces, Classical and Quantum Gravity26(2009), no. 8, 085018

    work page 2009

  5. [5]

    11, 111102

    Lars Andersson, Marc Mars, and Walter Simon,Local existence of dynamical and trapping horizons, Physical Review Letters95(2005), no. 11, 111102

    work page 2005

  6. [6]

    4, 853–888

    ,Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Advances in Theoretical and Mathematical Physics12(2008), no. 4, 853–888

    work page 2008

  7. [7]

    3, 941–972

    Lars Andersson and Jan Metzger,The area of horizons and the trapped region, Communications in Mathematical Physics290(2009), no. 3, 941–972

    work page 2009

  8. [8]

    2, L1–L7

    AbhayAshtekar, ChristopherBeetle, andStephenFairhurst,Isolated horizons: a generalization of black hole mechanics, Classical and Quantum Gravity16(1999), no. 2, L1–L7. 135

    work page 1999

  9. [9]

    Galloway,Some uniqueness results for dynamical horizons, Advances in Theoretical and Mathematical Physics9(2005), no

    Abhay Ashtekar and Gregory J. Galloway,Some uniqueness results for dynamical horizons, Advances in Theoretical and Mathematical Physics9(2005), no. 1, 1–30

    work page 2005

  10. [10]

    10, 104030

    AbhayAshtekarandBadriKrishnan,Dynamical horizons and their properties, PhysicalReview D68(2003), no. 10, 104030

    work page 2003

  11. [11]

    ,Isolated and dynamical horizons and their applications, Living Reviews in Relativity 7(2004), no. 10

    work page 2004

  12. [12]

    5, 661–693

    Robert Bartnik,The mass of an asymptotically flat manifold, Communications on Pure and Applied Mathematics39(1986), no. 5, 661–693

    work page 1986

  13. [13]

    2, 463–498

    Robert Beig and Niall Ó Murchadha,The Poincaré group as the symmetry group of canonical general relativity, Annals of Physics174(1987), no. 2, 463–498

    work page 1987

  14. [14]

    Ishai Ben-Dov,The Penrose inequality and apparent horizons, Physical Review D70(2004), 124031

    work page 2004

  15. [15]

    Bernal and Miguel Sánchez,Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Communications in Mathematical Physics257(2005), no

    Antonio N. Bernal and Miguel Sánchez,Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Communications in Mathematical Physics257(2005), no. 1, 43–50

    work page 2005

  16. [16]

    2, 183–197

    ,Further results on the smoothability of Cauchy hypersurfaces and Cauchy time func- tions, Letters in Mathematical Physics77(2006), no. 2, 183–197

    work page 2006

  17. [17]

    Bondi, M

    H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner,Gravitational waves in general relativity. VII. waves from axi-symmetric isolated systems, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences269(1962), no. 1336, 21–52

    work page 1962

  18. [18]

    Boyer and Richard W

    Robert H. Boyer and Richard W. Lindquist,Maximal analytic extension of the Kerr metric, Journal of Mathematical Physics8(1967), no. 2, 265–281

    work page 1967

  19. [19]

    Bray,Proof of the Riemannian Penrose inequality using the positive mass theorem, Journal of Differential Geometry59(2001), no

    Hubert L. Bray,Proof of the Riemannian Penrose inequality using the positive mass theorem, Journal of Differential Geometry59(2001), no. 2, 177–267

    work page 2001

  20. [20]

    Bray, Sean A

    Hubert L. Bray, Sean A. Hayward, Marc Mars, and Walter Simon,Generalized inverse mean curvature flows in spacetime, Communications in Mathematical Physics272(2007), no. 1, 119–138

    work page 2007

  21. [21]

    Bray and Jeffrey L

    Hubert L. Bray and Jeffrey L. Jauregui,Time flat surfaces and the monotonicity of the space- time Hawking mass, Communications in Mathematical Physics335(2015), no. 1, 285–307

    work page 2015

  22. [22]

