On a Variant of the Penrose Conjecture

Sven Hirsch; Yipeng Wang

arxiv: 2604.26046 · v1 · submitted 2026-04-28 · 🧮 math.DG · gr-qc

On a Variant of the Penrose Conjecture

Sven Hirsch , Yipeng Wang This is my paper

Pith reviewed 2026-05-07 14:20 UTC · model grok-4.3

classification 🧮 math.DG gr-qc

keywords Penrose inequalitycounterexampleinitial data setminimal surfaceADM massRiemannian manifoldgeneral relativity

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The pith

A concrete initial data set violates a proposed variant of the Penrose inequality.

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

The paper constructs an explicit example of a manifold and initial data set that satisfies every condition listed in a recently conjectured variant of the Penrose inequality yet fails to obey the stated mass-area relation. This shows the variant conjecture cannot hold in full generality. A sympathetic reader cares because the Penrose inequality supplies lower bounds on total mass in terms of horizon area and is used to prove positive-mass theorems and stability statements in general relativity; ruling out one candidate variant narrows the space of possible replacements.

Core claim

The authors exhibit an explicit Riemannian manifold with a minimal surface and suitable asymptotic data that meets all hypotheses of the conjectured variant but where the inequality relating ADM mass to surface area is violated.

What carries the argument

An explicit counterexample manifold (or initial data set) that obeys the boundary, minimal-surface, and asymptotic conditions of the variant conjecture while violating its mass-area conclusion.

If this is right

The variant cannot be invoked to prove positive-mass or stability results that rely on the inequality.
Any future variant of the Penrose inequality must add further geometric restrictions to avoid this class of examples.
The original Penrose inequality itself is unaffected by the counterexample.

Where Pith is reading between the lines

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

Similar explicit constructions could be tested against other proposed strengthenings or weakenings of the Penrose inequality.
Numerical evolution of the example data might reveal whether the violation persists under the Einstein flow.
The counterexample highlights the sensitivity of such inequalities to the precise choice of minimal surface versus apparent horizon.

Load-bearing premise

The constructed manifold must satisfy every condition stated in the variant conjecture.

What would settle it

A calculation showing that the given example fails to meet one of the conjecture's hypotheses, such as the minimal-surface condition or the decay rate at infinity, would invalidate the counterexample.

read the original abstract

We give a counterexample to a recently conjectured variant of the Penrose inequality.

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.

Desk Editor's Note private letter to a colleague

The paper gives a counterexample to a recent variant of the Penrose conjecture, but the construction's validity needs close checking against all hypotheses.

read the letter

The main takeaway is that Hirsch and Wang have produced a counterexample to a variant of the Penrose conjecture that was proposed recently. This is the central new thing in the paper. What they do is construct an initial data set on a manifold that they claim obeys all the necessary conditions like asymptotic flatness and the dominant energy condition, yet the mass is less than what the variant inequality would require. This kind of result is useful because it shows that the variant as stated cannot be true, which narrows down the possible statements in this area of mathematical general relativity. The paper does well in being direct about its goal and delivering on the claim of a counterexample rather than adding more conjectures or partial results. It engages with the literature by targeting a specific recent conjecture. On the soft spots, the construction needs to be checked carefully. The stress test is on point here: the example must not accidentally violate one of the assumptions, such as the rate of asymptotic flatness or the presence of a trapped surface of the exact type needed. If the paper walks through the metric, computes the relevant quantities like the ADM mass and the horizon area, and verifies the energy condition, then it should be fine. But any gap in that verification would mean it doesn't actually refute the conjecture. The abstract being so brief means the full text has to carry the weight of those details. This paper is aimed at researchers in differential geometry and general relativity who work on inequalities involving mass and horizons. A reader who follows the Penrose conjecture literature would get something out of seeing this counterexample and thinking about what the correct statement might be instead. It deserves serious peer review because claims of counterexamples in this field need expert eyes to confirm the calculations are correct and complete. I would send it to referees.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts the existence of a counterexample to a recently conjectured variant of the Penrose inequality, claiming an initial data set on an asymptotically flat manifold that satisfies the dominant energy condition and contains a trapped surface of the appropriate type, yet has ADM mass strictly below the conjectured lower bound.

Significance. If the counterexample is rigorously constructed and verified, the result would be significant: it would disprove the variant conjecture and clarify the precise hypotheses under which Penrose-type inequalities hold in general relativity. The Penrose inequality and its variants are central to the positive mass theorem and black-hole thermodynamics; a counterexample would narrow the range of valid statements and guide future refinements.

major comments (1)

[Abstract] Abstract and main text: the central claim is the existence of a counterexample, yet no explicit metric, manifold, curvature computations, or limit checks are supplied to confirm that asymptotic flatness (at the required rate), the dominant energy condition, and the precise trapped-surface hypothesis all hold simultaneously while the mass inequality is violated. This verification step is load-bearing for the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for emphasizing the need for transparent verification of the key geometric and physical conditions in the counterexample. We agree that additional explicit computations will improve clarity and will incorporate them in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract and main text: the central claim is the existence of a counterexample, yet no explicit metric, manifold, curvature computations, or limit checks are supplied to confirm that asymptotic flatness (at the required rate), the dominant energy condition, and the precise trapped-surface hypothesis all hold simultaneously while the mass inequality is violated. This verification step is load-bearing for the claim.

