Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces

Alejandro Pe\~nuela Diaz

REVIEW 4 major objections 5 minor 53 references

Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Hawking energy is nonnegative and rigid on its natural critical surfaces, with zero forcing a Euclidean or Minkowski region.

desk verdict Time-symmetric rigidity on Willmore surfaces is a genuine and clean result, but the dynamical theorem as advertised in the abstract is not proven; the technical sign condition is artificial and violated by the paper's own zero-energy examples. read the letter →

arxiv 2507.16588 v3 pith:3ULCTVZ6 submitted 2025-07-22 math-ph gr-qcmath.DGmath.MP

Alejandro Peñuela Diaz This is my paper · ORCID

classification math-phgr-qcmath.DGmath.MP MSC 53C2153C4283C4083C99 PACS 04.20.-q04.20.Cv

keywords Hawkingenergyquasi-localWillmoresurfacesrigiditypositivetheoreminitialdatasetsdominantcondition

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

This paper claims that the Hawking energy, the standard quasi-local measure of gravitational energy inside a closed 2-surface, becomes nonnegative and rigid when evaluated on its natural area-constrained critical surfaces rather than on arbitrary surfaces. In the time-symmetric case those surfaces are area-constrained Willmore surfaces, and zero Hawking energy forces the enclosed region to be a Euclidean ball with a round spherical boundary; charged, cosmological-constant, and higher-dimensional analogues are proved. In the fully dynamical case, where the critical surfaces are called Hawking surfaces, the paper gives the first nonnegativity and rigidity theorems under the dominant energy condition, with zero energy forcing the enclosed domain to be a spacelike hypersurface in Minkowski spacetime. The dynamical rigidity relies on a technical integral sign hypothesis that the paper itself identifies as neither optimal nor physically motivated, and the paper's own Minkowski-spacetime examples violate this hypothesis even though the surfaces there have zero Hawking energy.

What carries the argument

The central object is the area-constrained Euler-Lagrange equation of the Hawking functional $\int_\Sigma(H^2-P^2)\,d\mu$. For $k=0$ it reduces to the Willmore equation $0=\lambda H+\Delta_\Sigma H+H|\mathring B|^2+H\,\mathrm{Ric}_M(\nu,\nu)$; for general $k$ it becomes the Hawking-surface equation, which adds terms involving the second fundamental form $k$, its normal derivative, and the tangential gradient of $P$. Multiplying either equation by $H^{-1}$ and integrating by parts yields a master integral identity that separates nonnegative terms such as $|\nabla_\Sigma\log H|^2$ and $|\mathring B|^2$ from the integral controlling the Hawking energy; when the Hawking energy vanishes, all nonnegative terms must vanish, making $H$ constant, the surface umbilic, and the ambient Ricci tensor aligned, after which boundary rigidity theorems identify the region as Euclidean or Minkowski.

What would settle it

Compute the integral $\int_\Sigma(f_\beta-\lambda)\,d\mu$ on the zero-Hawking-energy round spheres in the hyperboloid and paraboloid hypersurfaces of Minkowski spacetime constructed in Examples 3.12 and 3.13; the paper shows it is positive, so the hypothesis is not necessary. To test the theorem directly, search for any initial data set satisfying the dominant energy condition that contains a positive-mean-curvature Hawking surface with $\int_\Sigma(f_\beta-\lambda)\,d\mu\le 0$ and $\int_\Sigma(H^2-P^2)\,d\mu=16\pi$ but whose enclosed region is not Minkowski; finding one refutes Theorem 3.6(ii).

