Area-charge inequality and local rigidity in charged initial data sets

Abra\~ao Mendes

REVIEW 2 major objections 4 minor 1 cited by

Area-charge inequality and local rigidity in charged initial data sets

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Equality in the area-charge inequality for a spherical marginally outer trapped surface forces an outer neighborhood to be a Riemannian product with constant electric and magnetic fields and zero cosmological constant.

desk verdict Genuine rigidity results for equality in the area-charge inequality; the proof is checkable but leans on two unstated external lemmas, one of which needs a sign check. read the letter →

arxiv 2505.20060 v1 pith:APMU7LJT submitted 2025-05-26 math.DG gr-qc

Abraão Mendes This is my paper · ORCID

classification math.DGgr-qc MSC 53C2153C8083C2283C57

keywords area-chargeinequalitymarginallyoutertrappedsurfaceEinstein-MaxwellinitialdatarigiditytheoremRiemannianproductchargeddominantenergyconditionMOTSstabilityoperatorcosmologicalconstant

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 asks what happens when the area-charge inequality $A \geq 4\pi(Q_{\rm E}^2+Q_{\rm M}^2)$, which holds for stable minimal spheres and for stable spherical marginally outer trapped surfaces (MOTS, the quasilocal notion of a black hole horizon), is saturated. It proves that equality is rigid: the surface must sit in a neighborhood isometric to an interval times a round two-sphere, the electric and magnetic fields must be constant and normal to the foliation, the energy and momentum densities must take the constant-field values $\mu=a^2+b^2$, $J=0$, and the cosmological constant must vanish. Two theorems cover the cases: a time-symmetric one for area-minimizing two-spheres, and a general one for weakly outermost spherical MOTS in initial data sets satisfying the charged dominant energy condition with two-convex extrinsic curvature. The explicit product model of Section 4 shows the rigid configuration is realized, so the characterization is sharp.

What carries the argument

The load-bearing mechanism is the MOTS stability operator, the linearization of the null expansion along normal variations, together with its principal eigenvalue. Proposition 3.1 combines the stability inequality with the charged dominant energy condition and Cauchy-Schwarz to show that saturation makes the principal eigenvalue zero and forces the surface data to be exactly that of a round sphere with constant field normal components. The zero eigenvalue then activates two foliation lemmas from the literature: one produces an outer foliation by constant null mean curvature surfaces, and the other converts an integrated first-variation inequality into $\theta(t)\le 0$, whose equality forces every leaf to be a MOTS of constant area. Chasing these equalities back through the variation formulas yields $\chi_+=\chi_-=0$, $E=a\nu_t$, $B=b\nu_t$, and ultimately $K=f\,dt^2$ with $J=0$.

What would settle it

The central claim would be falsified by an explicit weakly outermost spherical MOTS with $A=4\pi(Q_{\rm E}^2+Q_{\rm M}^2)$, divergence-free $E$ and $B$, two-convex $K$, and the charged dominant energy condition, whose outer neighborhood either fails to have $K=f\,dt^2$ or has a tangential electromagnetic component; such an example could be sought by direct construction of initial data with a nontrivial shear term.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that saturation of the area-charge inequality is an infinitesimal rigidity phenomenon. Proposition 3.1 shows that for a stable spherical MOTS, equality forces the principal eigenvalue of the stability operator to vanish, the null second fundamental form $\chi_+$ to vanish, the normal components $\langle E,\nu\rangle$ and $\langle B,\nu\rangle$ to be constants $a$ and $b$, and the Gaussian curvature of the surface to equal $a^2+b^2$. Theorem 1.2 then upgrades this to a full neighborhood statement: an outer neighborhood is isometric to $([0,\delta)\times\Sigma, dt^2+g_0)$ with $g_0$ a round metric of curvature $a^2+b^2$, the fields are $E=a\nu_t$, $B=b\nu_t$ for constants $a,b$, the second fundamental form has the form $K=f\,dt^2$, the energy density is $\mu=a^2+b^2$, the momentum density is $J=0$, and $\Lambda=0$. Theorem 1.1 is the time-symmetric analogue, with $K=0$ and the same product rigidity. The quasilocal infinitesimal rigidity is what carries the local conclusion.

