Lorentzian coarea inequality

Hikaru Kubota

arxiv: 2605.09101 · v3 · pith:UD3W232Unew · submitted 2026-05-09 · 🧮 math.MG

Lorentzian coarea inequality

Hikaru Kubota This is my paper

Pith reviewed 2026-05-20 23:06 UTC · model grok-4.3

classification 🧮 math.MG

keywords Lorentzian pre-length spacescoarea inequalityHausdorff measurecausal diamondscontrolling mapscovering lemmalocal causal enlargement property

0 comments

The pith

Locally uniformly d-controlling maps that preserve causal diamond diameters establish the coarea inequality for Lorentzian Hausdorff measure.

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

The paper introduces locally uniformly d-controlling maps between Lorentzian pre-length spaces that preserve the diameters of causal diamonds. These maps are used to prove the coarea inequality for the Lorentzian Hausdorff measure introduced by McCann and Sämänn. A covering lemma is also obtained for subsets satisfying the local causal enlargement property, which permits local enlargement of causal diamonds. This provides a version of the coarea formula adapted to the Lorentzian setting in geometric measure theory.

Core claim

Through the notion of locally uniformly d-controlling maps between Lorentzian pre-length spaces which preserve the diameters of causal diamonds, the coarea inequality for Lorentzian Hausdorff measure is established. A covering lemma for subsets in a Lorentzian pre-length space is obtained under the new local assumption of the local causal enlargement property, which enables enlargement of causal diamonds.

What carries the argument

Locally uniformly d-controlling map preserving diameters of causal diamonds, which carries the argument by allowing control of the Hausdorff measure.

Load-bearing premise

The local causal enlargement property on subsets of the Lorentzian pre-length space, which is invoked to obtain the covering lemma that supports the coarea inequality.

What would settle it

A Lorentzian pre-length space satisfying all other conditions but lacking the local causal enlargement property, in which the coarea inequality fails for the Hausdorff measure.

read the original abstract

In this article, we introduce the notion of locally uniformly d-controlling map between Lorentzian pre-length spaces which is preserving the diameters of causal diamonds, and through that we establish the coarea inequality for Lorentzian Hausdorff measure which is introduced by McCann and S\"{a}mann. Besides that we get a covering lemma for subsets in a Lorentzian pre-length space with a new local assumption named the local causal enlargement property, which enables us to enlarge causal diamonds.

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 introduces locally uniformly d-controlling maps to derive a coarea inequality for Lorentzian Hausdorff measure, but the result hinges on a new local causal enlargement assumption whose generality is unclear.

read the letter

Hi, the main thing to know is that this paper defines locally uniformly d-controlling maps between Lorentzian pre-length spaces that preserve causal diamond diameters and uses them to prove a coarea inequality for the Lorentzian Hausdorff measure from McCann and Sämann. It also gives a covering lemma that rests on a local causal enlargement property allowing subsets to enlarge causal diamonds locally. That combination looks like the actual new piece, since the abstract flags it as absent from the prior work. The setup is straightforward and builds directly on the existing Lorentzian measure framework without obvious circularity or extra fitted parameters. The definitions seem cleanly stated and the derivation path from the map notion to the inequality is presented as a direct application. The local causal enlargement property is the soft spot. It is introduced as a fresh local assumption on subsets rather than something derived from the controlling map or from standard properties of pre-length spaces. The stress-test note is on target here: if this property does not hold or is hard to verify in the spaces where the McCann-Sämann measure is typically used, then the covering lemma and the inequality become conditional on an extra hypothesis that may restrict the result more than the title suggests. Without concrete checks against common examples, it is difficult to judge how broad the claim really is. This is a specialized technical note aimed at people working in mathematical relativity and causal geometry. A reader already thinking about Hausdorff measures or coarea formulas in Lorentzian settings could pick up the new map definition and try to apply it elsewhere. The thinking is clear and the engagement with the literature is honest, so the paper is worth sending to a serious referee who can check the proofs and test the new assumption on standard examples. I would not cite it myself in the next year, but it is solid enough to deserve review rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The paper introduces locally uniformly d-controlling maps between Lorentzian pre-length spaces that preserve diameters of causal diamonds. Using this notion it derives a coarea inequality for the Lorentzian Hausdorff measure of McCann and Sämann, together with a covering lemma that rests on a new local assumption called the local causal enlargement property (allowing local enlargement of causal diamonds).