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Extended reading notes

Core claim

The paper's central claim is that the Hawking energy, although not positive on arbitrary surfaces (every non-round sphere in Euclidean space has negative Hawking energy), becomes nonnegative and rigid when evaluated on area-constrained critical surfaces of the Hawking functional. In the time-symmetric case these surfaces are area-constrained Willmore surfaces, and under $\mathrm{Sc}_M\ge 0$, positive mean curvature, and nonnegative Lagrange parameter, the Hawking energy is nonnegative; vanishing energy forces the enclosed region to be a Euclidean ball and the surface a round sphere. In the dynamical case, for Hawking surfaces under the dominant energy condition, the paper proves nonnegativity and, under an additional integral sign condition, rigidity: zero Hawking energy forces the enclosed domain to be isometric to a spacelike hypersurface in Minkowski spacetime, with vanishing second fundamental form on the round umbilic boundary. The same pattern is extended to charged Einstein-Maxwell data, to negative and positive cosmological-constant backgrounds, and to two higher-dimensional analogues, with rigidity against hyperbolic, Euclidean, spherical, or hemispherical reference geometries.

Load-bearing premise

The load-bearing premise in the dynamical rigidity theorem is the technical integral inequality $\int_\Sigma(f_\beta-\lambda)\,d\mu\le 0$, which the paper itself calls neither optimal nor physically motivated; if that integral is positive, the rigidity argument does not go through, as happens in the paper's own Minkowski-spacetime examples with zero Hawking energy.

Editorial extensions

If this is right

In any 3-manifold with nonnegative scalar curvature, an area-constrained Willmore sphere with positive mean curvature and nonnegative Lagrange parameter has nonnegative Hawking energy, and equality forces the enclosed domain to be a Euclidean ball with round spherical boundary.
In an asymptotically flat 3-manifold with nonnegative scalar curvature, every leaf of the canonical Willmore foliation has positive Hawking and Brown-York energy unless the manifold is Euclidean.
In the dynamical setting, a Hawking surface with positive mean curvature, a dominant energy condition, and $\int_\Sigma(f-\lambda)\,d\mu\le 0$ has nonnegative Hawking energy; if additionally $\int_\Sigma(f_\beta-\lambda)\,d\mu\le 0$ and the energy vanishes, the enclosed region is a spacelike hypersurface in Minkowski spacetime with $k=0$ on the round umbilic boundary.
The positivity-rigidity pattern extends to charged data, to both signs of the cosmological constant, and to two higher-dimensional Hawking-type energies, with the positive-$\Lambda$ minimal-surface endpoint forcing a hemisphere.
Under the technical dynamical hypothesis, Hawking surfaces in Minkowski spacetime have strictly positive Hawking energy unless the surface lies in a hyperplane, mirroring the over-positivity of the Kijowski-Liu-Yau energy.

Reading between the lines

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

The paper's own Minkowski examples suggest that the sign hypothesis on $f$ is stronger than needed; a plausible inference, not proven here, is that a sharper definition such as the paper's $\tilde f$ could yield the same rigidity under the dominant energy condition alone.
Because Hawking surfaces are defined relative to a chosen spacelike hypersurface, applying these results in numerical relativity would require checking the integral condition on the surfaces of evolved slices; the foliation construction advertised in the companion paper is the natural place to test it.
The master-integral strategy may transfer to other quasi-local quantities, such as charged mass or angular momentum, where analogous Euler-Lagrange identities could yield rigidity against Kerr or Reissner-Nordström reference geometries rather than flat space.
The positive-cosmological-constant hemisphere endpoint is a Ricci-strengthened rigidity of the same shape as the scalar-curvature hemisphere conjecture; if that conjecture were available, the spherical rigidity hypotheses could likely be weakened.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The short version: the time-symmetric part is genuinely good and worth knowing; the dynamical part as advertised in the abstract is overclaimed. The two halves should be judged separately.

What is new: Theorem 2.8 shows that a single area-constrained Willmore surface with nonnegative Lagrange parameter and zero Hawking energy forces the enclosed domain to be a Euclidean ball. That removes the even-symmetry and near-roundness assumptions needed in the stable CMC rigidity results of Sun and Shi-Tian-Wei. The proof is short, transparent, and correct: integrate the Willmore equation, use Gauss-Bonnet, then apply Shi-Tam rigidity. Corollary 2.11 gives positivity cleanly. The charged and cosmological-constant variants are routine but useful, and the higher-dimensional framework looks coherent even if I did not verify every constant.