Load-bearing premise

In the proof of Theorem 1.2, the argument depends on two cited lemmas, one producing a foliation by constant null mean curvature surfaces once a stability eigenvalue vanishes and one turning a differential inequality into the sign condition $\theta(t)\le 0$, and the paper does not restate these lemmas or check their hypotheses for the surfaces it considers; if either lemma does not apply, the conclusion that every leaf is marginally outer trapped can fail.

Editorial extensions

If this is right

For any spherical horizon saturating the bound, the geometry in an outer neighborhood is completely fixed up to the constants $a,b$ and the interval length; no other near-horizon geometry can attain equality under the charged dominant energy condition.
Saturation forces $\Lambda=0$, so the area-charge inequality cannot be sharp in the presence of a positive cosmological constant.
The electric and magnetic fields must be normal to the foliation and constant; a saturated horizon with tangential electromagnetic fields cannot exist.
In the time-symmetric setting, the saturated surface is a totally geodesic round sphere with ambient scalar curvature $R=2(a^2+b^2)$ on the surface.
The model of Section 4 realizes equality, so the rigidity results are sharp rather than vacuous.

Reading between the lines

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

One could try to extend the rigidity to non-spherical topologies: the proof invokes Gauss-Bonnet with $\chi=2$, so a version for higher genus would presumably carry a topological deficit term and may fail or need modification.
A quantitative stability estimate, bounding a geometric deviation from the product by $A - 4\pi(Q_{\rm E}^2+Q_{\rm M}^2)$, would follow if the two foliation lemmas can be made effective; the paper does not address such an estimate.
The role of two-convexity appears only through $\operatorname{tr}_{\Sigma}K\ge 0$ when comparing mean curvature with null expansion, so a weakly outermost MOTS theorem without two-convexity is a plausible target.
Saturation may serve as a quasi-local characterization of the constant-field product near-horizon geometry: any initial data whose horizon saturates the bound is locally indistinguishable from that model.

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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

Here's my take. The paper proves a real local rigidity statement: for spherical MOTS, saturation of A >= 4π(Q_E^2 + Q_M^2) forces an outer neighborhood to split as a product with constant normal electromagnetic fields, and the time-symmetric case gives the same product structure. The chain of inequalities in Proposition 3.1 is transparent, and I found no algebraic error in the equality analysis. The model in Section 4 is a useful sanity check. Credit where due: the inequality itself is prior work (Gibbons; Dain-Jaramillo-Reiris), but the rigidity analysis goes beyond those papers and is the genuinely new part.

Two soft spots. One is minor: the paper doesn't state how it differs from the closely titled 2025 papers [15] and [20]. The author should say that explicitly, because a referee will want to know the increment. The other is more load-bearing: the step "Using Lemma 3.2 in [21]" is applied to the differential inequality (3.10), which has the form θ'η − θζ ≤ ∫ θ ξ, with ζ involving ∫ τ. Two-convexity gives tr_Σ K ≥ 0, so H ≤ θ, but it does not control the sign of τ = tr_g K, which is what enters ζ. If the lemma requires ζ ≥ 0 or some bound relating ζ to η and ξ, that condition is not verified in the text. Likewise, [13, Lemma 2.3] is cited for the constant-null-mean-curvature foliation without stating its hypotheses. These are not necessarily fatal, but they are exactly the points a referee must check.

Overall, the central argument is coherent and the equality analysis is careful. If the author reproduces those two lemmas and verifies their hypotheses, I would expect the main theorems to stand. This is a solid contribution to MOTS rigidity that deserves a normal peer-review round, not a desk rejection. For a colleague working on geometric inequalities in GR, it is worth a read; for others, it's specialized. My recommendation: accept for peer review and ask for revision to fill the lemma gaps.