Significance. If the local causal enlargement property holds in the spaces where the McCann–Sämann measure is defined, the result would supply a useful coarea-type tool in Lorentzian geometric measure theory. The new controlling-map notion and the covering lemma are the main technical contributions; their value depends on the property being verifiable in standard examples rather than remaining an extra hypothesis.

major comments (1)

[Covering lemma and local causal enlargement property] The covering lemma (stated after the definition of the local causal enlargement property) is the key step supporting the coarea inequality. The lemma invokes the local causal enlargement property as a new local assumption on subsets, yet the manuscript does not verify that this property holds in Lorentzian pre-length spaces admitting the McCann–Sämann measure or in other standard examples. Without such verification or a proof that the property follows from the definitions of Lorentzian pre-length spaces, the coarea inequality remains conditional on an unverified extra hypothesis.

minor comments (2)

[Definition of locally uniformly d-controlling maps] Clarify whether the locally uniformly d-controlling condition is strictly stronger than, or equivalent to, existing notions of controlling maps in the literature on Lorentzian length spaces.
[Introduction and applications] Add a short discussion or example showing that the local causal enlargement property is satisfied (or easily checked) in at least one non-trivial class of spaces to which the coarea inequality is meant to apply.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The main concern is the status of the local causal enlargement property as an additional hypothesis. We address this point directly below and indicate the revisions we will make.

read point-by-point responses

Referee: The covering lemma (stated after the definition of the local causal enlargement property) is the key step supporting the coarea inequality. The lemma invokes the local causal enlargement property as a new local assumption on subsets, yet the manuscript does not verify that this property holds in Lorentzian pre-length spaces admitting the McCann–Sämann measure or in other standard examples. Without such verification or a proof that the property follows from the definitions of Lorentzian pre-length spaces, the coarea inequality remains conditional on an unverified extra hypothesis.

Authors: We agree that the local causal enlargement property is introduced as a new local assumption and is not derived from the axioms of Lorentzian pre-length spaces. It is a sufficient condition that enables the covering lemma and, consequently, the coarea inequality. We do not claim it holds universally; rather, the results are stated under this hypothesis. In the revised version we will add a new subsection that verifies the property in standard examples where the McCann–Sämann measure is typically defined, including Minkowski space, smooth Lorentzian manifolds with the usual causal structure, and certain Lorentzian length spaces satisfying local compactness and local causal convexity. We will also include a brief discussion of the geometric meaning of the property and note that it is satisfied whenever causal diamonds can be enlarged by a controlled factor in a neighborhood, which holds in all currently studied settings for the McCann–Sämann measure. This addresses the conditional nature of the result while keeping the statement accurate. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from newly introduced definitions and assumptions

full rationale

The paper defines locally uniformly d-controlling maps that preserve causal diamond diameters and invokes the local causal enlargement property as a new local assumption to obtain a covering lemma, from which the coarea inequality for Lorentzian Hausdorff measure follows. No quoted equations, definitions, or self-citations reduce the target inequality to a fitted input, self-referential quantity, or prior result by the same authors. The central claim is presented as a derivation from these constructions rather than a renaming or tautological restatement of inputs, making the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence and properties of Lorentzian pre-length spaces together with the newly introduced controlling-map condition and the local causal enlargement property; no numerical free parameters appear.

axioms (1)

domain assumption Lorentzian pre-length spaces are equipped with a causal preorder and a metric compatible with the causal structure.
Invoked throughout the definitions of causal diamonds and the Hausdorff measure.

invented entities (1)

locally uniformly d-controlling map no independent evidence
purpose: Map that preserves diameters of causal diamonds to enable the coarea inequality.
Newly defined notion introduced in the paper to bridge the classical coarea formula to the Lorentzian setting.