The soft spot is the dynamical section. Theorem 3.6 is explicitly conditional on an integral inequality involving f_beta and lambda. The paper itself concedes in Remark 3.8 that f_beta is not physically motivated and may bias the Hawking energy toward positivity. More seriously, Examples 3.12 and 3.13 exhibit round Hawking spheres in Minkowski spacetime with zero Hawking energy that violate the inequality. So the extra hypothesis excludes exactly the rigidity examples one would want to capture. The abstract and the concluding summary state that under the dominant energy condition the Hawking energy is nonnegative and rigid on Hawking surfaces, without flagging the condition. That is not supported by the theorems. The body is more honest, but the advertisement is what most readers will see.

A smaller concern: equation (21) defining E_{n,Lambda,1} looks odd; in n=3 it does not seem to reduce to the standard Hawking energy with cosmological constant as claimed. I would check the normalization before citing the higher-dimensional results.

Who gets value from this? Geometric relativists working on quasi-local mass. The time-symmetric rigidity will be cited. The dynamical theorem, in its honest conditional form, is a real contribution but not the breakthrough the abstract suggests.

My recommendation: this deserves peer review, not desk rejection. The time-symmetric result alone justifies referee time, and a good referee can force the abstract to match the theorems. But the current version should not be accepted as is; it needs a revision that either narrows the abstract or finds a physically motivated condition that does not exclude the Minkowski examples.

Referee Report

4 major / 5 minor

Summary. The paper studies the Hawking quasi-local energy on area-constrained critical surfaces in initial data sets. In the time-symmetric case these are area-constrained Willmore surfaces, and the author proves nonnegativity and rigidity theorems under nonnegative scalar curvature (Theorems 2.8 and 2.11), with extensions to charged manifolds (Corollary 2.18), cosmological constant settings (Theorems 2.23 and 2.26), and higher dimensions (Theorems 2.29, 2.30, 2.33, 2.35). In the dynamical case (k≠0), critical surfaces are called Hawking surfaces (equation (35)), and the main results (Theorem 3.6, Corollary 3.9) assert positivity and rigidity for the Hawking energy under the dominant energy condition plus an additional integral hypothesis involving a technically constructed function f_beta. The paper also gives examples in Minkowski spacetime of Hawking surfaces with zero Hawking energy for which this integral hypothesis fails (Examples 3.12 and 3.13).

Significance. The time-symmetric results are a solid contribution: they combine the Willmore equation with Shi-Tam and Hang-Wang rigidity to obtain clean rigidity statements on area-constrained Willmore surfaces, and the charged, cosmological, and higher-dimensional generalizations are natural and appear correct. The dynamical results, however, are not established in the unconditional form promised by the abstract and conclusion: Theorem 3.6 requires an ad hoc integral condition on the surface, and the paper itself concedes in Remark 3.8 that this condition is neither optimal nor physically motivated, while Examples 3.12 and 3.13 show that the condition is violated by natural zero-energy Hawking surfaces in Minkowski spacetime. The conditional theorems are still meaningful, but the advertised claim that the Hawking energy is nonnegative and rigid under the dominant energy condition on its critical surfaces is not supported by the body of the paper.

major comments (4)