Referee Report

2 major / 4 minor

Summary. The paper proves area-charge inequalities A >= 4π(Q_E^2 + Q_M^2) for spherical minimal surfaces (Theorem 1.1) and for spherical weakly outermost MOTS (Theorem 1.2) under charged dominant energy conditions, and characterizes the equality case: in a neighborhood of the surface the metric splits as dt^2 + g0, the electric and magnetic fields are constant multiples of the normal, K = f dt^2, μ = a^2 + b^2, J = 0, and Λ = 0. The proofs combine a quasi-local infinitesimal rigidity proposition (Proposition 3.1) with foliation arguments and a comparison lemma cited from prior work.

Significance. If fully substantiated, the rigidity theorems are natural and sharp; the dyonic Bertotti-Robinson model in Section 4 demonstrates that the inequalities and rigidity are saturated. The paper's main contribution is the equality-case analysis, which is carefully traced through a chain of inequalities, and Proposition 3.1 is a clean quasi-local statement. The area-charge inequality itself is derived in a self-contained way from standard stability facts, with no fitted parameters or definitional circularity. The principal caveat is the reliance on two external lemmas whose hypotheses are not stated or verified; this is addressable and does not undermine the plausibility of the results, but it is load-bearing for the rigidity conclusions.

major comments (2)

[§3, Theorem 1.2 proof, Eq. (3.10)] The step 'Using Lemma 3.2 in [21], we conclude that θ(t) ≤ 0' is load-bearing and is not justified. Lemma 3.2 is not stated, and the differential inequality (3.10), θ′η − θζ ≤ ∫₀ᵗ θξ, has coefficient ζ(t) = A(t)/(4π) ∫_{Σ_t} τ whose sign and size are not controlled. Two-convexity of K gives tr_{Σ_t}K ≥ 0 and hence H ≤ θ, but τ = tr_{Σ_t}K + K(ν,ν) can be negative; if Lemma 3.2 requires ζ ≤ 0 or a relation among η, ζ, ξ, that hypothesis is not verified. The later conclusions that θ(t)=0, that every leaf is a MOTS, that A(t)=A(0), and hence the product rigidity all depend on this step. The same unverified application occurs for H(t) in the proof of Theorem 1.1.
[§3, Theorem 1.2 proof, first paragraph after Proposition 3.1] The existence of the foliation by constant null mean curvature surfaces is delegated to [13, Lemma 2.3], whose content and hypotheses are not given. The proof only establishes λ1(L)=0 via Proposition 3.1; it does not check that a weakly outermost spherical MOTS in a charged initial data set with two-convex K satisfies the additional conditions of [13, Lemma 2.3] (for instance, any strict stability or nondegeneracy assumption). If the lemma does not apply, the family {Σ_t} on which equations (3.7)–(3.15) are integrated does not exist, and the proof of Theorem 1.2 collapses.

minor comments (4)

[Section 2] The same symbol L is used for the MOTS stability operator and for its symmetrized version, so λ1(L) and λ1(L) are hard to distinguish in Proposition 3.1; a different notation such as L_sym would improve clarity.
[Proof of Theorem 1.2] The phrase 'a e b são constantes' appears as 'a e b are constant'; this should read 'a and b are constant' for English prose.
[Proof of Theorem 1.2, after Eq. (3.11)] The sentence 'Equalities in (3.11) give tr_{Σ_t}K = H_{Σ_t} = 0' is compressed; it should explicitly use the already-established fact A(t)=A(0) to convert the integral inequality into pointwise vanishing of H.
[Eq. (3.9)] The Cauchy-Schwarz step in (3.9) is correct but terse; stating it as two separate applications, one for E and one for B, would make the inequality easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the area-charge inequality is derived, and the two cited foliation/sign lemmas are independent prior results, not inputs that encode the rigidity conclusion.