pith-pipeline@v0.9.0 · 5584 in / 1327 out tokens · 48952 ms · 2026-05-20T23:06:47.636960+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We introduce the notion of locally uniformly d-controlling map between Lorentzian pre-length spaces which is preserving the diameters of causal diamonds, and through that we establish the coarea inequality for Lorentzian Hausdorff measure

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

41 extracted references · 41 canonical work pages

[1]

S. B. Alexander, M. Graf, M. Kunzinger, and C. S¨ amann.Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom.31(2023), no. 6, 1469–1528; MR4785565

work page 2023
[2]

J., Ohanyan, A., Rott, F., and S ¨amann, C

T. Beran, M. Braun, M. Calisti, N. Gigli, R. J. McCann, A. Ohanyan, F. Rott, C. S¨ amann. A nonlinear d’Alembert comparison theorem and causal differential calculus on metric measure spacetimes. arXiv:2408.15968

work page arXiv
[3]

Beran, M

T. Beran, M. Kunzinger, A. Ohanyan, and F. Rott.The equivalence of smooth and synthetic notions of timelike sectional curvature bounds. Proc. Amer. Math. Soc.153(2025), no. 2, 783–797; MR4852799

work page 2025
[4]

Beran, M

T. Beran, M. Kunzinger, and F. Rott.On curvature bounds in Lorentzian length spaces. J. Lond. Math. Soc. (2)110(2024), no. 2, Paper No. e12971, 41 pp.; MR4781260

work page 2024
[5]

Beran, A

T. Beran, A. Ohanyan, F. Rott, and D. A. Solis.The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett. Math. Phys.113(2023), no. 2, Paper No. 48, 47 pp.; MR4579262

work page 2023
[6]

Beran and C

T. Beran and C. S¨ amann.Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds. J. Lond. Math. Soc. (2)107(2023), no. 5, 1823–1880; MR4585303

work page 2023
[7]

Braun.R´ enyi’s entropy on Lorentzian spaces

M. Braun.R´ enyi’s entropy on Lorentzian spaces. Timelike curvature-dimension conditions. J. Math. Pures Appl177(2023), Pages 46-128, ISSN 0021-7824

work page 2023
[8]

Braun and M.-S

M. Braun and M.-S. Candal.Comparison theory for Lipschitz spacetimes. arXiv:2603.24195

work page arXiv
[9]

Braun, N

M. Braun, N. Gigli, R. J. McCann, A. Ohanyan, and C. S¨ amann.A Lorentzian splitting theorem for continuously differentiable metrics and weights. arXiv:2507.06836

work page arXiv
[10]

Braun and S

M. Braun and S. Ohta.Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes. Trans. Amer. Math. Soc.377(2024), no. 5, 3529–3576; MR4744787. 22

work page 2024
[11]

Braun and C

M. Braun and C. S¨ amann.Gromov’s reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry. arXiv:2506.10852

work page arXiv
[12]

Cavalletti, D

F. Cavalletti, D. Manini and A. Mondino.Optimal transport on null hypersurfaces and the null energy condition. Comm. Math. Phys.406(2025), no. 9, Paper No. 212, 62 pp.; MR4940202

work page 2025
[13]

Cavalletti and A

F. Cavalletti and A. Mondino.Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Camb. J. Math.12(2024), no. 2, 417–534; MR4779676

work page 2024
[14]

Cavalletti and A

F. Cavalletti and A. Mondino.A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds. arXiv:2401.03949

work page arXiv
[15]

P. T. Chru´ sciel and J. D. E. Grant.On Lorentzian causality with continuous metrics. Classical Quantum Gravity29(2012), no. 14, 145001, 32 pp.; MR2949547

work page 2012
[16]

D. L. Cohn.Measure theory, 2nd edition. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher, Birkh¨ auser/Springer, New York, 2013; MR3098996

work page 2013
[17]

Esmayli and P

B. Esmayli and P. Haj lasz.The coarea inequality. Ann. Fenn. Math.46(2021), no. 2, 965–991; MR4307012

work page 2021
[18]