[Abstract and §1.3] The abstract and the concluding summary state that, under the dominant energy condition, the Hawking energy is nonnegative and rigid on its natural critical surfaces. This is not what Theorem 3.6 proves: both parts (i) and (ii) require the integral condition ∫Σ(fβ−λ)dμ≤0 (or the weaker ∫Σ(f−λ)dμ≤0), which is not a consequence of the dominant energy condition, as the paper itself notes in Remark 3.8. The advertised claim is therefore unsupported and should be revised to state the conditional nature of the dynamical results.
[Examples 3.12 and 3.13] The two Minkowski examples are Hawking surfaces with zero Hawking energy for which ∫Σ(f−λ)dμ = ∫Σ(P/H)^2|k|^2 dμ > 0, so the technical hypothesis of Theorem 3.6 fails exactly in the rigidity regime one would most want to capture. This shows that the integral condition is not merely a technical convenience but actively excludes natural flat-space surfaces, undermining the interpretation that the theorem establishes 'positive mass' behavior for the Hawking energy on its critical surfaces in the dynamical setting.
[Remark 3.7 and Remark 3.14] The alternative function f̃ introduced in Remark 3.7 is said to satisfy the needed sign condition automatically on the Examples 3.12 and 3.13, but no theorem is proved using f̃, and it is not shown that the f̃-condition is implied by any physical hypothesis. As a result, the paper does not supply a positive dynamical theorem whose hypotheses hold on any nontrivial class of Hawking surfaces broader than those already covered by the fβ-condition; the discussion in Remark 3.14 remains heuristic.
[Corollary 3.9] Corollary 3.9 is stated as a positive energy theorem for the dynamical setting, but its hypothesis is the same ad hoc integral condition. Given that the condition can be violated by zero-energy Hawking surfaces in Minkowski spacetime, the corollary should be framed as a conditional statement rather than as a confirmation that the Hawking energy satisfies the basic physical principles of positivity and rigidity without further assumptions.

minor comments (5)

[Equation (35)] In the Hawking surface equation, the term 2k(∇ΣP,ν) appears with a sign that differs from the sign in the derivation in the proof of Lemma 3.1; please check the overall sign convention and ensure consistency between equation (35) and equation (38).
[Theorem 2.8] The phrase 'the rest of components have positive scalar curvature' should be clarified: for 2-dimensional boundary components, positive scalar curvature means positive Gauss curvature, which is what is needed for the Weyl-Nirenberg-Pogorelov embedding theorem; please make this explicit to avoid ambiguity.
[Theorem 3.15] The proof of Theorem 3.15 is only a sentence saying it is a direct combination of Theorems 2.30 and 3.6; since the higher-dimensional Hawking equation (56) contains an extra term involving n, the proof should indicate how the dimensional constants enter the inequalities.
[General] There are several typographical issues, including inconsistent use of 'its' and 'it's', missing spaces around equations, and the variable 'a' in Remark 3.14 referring to the hyperboloid parameter while 'a' is also used for the constant in Example 3.12; a careful proofreading pass is recommended.
[References] The reference list is generally complete, but the companion paper [40] is cited as 'Communications in Analysis and Geometry 34 (2026)' while other citations from the same venue use different formatting; please verify all bibliographic details.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central derivations are self-contained modulo standard external rigidity theorems, and the dynamical sign condition is an explicit, non-fitted hypothesis rather than a conclusion restated as an input.

full rationale

The paper's derivation chain is not circular. The Willmore and Hawking surface equations are obtained by direct first-variation computations (Section 3, Lemma 3.1), not imported as an unproved ansatz from a citation. The positivity and rigidity results in Theorems 2.8, 2.11, and 3.6 proceed by integrating the Euler-Lagrange equation, using the Gauss equation and Gauss-Bonnet, and then invoking external rigidity theorems such as Shi-Tam (Theorem 2.4/2.6), Liu-Yau (Theorem 3.4), Hang-Wang, and Ros. Those external theorems are independent of the present paper's claims and do not presuppose Hawking-energy positivity or rigidity. The extra condition in the dynamical theorem, ∫(f_β - λ)dμ ≤ 0, is an explicit additional hypothesis; it is not a fitted parameter and it is not equivalent to the desired inequality by construction. The paper itself states in Remark 3.8 that the condition is 'neither optimal nor physically motivated' and that f_β was introduced for a technical reason, and Examples 3.12-3.13 show the condition is not automatic by exhibiting zero-Hawking-energy Hawking surfaces violating it. This means the abstract's unconditional dominant-energy claim is stronger than what Theorem 3.6 proves, but that is an overstatement or support gap, not circularity. Self-citations to [38]-[40] appear in contextual remarks (small-sphere limits, the author's thesis, and a companion paper on foliations) and are not load-bearing for the theorems established here. No equation in the paper reduces to its own input by definition, and no prediction is secretly a fitted quantity.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central time-symmetric results rest on standard rigidity theorems (Shi-Tam, Liu-Yau, Hang-Wang, Ros) and on the nonnegative scalar curvature assumption. The dynamical results require an additional non-physical condition involving f_beta, which is the strongest extra input beyond the dominant energy condition. No new physical entities are introduced. The Lagrange multiplier lambda is part of the surface equation rather than a fitted parameter; beta is the only genuinely ad hoc auxiliary parameter.