full rationale

No step in the derivation reduces to its own input by construction. Proposition 3.1 proves the area-charge inequality from the charged dominant energy condition, stability, Gauss-Bonnet, and Cauchy-Schwarz, rather than assuming it; the equality analysis then derives roundness and constancy of the field normal components. The proof of Theorem 1.2 invokes [13, Lemma 2.3] only to obtain a constant-null-mean-curvature foliation once λ1(L)=0, and [21, Lemma 3.2] only as a general differential-inequality lemma applied to inequality (3.10), which is derived from the energy condition and Gauss-Bonnet. The self-citation [21] is load-bearing in the sign step θ≤0, but it is a previously published lemma with its own proof; it does not assume the product-isometry conclusion or the area-charge equality. Similarly, Theorem 1.1 uses the same lemma for H(t) after deriving H′η≤∫Hξds. There are no fitted parameters, no quantity is renamed as a prediction, and no uniqueness theorem from the author's prior work is used to forbid alternatives. The only weakness is that the hypotheses of the two cited lemmas are not reproduced or checked case-by-case; this is a proof-completeness and correctness concern, not circularity. Hence score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The derivation is built on standard differential geometry and on two cited external lemmas about MOTS foliations and sign control, which are load-bearing but not proved in the paper.

assumptions (6)

standard math Gauss-Bonnet theorem
Used to bound the integral of the spherical Gaussian curvature by 4π in Propositions 3.1 and 3.2.
standard math Principal eigenvalue comparison for the stability operator
Invoked at the start of Proposition 3.1 to connect λ1 of the symmetrized operator to the stability inequality; cited to [4,12,16].
domain assumption Existence of a constant null mean curvature foliation near a stable MOTS
Theorem 1.2 uses [13, Lemma 2.3] to obtain the outer neighborhood foliation once λ1(L) = 0.
domain assumption Sign lemma for the area growth inequality
The proof uses [21, Lemma 3.2] to conclude θ(t) ≤ 0 in Theorem 1.2 and H(t) ≤ 0 or H(t) ≥ 0 in Theorem 1.1.
domain assumption Classical CMC foliation near a minimal surface
Theorem 1.1 invokes [2,5,23,24] to construct a constant mean curvature foliation once the Jacobi operator reduces to -Δ.
domain assumption Charged dominant energy condition (1.3) or (1.4)
This is a hypothesis of Theorems 1.1 and 1.2, derived in Section 2 from the dominant energy condition for the matter tensor.

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Cite this review

Pith. "Pith review of Area-charge inequality and local rigidity in charged initial data sets." pith.science (2026). https://pith.science/paper/APMU7LJT

@misc{pith2026250520060,
  author       = {Pith},
  title        = {Pith review of: Area-charge inequality and local rigidity in charged initial data sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/APMU7LJT}},
  note         = {Machine review of arXiv:2505.20060}
}

abstract

This paper investigates the geometric consequences of equality in area-charge inequalities for spherical minimal surfaces and, more generally, for marginally outer trapped surfaces (MOTS), within the framework of the Einstein-Maxwell equations. We show that, under appropriate energy and curvature conditions, saturation of the inequality $\mathcal{A} \geq 4\pi(\mathcal{Q}_{\rm E}^2 + \mathcal{Q}_{\rm M}^2)$ imposes a rigid geometric structure in a neighborhood of the surface. In particular, the electric and magnetic fields must be normal to the foliation, and the local geometry is isometric to a Riemannian product. We establish two main rigidity theorems: one in the time-symmetric case and another for initial data sets that are not necessarily time-symmetric. In both cases, equality in the area-charge bound leads to a precise characterization of the intrinsic and extrinsic geometry of the initial data near the critical surface.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Area-charge inequalities and rigidity of time-symmetric initial data sets
gr-qc 2025-07 conditional novelty 6.0 of 10