Garc´ ıa-Heveling and E

L. Garc´ ıa-Heveling and E. Soultanis.Causal bubbles in globally hyperbolic spacetime. Gen. Relativity Gravitation54(2022), no. 12, Paper No. 155, 7 pp.; MR4517255

work page 2022
[19]

Graf.Singularity theorems forC 1-Lorentzian metric

M. Graf.Singularity theorems forC 1-Lorentzian metric. Comm. Math. Phys.378(2020), no. 2, 1417–1450; MR4134950

work page 2020
[20]

M. Graf, E. Hafemann, M. Kunzinger, and R. Steinbauer.Hawking’s singularity theorem for Lipschitz Lorentzian metrics. Comm. Math. Phys.406(2025), no. 9, Paper No. 207, 31 pp.; MR4940197

work page 2025
[21]

Graf, E.-A

M. Graf, E.-A. Kontou, A. Ohanyan, and Y. Schinnerl.Hawking-type singularity theo- rems for worldvolume energy inequalities. Ann. Henri Poincar´ e26(2025), no. 11, 3871–3906; MR4970217

work page 2025
[22]

J. D. E. Grant, M. Kunzinger, C. S¨ amann, and R. Steinbauer.The future is not always open. Lett. Math. Phys.110(2020), no. 1, 83–103; MR4047145

work page 2020
[23]

Korte and P

R. Korte and P. Lahti.Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire31(2014), no. 1, 129–154; MR3165282

work page 2014
[24]

Kubota.Volume comparison by timelike Lipschitz maps

H. Kubota.Volume comparison by timelike Lipschitz maps. Lett. Math. Phys.116(2026), no. 1, Paper No. 6, 24 pp.; MR5006726. 23

work page 2026
[25]

Kunzinger, A

M. Kunzinger, A. Ohanyan, B. Schinnerl, and R. Steinbauer.The Hawking–Penrose Singu- larity Theorem forC 1-Lorentzian Metrics. Comm. Math. Phys.391(2022), no. 3, 1143–1179; MR4405569

work page 2022
[26]

Kunzinger and C

M. Kunzinger and C. S¨ amann.Lorentzian length spaces. Ann. Global Anal. Geom.54(2018), no. 3, 399–447; MR3867652

work page 2018
[27]

Kunzinger and R

M. Kunzinger and R. Steinbauer.Null Distance and Convergence of Lorentzian Length Spaces. Ann. Henri Poincar´ e23(2022), no. 12, 4319–4342; MR4512238

work page 2022
[28]

Ling.Aspects ofC 0 causal theory

E. Ling.Aspects ofC 0 causal theory. Gen. Relativity Gravitation52(2020), no. 6, Paper No. 57, 40 pp.; MR4111861

work page 2020
[29]

Lott and C

J. Lott and C. VillaniRicci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2)169(2009), no. 3, 903–991; MR2480619

work page 2009
[30]

R. J. McCann.Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity. Camb. J. Math.8(2020), no. 3, 609–681; MR4192570

work page 2020
[31]

R. J. McCann.A synthetic null energy condition. Comm. Math. Phys.405(2024), no. 2, Paper No. 38, 24 pp.; MR4703452

work page 2024
[32]

R. J. McCann and C. S¨ amann.A Lorentzian analog for Hausdorff dimension and measure. Pure Appl. Anal.4(2022), no. 2, 367–400; MR4496090

work page 2022
[33]

Minguzzi and S

E. Minguzzi and S. Suhr.Lorentzian metric spaces and their Gromov–Hausdorff convergence. Lett. Math. Phys.114(2024), no. 3, Paper No. 73, 63 pp.; MR4752400

work page 2024
[34]

Mondino and C

A. Mondino and C. S¨ amann.Lorentzian Gromov-Hausdorff convergence and pre-compactness. arXiv:2504.10380

work page arXiv
[35]

Mondino and S

A. Mondino and S. Suhr.An optimal transport formulation of the Einstein equations of general relativity. J. Eur. Math. Soc. (JEMS)25(2023), no. 3, 933–994; MR4577957

work page 2023
[36]