free parameters (1)

beta (technical parameter) = any number in [0, 1/2)
Appears only in the technical condition integral(f_beta - lambda) <= 0; introduced so that vanishing Hawking energy forces k=0 on Sigma. No physical motivation (Remark 3.8).

assumptions (8)

domain assumption Dominant energy condition mu >= |J|
Used in Theorem 3.6 to make the term -(mu - |J|) nonpositive in the energy estimate.
domain assumption Nonnegative scalar curvature Sc>=0 or Sc>=2Lambda
A restatement of the dominant energy condition in the time-symmetric setting, used throughout Section 2.
standard math Shi-Tam Brown-York rigidity (and its higher-dimensional version), relying on the positive mass theorem
Used in Theorem 2.8 and Theorem 2.30 to conclude that zero Brown-York energy forces a Euclidean ball.
standard math Liu-Yau Kijowski-Liu-Yau rigidity
The dynamical rigidity in Theorem 3.6(ii) reduces to vanishing Kijowski-Liu-Yau energy, then invokes the Liu-Yau rigidity theorem.
standard math Hang-Wang rigidity theorems for positive Ricci curvature with boundary
Used for the spherical Lambda>0 rigidity in Theorems 2.26, 2.33, and 2.35.
ad hoc to paper Integral condition integral_Sigma (f_beta - lambda) dmu <= 0
This is the load-bearing extra assumption in the dynamical setting. It is not derived from physical principles and is violated by the zero-Hawking-energy examples in Minkowski spacetime (Examples 3.12 and 3.13).
standard math Ros constant-scalar-curvature rigidity and Weyl-Nirenberg-Pogorelov embedding
Used in the higher-dimensional rigidity proofs to conclude that surfaces with constant scalar curvature and convex Euclidean embeddings are round.
standard math Alexandrov-Fenchel inequality for convex domains
Used in Theorem 2.35 to bound the total scalar curvature of the convex boundary from below.

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Cite this review

Pith. "Pith review of Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces." pith.science (2026). https://pith.science/paper/3ULCTVZ6

@misc{pith2026250716588,
  author       = {Pith},
  title        = {Pith review of: Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ULCTVZ6}},
  note         = {Machine review of arXiv:2507.16588}
}

read the original abstract

A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, also known as the Hawking mass, satisfies these properties under the dominant energy condition when evaluated on its natural area-constrained critical surfaces within a spacelike hypersurface (initial data set). In the time-symmetric case, these critical surfaces coincide with area-constrained Willmore surfaces, and we obtain positivity and rigidity theorems for the Hawking energy on such surfaces, including charged and cosmological constant (hyperbolic and spherical) variants as well as higher-dimensional analogues. In the fully dynamical (non-time-symmetric) case, we establish the first nonnegativity and rigidity theorems for the Hawking energy in this general setting. These results confirm the Hawking energy's consistency with basic physical principles and address several longstanding ambiguities and criticisms.