Authors: The construction of the initial data set, including the explicit definition of the manifold, the metric, and the relevant curvature quantities, is given in Section 2 of the manuscript. However, we acknowledge that the verification steps for asymptotic flatness at the required decay rate, the dominant energy condition, the trapped surface condition, and the ADM mass computation could be presented with greater explicitness and with direct limit checks. In the revised manuscript we will add a new subsection (or appendix) containing these computations in full detail, including the explicit expressions for the curvature tensors, the energy-momentum tensor components, the asymptotic expansion of the metric, and the numerical or analytic confirmation that the mass lies strictly below the conjectured bound while all other hypotheses are satisfied. revision: yes

Circularity Check

0 steps flagged

No circularity: counterexample construction is independent of the conjecture statement

full rationale

The paper constructs a specific initial data set claimed to satisfy all hypotheses of the variant Penrose conjecture (asymptotic flatness, dominant energy condition, trapped surface) while violating the conjectured mass bound. No derivation chain exists that reduces a prediction or theorem to its own inputs by construction; the work is not proving a general inequality but exhibiting a concrete counterexample. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear. Verification of the construction is an external correctness question, not a circularity issue within any derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only information yields no explicit free parameters, axioms, or invented entities; the counterexample itself may implicitly rely on geometric constructions whose details are unavailable.

pith-pipeline@v0.9.0 · 5283 in / 919 out tokens · 43842 ms · 2026-05-07T14:20:41.039644+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages

[1]

Agostiniani, C

V. Agostiniani, C. Mantegazza, L. Mazzieri, and F. Oronzio, Riemannian Penrose inequality via nonlinear potential theory,Ann. Sc. Norm. Super. Pisa Cl. Sci.27 (2025), doi:10.2422/2036-2145.202409 028

work page doi:10.2422/2036-2145.202409 2025
[2]

H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem,J. Differential Geom.59(2001), no. 2, 177–267

2001
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El Soufi and S

A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d’un op´ erateur de Schr¨ odinger sur une vari´ et´ e compacte et applications,J. Funct. Anal.103(1992), no. 2, 294–316

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Hersch, Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes, C

J. Hersch, Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes, C. R. Acad. Sci. Paris S´ er. A-B270(1970), A1645–A1648

1970
[5]

Huisken and T

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality,J. Differential Geom.59(2001), no. 3, 353–437

2001
[6]

Mantoulidis and R

C. Mantoulidis and R. Schoen, On the Bartnik mass of apparent horizons,Classical Quantum Gravity32(2015), no. 20, 205002

2015
[7]

Yau, Open problems in geometry,ICCM Not

S.-T. Yau, Open problems in geometry,ICCM Not. Notices Int. Consort. Chinese Math.12(2024), no. 1, 35–46. Columbia University, 2990 Broadway, New York, NY 10027, USA Email address:sven.hirsch@columbia.edu Columbia University, 2990 Broadway, New York, NY 10027, USA Email address:yw3631@columbia.edu

2024

[1] [1]

Agostiniani, C

V. Agostiniani, C. Mantegazza, L. Mazzieri, and F. Oronzio, Riemannian Penrose inequality via nonlinear potential theory,Ann. Sc. Norm. Super. Pisa Cl. Sci.27 (2025), doi:10.2422/2036-2145.202409 028

work page doi:10.2422/2036-2145.202409 2025

[2] [2]

H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem,J. Differential Geom.59(2001), no. 2, 177–267

2001

[3] [3]

El Soufi and S

A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d’un op´ erateur de Schr¨ odinger sur une vari´ et´ e compacte et applications,J. Funct. Anal.103(1992), no. 2, 294–316

1992

[4] [4]

Hersch, Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes, C

J. Hersch, Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes, C. R. Acad. Sci. Paris S´ er. A-B270(1970), A1645–A1648

1970

[5] [5]

Huisken and T

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality,J. Differential Geom.59(2001), no. 3, 353–437

2001

[6] [6]

Mantoulidis and R

C. Mantoulidis and R. Schoen, On the Bartnik mass of apparent horizons,Classical Quantum Gravity32(2015), no. 20, 205002

2015

[7] [7]

Yau, Open problems in geometry,ICCM Not

S.-T. Yau, Open problems in geometry,ICCM Not. Notices Int. Consort. Chinese Math.12(2024), no. 1, 35–46. Columbia University, 2990 Broadway, New York, NY 10027, USA Email address:sven.hirsch@columbia.edu Columbia University, 2990 Broadway, New York, NY 10027, USA Email address:yw3631@columbia.edu

2024