M¨ uller.Lorentzian Gromov–Hausdorff theory and finiteness results

O. M¨ uller.Lorentzian Gromov–Hausdorff theory and finiteness results. Gen. Relativity Grav- itation54(2022), no. 10, Paper No. 117, 17 pp.; MR4493631

work page 2022
[37]

Noldus.A Lorentzian Gromov-Hausdorff notion of distance

J. Noldus.A Lorentzian Gromov-Hausdorff notion of distance. Classical Quantum Gravity 21(2004), no. 4, 839–850; MR2036128

work page 2004
[38]

S¨ amann.Global hyperbolicity for spacetimes with continuous metrics

C. S¨ amann.Global hyperbolicity for spacetimes with continuous metrics. Ann. Henri Poincar´ e 17(2016), no. 6, 1429–1455; MR3500220

work page 2016
[39]

Sormani and C

C. Sormani and C. Vega.Null distance on a spacetime. Classical Quantum Gravity33(2016), no. 8, 085001, 29 pp.; MR3476515. 24

work page 2016
[40]

K. T. SturmOn the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65–131

work page 2006
[41]

K. T. SturmOn the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133–177. 25

work page 2006

[1] [1]

S. B. Alexander, M. Graf, M. Kunzinger, and C. S¨ amann.Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom.31(2023), no. 6, 1469–1528; MR4785565

work page 2023

[2] [2]

J., Ohanyan, A., Rott, F., and S ¨amann, C

T. Beran, M. Braun, M. Calisti, N. Gigli, R. J. McCann, A. Ohanyan, F. Rott, C. S¨ amann. A nonlinear d’Alembert comparison theorem and causal differential calculus on metric measure spacetimes. arXiv:2408.15968

work page arXiv

[3] [3]

Beran, M

T. Beran, M. Kunzinger, A. Ohanyan, and F. Rott.The equivalence of smooth and synthetic notions of timelike sectional curvature bounds. Proc. Amer. Math. Soc.153(2025), no. 2, 783–797; MR4852799

work page 2025

[4] [4]

Beran, M

T. Beran, M. Kunzinger, and F. Rott.On curvature bounds in Lorentzian length spaces. J. Lond. Math. Soc. (2)110(2024), no. 2, Paper No. e12971, 41 pp.; MR4781260

work page 2024

[5] [5]

Beran, A

T. Beran, A. Ohanyan, F. Rott, and D. A. Solis.The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett. Math. Phys.113(2023), no. 2, Paper No. 48, 47 pp.; MR4579262

work page 2023

[6] [6]

Beran and C

T. Beran and C. S¨ amann.Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds. J. Lond. Math. Soc. (2)107(2023), no. 5, 1823–1880; MR4585303

work page 2023

[7] [7]

Braun.R´ enyi’s entropy on Lorentzian spaces

M. Braun.R´ enyi’s entropy on Lorentzian spaces. Timelike curvature-dimension conditions. J. Math. Pures Appl177(2023), Pages 46-128, ISSN 0021-7824

work page 2023

[8] [8]

Braun and M.-S

M. Braun and M.-S. Candal.Comparison theory for Lipschitz spacetimes. arXiv:2603.24195

work page arXiv

[9] [9]

Braun, N

M. Braun, N. Gigli, R. J. McCann, A. Ohanyan, and C. S¨ amann.A Lorentzian splitting theorem for continuously differentiable metrics and weights. arXiv:2507.06836

work page arXiv

[10] [10]

Braun and S

M. Braun and S. Ohta.Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes. Trans. Amer. Math. Soc.377(2024), no. 5, 3529–3576; MR4744787. 22

work page 2024

[11] [11]

Braun and C

M. Braun and C. S¨ amann.Gromov’s reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry. arXiv:2506.10852

work page arXiv

[12] [12]

Cavalletti, D

F. Cavalletti, D. Manini and A. Mondino.Optimal transport on null hypersurfaces and the null energy condition. Comm. Math. Phys.406(2025), no. 9, Paper No. 212, 62 pp.; MR4940202

work page 2025

[13] [13]