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Reference graph

Works this paper leans on

53 extracted references · 49 canonical work pages

[1]

H. Baltazar, Abdênago Barros, and Rondinelle Batista,A local rigidity theorem for minimal two-spheres in charged time-symmetric initial data set, Letters in Mathematical Physics113(2023)

work page 2023
[2]

A Barros, R Batista, and T Cruz,Hawking mass and local rigidity of minimal surfaces in three-manifolds, Communications in Analysis and Geometry25(2017), no. 1, 1–23

work page 2017
[3]

Hubert Bray, Sean Hayward, Marc Mars, and Walter Simon,Generalized inverse mean curvature flows in spacetime, Communications in mathematical physics272(2007), 119–138

work page 2007
[4]

Bray,On the positive mass, penrose, an zas inequalities in general dimension, surveys in geo- metric analysis and relativity, adv

Hubert L. Bray,On the positive mass, penrose, an zas inequalities in general dimension, surveys in geo- metric analysis and relativity, adv. lect. math.(alm), 20, Int. Press, Somerville, MA (2011)

work page 2011
[5]

Bray and Jeffrey L

Hubert L. Bray and Jeffrey L. Jauregui,A geometric theory of zero area singularities in general relativity, Asian Journal of Mathematics17(2013), no. 3, 525 – 560

work page 2013
[6]

1, 175–197

Simon Brendle, Fernando C Marques, and Andre Neves,Deformations of the hemisphere that increase scalar curvature, Inventiones mathematicae185(2011), no. 1, 175–197

work page 2011
[7]

J David Brown and James W York Jr,Quasilocal energy and conserved charges derived from the gravita- tional action, Physical Review D47(1993), no. 4, 1407

work page 1993
[8]

Armando J. Cabrera Pacheco, Carla Cederbaum, Penelope Gehring, and Alejandro Peñuela Diaz,Con- structing electrically charged riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass, Journal of Geometry and Physics185(2023), 104746

work page 2023

Show all 53 references

[9]

Math., vol

Demetrios Christodoulou and Shing-Tung Yau,Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 9–14. MR 954405 32 PEÑUELA DIAZ

1986
[10]

24, 245019

Marcelo M Disconzi and Marcus A Khuri,On the penrose inequality for charged black holes, Classical and Quantum Gravity29(2012), no. 24, 245019

2012
[11]

1, 105–160

Michael Eichmair and Thomas Koerber,Large area-constrained willmore surfaces in asymptotically schwarzschild3-manifolds, Journal of Differential Geometry127(2024), no. 1, 105–160

2024
[12]

1, 159–186

Michael Eichmair and Jan Metzger,Large isoperimetric surfaces in initial data sets, Journal of Differential Geometry94(2013), no. 1, 159–186

2013
[13]

,Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions, Invent. Math. 194(2013), no. 3, 591–630. MR 3127063

2013
[14]

Alexander Friedrich,Concentration of small hawking type surfaces, Differential Geometry and its Appli- cations85(2022), 101927

2022
[15]

5, 1725–1732

PengfeiGuanandJunfangLi,The quermassintegral inequalities for k-convex starshaped domains, Advances in Mathematics221(2009), no. 5, 1725–1732

2009
[16]

Fengbo Hang and Xiaodong Wang,Rigidity theorems for compact manifolds with boundary and positive ricci curvature, Journal of Geometric Analysis19(2009), 628–642

2009
[17]

Hawking,Gravitational radiation in an expanding universe, J

Stephen W. Hawking,Gravitational radiation in an expanding universe, J. Mathematical Phys.9(1968), no. 4, 598–604. MR 3960907

1968
[18]

3, 307–343

JerzyKijowski,A simple derivation of canonical structure and quasi-local hamiltonians in general relativity, General Relativity and Gravitation29(1997), no. 3, 307–343

1997
[19]

Thomas Koerber,The area preserving Willmore flow and local maximizers of the Hawking mass in asymp- totically Schwarzschild manifolds, J. Geom. Anal.31(2021), no. 4, 3455–3497. MR 4236532

2021
[20]

Ann.350(2011), no

Tobias Lamm, Jan Metzger, and Felix Schulze,Foliations of asymptotically flat manifolds by surfaces of Willmore type, Math. Ann.350(2011), no. 1, 1–78. MR 2785762

2011
[21]

BeneditoLeandro andGuilhermeSabo,Sharp lower bound for the charged hawking mass in the electrostatic space, arXiv preprint arXiv:2507.03353 (2025)

2025 arXiv
[22]