    Bray, Jeffrey L

    Hubert L. Bray, Jeffrey L. Jauregui, and Marc Mars,Time flat surfaces and the monotonicity of the spacetime Hawking mass II, Annales Henri Poincaré17(2016), 1457–1475

    work page 2016

  23. [23]

    Bray and Marcus A

    Hubert L. Bray and Marcus A. Khuri,A Jang equation approach to the Penrose inequality, Discrete and Continuous Dynamical Systems27(2010), no. 2, 741–766

    work page 2010

  24. [24]

    4, 557–610

    ,P.D.E.’s which imply the Penrose conjecture, Asian Journal of Mathematics15(2011), no. 4, 557–610. 136

    work page 2011

  25. [25]

    Bray and Dan A

    Hubert L. Bray and Dan A. Lee,On the Riemannian Penrose inequality in dimensions less than eight, Duke Mathematical Journal148(2009), no. 1, 81–106

    work page 2009

  26. [26]

    Armando J. Cabrera Pacheco, Carla Cederbaum, Penelope Gehring, and Alejandro Peñuela Díaz,Constructing electrically charged Riemannian manifolds with minimal bound- ary, prescribed asymptotics, and controlled mass, Journal of Geometry and Physics185(2023), 104746

    work page 2023

  27. [27]

    Alberto Carrasco and Marc Mars,A counter-example to a recent version of the Penrose con- jecture, Classical and Quantum Gravity27(2010), 062001

    work page 2010

  28. [28]

    Chruściel,On the invariant mass conjecture in general relativity, Communications in Mathematical Physics120(1988), 233–248

    Piotr T. Chruściel,On the invariant mass conjecture in general relativity, Communications in Mathematical Physics120(1988), 233–248

    work page 1988

  29. [29]

    Piotr T. Chruściel and Erwann Delay,On mapping properties of the general relativistic con- straints operator in weighted function spaces, with applications, Mémoires de la Société Math- ématique de France, no. 94, Société Mathématique de France, 2003

    work page 2003

  30. [30]

    Chruściel, Jacek Jezierski, and Szymon Łęski,The Trautman-Bondi mass of hyper- boloidal initial data sets, Advances in Theoretical and Mathematical Physics8(2004), no

    Piotr T. Chruściel, Jacek Jezierski, and Szymon Łęski,The Trautman-Bondi mass of hyper- boloidal initial data sets, Advances in Theoretical and Mathematical Physics8(2004), no. 1, 83–139

    work page 2004

  31. [31]

    1, 137–189

    Justin Corvino,Scalar curvature deformation and a gluing construction for the Einstein con- straint equations, Communications in Mathematical Physics214(2000), no. 1, 137–189

    work page 2000

  32. [32]

    Schoen,On the asymptotics for the vacuum Einstein constraint equations, Journal of Differential Geometry73(2006), no

    Justin Corvino and Richard M. Schoen,On the asymptotics for the vacuum Einstein constraint equations, Journal of Differential Geometry73(2006), no. 2, 185–217

    work page 2006

  33. [33]

    1, 1–214

    Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski,The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Mathematica222(2019), no. 1, 1–214

    work page 2019

  34. [34]

    Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor,The non-linear sta- bility of the Schwarzschild family of black holes, 2021, Preprint, arXiv:2104.08222 [gr-qc]

    work page arXiv 2021

  35. [35]

    1, 37–72

    Xu-Qian Fan, Yuguang Shi, and Luen-Fai Tam,Large-sphere and small-sphere limits of the Brown-York mass, Communications in Analysis and Geometry17(2009), no. 1, 37–72

    work page 2009

  36. [36]

    10, 101101

    Jörg Frauendiener,On the Penrose inequality, Physical Review Letters87(2001), no. 10, 101101

    work page 2001

  37. [37]

    ElenaGiorgi,Stable black holes: in vacuum and beyond, BulletinoftheAmericanMathematical Society60(2023), no. 1, 1–27

    work page 2023

  38. [38]