In charged Einstein-Maxwell initial data sets, a boundary surface must have area at least a sharp function of its electric charge and the cosmological constant, with equality only for product geometries.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages · cited by 1 Pith paper

[15]

Galloway and Abraão Mendes

Gregory J. Galloway and Abraão Mendes. Some rigidity results for charged initial data sets. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 256:9, 2025. Id/No 113780

work page 2025
[20]

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(2):15, 2025. Id/No 41

work page 2025
[21]

Rigidity of marginally outer trapped (hyper)surfaces with negative σ- constant

Abraão Mendes. Rigidity of marginally outer trapped (hyper)surfaces with negative σ- constant. Trans. Am. Math. Soc., 372(8):5851–5868, 2019

work page 2019
[1]

Geometric inequalities for quasi-local masses

Aghil Alaee, Marcus Khuri, and Shing-Tung Yau. Geometric inequalities for quasi-local masses. Commun. Math. Phys., 378(1):467–505, 2020

work page 2020
[2]

Galloway

Lars Andersson, Mingliang Cai, and Gregory J. Galloway. Rigidity and positivity of mass for asymptotically hyperbolic manifolds.Ann. Henri Poincaré, 9(1):1–33, 2008

work page 2008
[3]

Local existence of dynamical and trapping horizons

Lars Andersson, Marc Mars, and Walter Simon. Local existence of dynamical and trapping horizons. Phys. Rev. Lett., 95:111102, Sep 2005

work page 2005
[4]

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes.Adv

Lars Andersson, Marc Mars, and Walter Simon. Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes.Adv. Theor. Math. Phys., 12(4):853–888, 2008

work page 2008
[5]

Rigidity of area-minimizing two-spheres in three-manifolds

Hubert Bray, Simon Brendle, and Andre Neves. Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom., 18(4):821–830, 2010

work page 2010

Show all 25 references

[6]

Bryden and Marcus A

Edward T. Bryden and Marcus A. Khuri. The area-angular momentum-charge inequality for black holes with positive cosmological constant.Classical Quantum Gravity, 34(12):24, 2017. Id/No 125017

2017
[7]

Vitor Cardoso, Óscar J. C. Dias, and José P. S. Lemos. Nariai, Bertotti-Robinson, and anti- Nariai solutions in higher dimensions.Phys. Rev. D, 70:024002, Jul 2004

2004
[8]

Min-max minimal surfaces, horizons and electrostatic systems.J

Tiarlos Cruz, Vanderson Lima, and Alexandre de Sousa. Min-max minimal surfaces, horizons and electrostatic systems.J. Differ. Geom., 128(2):583–637, 2024

2024
[9]

Area-charge inequality for black holes

Sergio Dain, José Luis Jaramillo, and Martín Reiris. Area-charge inequality for black holes. Classical Quantum Gravity, 29(3):15, 2012. Id/No 035013

2012
[10]

Rigidity results for free boundary hypersurfaces in initial data sets with boundary

Deivid de Almeida and Abraão Mendes. Rigidity results for free boundary hypersurfaces in initial data sets with boundary. Preprint, arXiv:2502.09433 [math.DG] (2025), 2025

2025 arXiv
[11]

Galloway, and Abraão Mendes

Michael Eichmair, Gregory J. Galloway, and Abraão Mendes. Initial data rigidity results. Commun. Math. Phys., 386(1):253–268, 2021

2021
[12]

Galloway

Gregory J. Galloway. Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity. InSurveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday, pages 221–239. Somerville, MA: International Press; Beijin...