Jihyeon Lee and Sanghun Lee,Modified hawking mass and rigidity of three-manifolds with boundary, arXiv preprint arXiv:2505.08301 (2025)

2025 arXiv
[23]

23, 231102

Chiu-Chu Melissa Liu and Shing-Tung Yau,Positivity of quasilocal mass, Physical review letters90(2003), no. 23, 231102

2003
[24]

1, 181–204

,Positivity of quasi-local mass ii, Journal of the American Mathematical Society19(2006), no. 1, 181–204

2006
[25]

Joachim Lohkamp,The higher dimensional positive mass theorem i, arXiv preprint arXiv:math/0608795 (2016)

2016 arXiv
[26]

,The higher dimensional positive mass theorem ii, arXiv preprint arXiv:1612.07505 (2016)

2016 arXiv
[27]

1, 015007

Stephen McCormick,On the charged riemannian penrose inequality with charged matter, Classical and quantum gravity37(2019), no. 1, 015007

2019
[28]

12, 5373–5380

Luiz Ricardo Melo,On the hawking mass for cmc surfaces in positive curved 3-manifolds, Proceedings of the American Mathematical Society152(2024), no. 12, 5373–5380

2024
[29]

Differential Geom.77(2007), no

Jan Metzger,Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature, J. Differential Geom.77(2007), no. 2, 201–236. MR 2355784

2007
[30]

2, 437–459

Pengzi Miao, Yuguang Shi, and Luen-Fai Tam,On geometric problems related to brown-york and liu-yau quasilocal mass, Communications in mathematical physics298(2010), no. 2, 437–459

2010
[31]

2, 1323–1354

Pengzi Miao, Luen-Fai Tam, and Naqing Xie,Quasi-local mass integrals and the total mass, The Journal of Geometric Analysis27(2017), no. 2, 1323–1354

2017
[32]

Pengzi Miao, Yaohua Wang, and Naqing Xie,On Hawking mass and Bartnik mass of CMC surfaces, Math. Res. Lett.27(2020), no. 3, 855–885. MR 4216572

2020
[33]

Maung Min-Oo,Scalar curvature rigidity of certain symmetric spaces, Geometry, topology, and dynamics (Montreal, 1995)127137(1998)

1998
[34]

3, 1844–1896

Andrea Mondino and Aidan Templeton-Browne,Some rigidity results for the hawking mass and a lower bound for the bartnik capacity, Journal of the London Mathematical Society106(2022), no. 3, 1844–1896

2022
[35]

DaviMáximoandIvaldoNunes,Hawking mass and local rigidity of minimal two-spheres in three-manifolds, Communications in Analysis and Geometry21(2012)

2012
[36]

Christopher Nerz,Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry, Calc. Var. Partial Differential Equations54(2015), no. 2, 1911–1946. MR 3396437

2015
[37]

3, 337–394

Louis Nirenberg,The weyl and minkowski problems in differential geometry in the large, Communications on pure and applied mathematics6(1953), no. 3, 337–394. 33

1953
[38]

3, 035002

Alejandro Peñuela Díaz,Local foliations by critical surfaces of the hawking energy and small sphere limit, Classical and Quantum Gravity40(2023), no. 3, 035002

2023
[39]

thesis, Universität Potsdam, 2025

,Geometrically defined surfaces in general relativity and their relation to physical invariants, Ph.D. thesis, Universität Potsdam, 2025

2025
[40]

2, 505–570

,Foliations by critical surfaces of the hawking energy in asymptotically flat initial data sets, Com- munications in Analysis and Geometry34(2026), no. 2, 505–570

2026
[41]

35, American Mathematical Soc., 1972

Viktor Andreevich Pogorelov,Extrinsic geometry of convex surfaces, vol. 35, American Mathematical Soc., 1972

1972
[42]

2, 215–220

Antonio Ros,Compact hypersurfaces with constant scalar curvature and a congruence theorem, Journal of Differential Geometry27(1988), no. 2, 215–220

1988
[43]