    7, 2865–3849

    Elena Giorgi, Sergiu Klainerman, and Jérémie Szeftel,Wave equations estimates and the non- linear stability of slowly rotating Kerr black holes, Pure and Applied Mathematics Quarterly 20(2024), no. 7, 2865–3849

    work page 2024

  39. [39]

    846, Springer, Berlin, Heidelberg, 2012

    Éric Gourgoulhon,3+1 formalism in general relativity: Bases of numerical relativity, Lecture Notes in Physics, vol. 846, Springer, Berlin, Heidelberg, 2012. 137

    work page 2012

  40. [40]

    Hawking,Gravitational radiation in an expanding universe, Journal of Mathemat- ical Physics9(1968), no

    Stephen W. Hawking,Gravitational radiation in an expanding universe, Journal of Mathemat- ical Physics9(1968), no. 4, 598–604

    work page 1968

  41. [41]

    Hayward,Gravitational energy in spherical symmetry, Physical Review D53(1996), 1938–1949

    Sean A. Hayward,Gravitational energy in spherical symmetry, Physical Review D53(1996), 1938–1949

    work page 1996

  42. [42]

    4, 535–558

    Sven Hirsch,Hawking mass monotonicity for initial data sets, Asian Journal of Mathematics 29(2025), no. 4, 535–558

    work page 2025

  43. [43]

    3, 353–437

    Gerhard Huisken and Tom Ilmanen,The inverse mean curvature flow and the Riemannian Penrose inequality, Journal of Differential Geometry59(2001), no. 3, 353–437

    work page 2001

  44. [44]

    Kunduri,The spacetime Penrose inequality for cohomogeneity one initial data, Advances in Theoretical and Mathematical Physics29(2025), no

    Marcus Khuri and Hari K. Kunduri,The spacetime Penrose inequality for cohomogeneity one initial data, Advances in Theoretical and Mathematical Physics29(2025), no. 7, 1905–1943

    work page 2025

  45. [45]

    Mécanique353(2025), 555–581

    Sergiu Klainerman,The black hole stability problem, Comptes Rendus. Mécanique353(2025), 555–581

    work page 2025

  46. [46]

    3, 791–1678

    Sergiu Klainerman and Jérémie Szeftel,Kerr stability for small angular momentum, Pure and Applied Mathematics Quarterly19(2023), no. 3, 791–1678

    work page 2023

  47. [47]

    N. V. Krylov,Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996

    work page 1996

  48. [48]

    23, American Mathematical Society, Providence, RI, 1968

    O.A.Ladyzhenskaya, V.A.Solonnikov, andN.N.Ural’tseva,Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968

    work page 1968

  49. [49]

    4, 1667–1696

    Pengyu Le,Null Penrose inequality in a perturbed Schwarzschild spacetime, Pure and Applied Mathematics Quarterly20(2024), no. 4, 1667–1696

    work page 2024

  50. [50]

    Lieberman,Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996

    Gary M. Lieberman,Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996

    work page 1996

  51. [51]

    12, 33528

    Thomas Mädler and Jeffrey Winicour,Bondi-Sachs formalism, Scholarpedia11(2016), no. 12, 33528

    work page 2016

  52. [52]

    12, 6931–6934

    Edward Malec and Niall Ó Murchadha,Trapped surfaces and the Penrose inequality in spher- ically symmetric geometries, Physical Review D49(1994), no. 12, 6931–6934

    work page 1994

  53. [53]

    24, 5777–5787

    ,The Jang equation, apparent horizons, and the Penrose inequality, Classical and Quan- tum Gravity21(2004), no. 24, 5777–5787

    work page 2004

  54. [54]

    19, 193001

    Marc Mars,Present status of the Penrose inequality, Classical and Quantum Gravity26(2009), no. 19, 193001

    work page 2009

  55. [55]

    18, 185020

    Marc Mars and Alberto Soria,The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces, Classical and Quantum Gravity32(2015), no. 18, 185020

    work page 2015

  56. [56]

    11, 115019

    ,On the Penrose inequality along null hypersurfaces, Classical and Quantum Gravity 33(2016), no. 11, 115019. 138

    work page 2016

  57. [57]