2011
[13]

Galloway

Gregory J. Galloway. Rigidity of outermost MOTS: the initial data version.Gen. Relativ. Gravitation, 50(3):7, 2018. Id/No 32

2018
[14]

Galloway and Abraão Mendes

Gregory J. Galloway and Abraão Mendes. Some rigidity results for compact initial data sets. Trans. Am. Math. Soc., 377(3):1989–2007, 2024

1989
[16]

Galloway and Richard Schoen

Gregory J. Galloway and Richard Schoen. A generalization of Hawking’s black hole topology theorem to higher dimensions.Commun. Math. Phys., 266(2):571–576, 2006

2006
[17]

G. W. Gibbons. Some comments on gravitational entropy and the inverse mean curvature flow. Classical Quantum Gravity, 16(6):1677–1687, 1999

1999
[18]

Proof of the Riemannian Penrose inequality with charge for multiple black holes.J

Marcus Khuri, Gilbert Weinstein, and Sumio Yamada. Proof of the Riemannian Penrose inequality with charge for multiple black holes.J. Differ. Geom., 106(3):451–498, 2017

2017
[19]

Marcus A. Khuri. Inequalities between size and charge for bodies and the existence of black holes due to concentration of charge.J. Math. Phys., 56(11):112503, 9, 2015

2015
[22]

Rigidity of free boundary MOTS.Nonlinear Anal., Theory Methods Appl., Ser

Abraão Mendes. Rigidity of free boundary MOTS.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 220:15, 2022. Id/No 112841. AREA-CHARGE INEQUALITY AND LOCAL RIGIDITY 15

2022
[23]

Splitting of 3-manifolds and rigidity of area-minimising surfaces

Mario Micallef and Vlad Moraru. Splitting of 3-manifolds and rigidity of area-minimising surfaces. Proc. Am. Math. Soc., 143(7):2865–2872, 2015

2015
[24]

Rigidity of area-minimizing hyperbolic surfaces in three-manifolds.J

Ivaldo Nunes. Rigidity of area-minimizing hyperbolic surfaces in three-manifolds.J. Geom. Anal., 23(3):1290–1302, 2013

2013
[25]

On a Penrose inequality with charge.Commun

Gilbert Weinstein and Sumio Yamada. On a Penrose inequality with charge.Commun. Math. Phys., 257(3):703–723, 2005. Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, Brazil. Email address: abraao.mendes@im.ufal.br

2005

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [15]

Galloway and Abraão Mendes

Gregory J. Galloway and Abraão Mendes. Some rigidity results for charged initial data sets. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 256:9, 2025. Id/No 113780

work page 2025

[2] [20]

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(2):15, 2025. Id/No 41

work page 2025

[3] [21]

Rigidity of marginally outer trapped (hyper)surfaces with negative σ- constant

Abraão Mendes. Rigidity of marginally outer trapped (hyper)surfaces with negative σ- constant. Trans. Am. Math. Soc., 372(8):5851–5868, 2019

work page 2019

[4] [1]

Geometric inequalities for quasi-local masses

Aghil Alaee, Marcus Khuri, and Shing-Tung Yau. Geometric inequalities for quasi-local masses. Commun. Math. Phys., 378(1):467–505, 2020

work page 2020

[5] [2]

Galloway

Lars Andersson, Mingliang Cai, and Gregory J. Galloway. Rigidity and positivity of mass for asymptotically hyperbolic manifolds.Ann. Henri Poincaré, 9(1):1–33, 2008

work page 2008

[6] [3]

Local existence of dynamical and trapping horizons

Lars Andersson, Marc Mars, and Walter Simon. Local existence of dynamical and trapping horizons. Phys. Rev. Lett., 95:111102, Sep 2005

work page 2005

[7] [4]

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes.Adv

Lars Andersson, Marc Mars, and Walter Simon. Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes.Adv. Theor. Math. Phys., 12(4):853–888, 2008

work page 2008

[8] [5]

Rigidity of area-minimizing two-spheres in three-manifolds

Hubert Bray, Simon Brendle, and Andre Neves. Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom., 18(4):821–830, 2010

work page 2010