Richard Schoen and Shing-Tung Yau,Positive scalar curvature and minimal hypersurface singularities, arXiv preprint arXiv:1704.05490 (2017)

2017 arXiv
[44]

22, 7038–7050

Yuguang Shi,The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curva- ture, International Mathematics Research Notices2016(2016), no. 22, 7038–7050

2016
[45]

Yuguang Shi, Jiacheng Sun, Gang Tian, and Dongyi Wei,Uniqueness of the mean field equation and rigidity of hawking mass, Calculus of Variations and Partial Differential Equations58(2019), 1–16

2019
[46]

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

2002
[47]

,Rigidity of compact manifolds and positivity of quasi-local mass, Classical and Quantum Gravity 24(2007), no. 9, 2357

2007
[48]

4, 419–441

Yuguang Shi, Guofang Wang, and Jie Wu,On the behavior of quasi-local mass at the infinity along nearly round surfaces, Annals of Global Analysis and Geometry36(2009), no. 4, 419–441

2009
[49]

Paulo A Sousa and Alexandre B Lima,Charged hawking mass and local rigidity of three-manifolds, The Journal of Geometric Analysis33(2023), no. 1, 11

2023
[50]

1, 257–282

Jiacheng Sun,Rigidity of hawking mass for surfaces in three manifolds, Pacific Journal of Mathematics 292(2017), no. 1, 257–282

2017
[51]

Relativity 7(2004), no

László Benő Szabados,Quasi-local energy-momentum and angular momentum in gr, Living Rev. Relativity 7(2004), no. 4

2004
[52]

6, 5837–5853

Guodong Wei,On the minimizers of curvature functionals in asymptotically flat manifolds, The Journal of Geometric Analysis31(2021), no. 6, 5837–5853

2021
[53]

positivity of quasilocal mass

Niall Ó Murchadha, László Benő Szabados, and Paul Tod,Comment on “positivity of quasilocal mass”, Physical review letters92(2004), no. 25, 259001. University of Potsdam, 14476 Potsdam, Germany Email address:alejandro.penuela.diaz@uni-potsdam.de

2004

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

H. Baltazar, Abdênago Barros, and Rondinelle Batista,A local rigidity theorem for minimal two-spheres in charged time-symmetric initial data set, Letters in Mathematical Physics113(2023)

work page 2023

[2] [2]

A Barros, R Batista, and T Cruz,Hawking mass and local rigidity of minimal surfaces in three-manifolds, Communications in Analysis and Geometry25(2017), no. 1, 1–23

work page 2017

[3] [3]

Hubert Bray, Sean Hayward, Marc Mars, and Walter Simon,Generalized inverse mean curvature flows in spacetime, Communications in mathematical physics272(2007), 119–138

work page 2007

[4] [4]

Bray,On the positive mass, penrose, an zas inequalities in general dimension, surveys in geo- metric analysis and relativity, adv

Hubert L. Bray,On the positive mass, penrose, an zas inequalities in general dimension, surveys in geo- metric analysis and relativity, adv. lect. math.(alm), 20, Int. Press, Somerville, MA (2011)

work page 2011

[5] [5]

Bray and Jeffrey L

Hubert L. Bray and Jeffrey L. Jauregui,A geometric theory of zero area singularities in general relativity, Asian Journal of Mathematics17(2013), no. 3, 525 – 560

work page 2013

[6] [6]

1, 175–197

Simon Brendle, Fernando C Marques, and Andre Neves,Deformations of the hemisphere that increase scalar curvature, Inventiones mathematicae185(2011), no. 1, 175–197

work page 2011

[7] [7]

J David Brown and James W York Jr,Quasilocal energy and conserved charges derived from the gravita- tional action, Physical Review D47(1993), no. 4, 1407

work page 1993

[8] [8]

Armando J. Cabrera Pacheco, Carla Cederbaum, Penelope Gehring, and Alejandro Peñuela Diaz,Con- structing electrically charged riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass, Journal of Geometry and Physics185(2023), 104746

work page 2023