    6, 1163–1182

    Pengzi Miao,Positive mass theorem on manifolds admitting corners along a hypersurface, Advances in Theoretical and Mathematical Physics6(2002), no. 6, 1163–1182

    work page 2002

  58. [58]

    3, 337–394

    Louis Nirenberg,The Weyl and Minkowski problems in differential geometry in the large, Com- munications on Pure and Applied Mathematics6(1953), no. 3, 337–394

    work page 1953

  59. [59]

    Barrett O’Neill,The geometry of Kerr black holes, A K Peters, Wellesley, MA, 1995

    work page 1995

  60. [60]

    1, 125–134

    Roger Penrose,Naked singularities, Annals of the New York Academy of Sciences224(1973), no. 1, 125–134

    work page 1973

  61. [61]

    Novaya Seriya31(73)(1952), no

    A.V.Pogorelov,Regularity of a convex surface with given Gaussian curvature, Matematicheskii Sbornik. Novaya Seriya31(73)(1952), no. 1, 88–103 (Russian)

    work page 1952

  62. [62]

    6, 1937–1978

    Henri Roesch,Quasi-round MOTSs and stability of the Schwarzschild null Penrose inequality, Annales Henri Poincaré22(2021), no. 6, 1937–1978

    work page 2021

  63. [63]

    Roesch,Proof of a null Penrose conjecture using a new quasi-local mass, Communi- cations in Analysis and Geometry29(2021), no

    Henri P. Roesch,Proof of a null Penrose conjecture using a new quasi-local mass, Communi- cations in Analysis and Geometry29(2021), no. 8, 1847–1915

    work page 2021

  64. [64]

    R. K. Sachs,Gravitational waves in general relativity. VIII. waves in asymptotically flat space- time, ProceedingsoftheRoyalSocietyofLondon.SeriesA.MathematicalandPhysicalSciences 270(1962), no. 1340, 103–126

    work page 1962

  65. [65]

    thesis, ETH Zürich, 2008

    Johannes Sauter,Foliations of null hypersurfaces and the penrose inequality, Ph.D. thesis, ETH Zürich, 2008

    work page 2008

  66. [66]

    1, 45–76

    Richard Schoen and Shing-Tung Yau,On the proof of the positive mass conjecture in general relativity, Communications in Mathematical Physics65(1979), no. 1, 45–76

    work page 1979

  67. [67]

    II, Communications in Mathematical Physics79 (1981), no

    ,Proof of the positive mass theorem. II, Communications in Mathematical Physics79 (1981), no. 2, 231–260

    work page 1981

  68. [68]

    1, Paper No

    Dawei Shen,Construction of GCM hypersurfaces in perturbations of Kerr, Annals of PDE9 (2023), no. 1, Paper No. 11, 112 pp

    work page 2023

  69. [69]

    1, 79–125

    Yuguang Shi and Luen-Fai Tam,Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, Journal of Differential Geometry62(2002), no. 1, 79–125

    work page 2002

  70. [70]

    Szabados,On the roots of the Poincaré structure of asymptotically flat spacetimes, Classical and Quantum Gravity20(2003), no

    László B. Szabados,On the roots of the Poincaré structure of asymptotically flat spacetimes, Classical and Quantum Gravity20(2003), no. 13, 2627–2662

    work page 2003

  71. [71]

    ,Quasi-local energy-momentum and angular momentum in general relativity, Living Reviews in Relativity12(2009), no. 4

    work page 2009

  72. [72]

    Taylor,Partial differential equations III: Nonlinear equations, 3 ed., Applied Math- ematical Sciences, vol

    Michael E. Taylor,Partial differential equations III: Nonlinear equations, 3 ed., Applied Math- ematical Sciences, vol. 117, Springer, Cham, 2023

    work page 2023

  73. [73]

    3, 381–402

    Edward Witten,A new proof of the positive energy theorem, Communications in Mathematical Physics80(1981), no. 3, 381–402. 139

    work page 1981