Cavalletti and A

F. Cavalletti and A. Mondino.Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Camb. J. Math.12(2024), no. 2, 417–534; MR4779676

work page 2024

[14] [14]

Cavalletti and A

F. Cavalletti and A. Mondino.A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds. arXiv:2401.03949

work page arXiv

[15] [15]

P. T. Chru´ sciel and J. D. E. Grant.On Lorentzian causality with continuous metrics. Classical Quantum Gravity29(2012), no. 14, 145001, 32 pp.; MR2949547

work page 2012

[16] [16]

D. L. Cohn.Measure theory, 2nd edition. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher, Birkh¨ auser/Springer, New York, 2013; MR3098996

work page 2013

[17] [17]

Esmayli and P

B. Esmayli and P. Haj lasz.The coarea inequality. Ann. Fenn. Math.46(2021), no. 2, 965–991; MR4307012

work page 2021

[18] [18]

Garc´ ıa-Heveling and E

L. Garc´ ıa-Heveling and E. Soultanis.Causal bubbles in globally hyperbolic spacetime. Gen. Relativity Gravitation54(2022), no. 12, Paper No. 155, 7 pp.; MR4517255

work page 2022

[19] [19]

Graf.Singularity theorems forC 1-Lorentzian metric

M. Graf.Singularity theorems forC 1-Lorentzian metric. Comm. Math. Phys.378(2020), no. 2, 1417–1450; MR4134950

work page 2020

[20] [20]

M. Graf, E. Hafemann, M. Kunzinger, and R. Steinbauer.Hawking’s singularity theorem for Lipschitz Lorentzian metrics. Comm. Math. Phys.406(2025), no. 9, Paper No. 207, 31 pp.; MR4940197

work page 2025

[21] [21]

Graf, E.-A

M. Graf, E.-A. Kontou, A. Ohanyan, and Y. Schinnerl.Hawking-type singularity theo- rems for worldvolume energy inequalities. Ann. Henri Poincar´ e26(2025), no. 11, 3871–3906; MR4970217

work page 2025

[22] [22]

J. D. E. Grant, M. Kunzinger, C. S¨ amann, and R. Steinbauer.The future is not always open. Lett. Math. Phys.110(2020), no. 1, 83–103; MR4047145

work page 2020

[23] [23]

Korte and P

R. Korte and P. Lahti.Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire31(2014), no. 1, 129–154; MR3165282

work page 2014

[24] [24]

Kubota.Volume comparison by timelike Lipschitz maps

H. Kubota.Volume comparison by timelike Lipschitz maps. Lett. Math. Phys.116(2026), no. 1, Paper No. 6, 24 pp.; MR5006726. 23

work page 2026

[25] [25]

Kunzinger, A

M. Kunzinger, A. Ohanyan, B. Schinnerl, and R. Steinbauer.The Hawking–Penrose Singu- larity Theorem forC 1-Lorentzian Metrics. Comm. Math. Phys.391(2022), no. 3, 1143–1179; MR4405569

work page 2022

[26] [26]

Kunzinger and C

M. Kunzinger and C. S¨ amann.Lorentzian length spaces. Ann. Global Anal. Geom.54(2018), no. 3, 399–447; MR3867652

work page 2018

[27] [27]

Kunzinger and R

M. Kunzinger and R. Steinbauer.Null Distance and Convergence of Lorentzian Length Spaces. Ann. Henri Poincar´ e23(2022), no. 12, 4319–4342; MR4512238

work page 2022

[28] [28]

Ling.Aspects ofC 0 causal theory

E. Ling.Aspects ofC 0 causal theory. Gen. Relativity Gravitation52(2020), no. 6, Paper No. 57, 40 pp.; MR4111861

work page 2020

[29] [29]

Lott and C

J. Lott and C. VillaniRicci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2)169(2009), no. 3, 903–991; MR2480619

work page 2009

[30] [30]

R. J. McCann.Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity. Camb. J. Math.8(2020), no. 3, 609–681; MR4192570

work page 2020

[31] [31]

R. J. McCann.A synthetic null energy condition. Comm. Math. Phys.405(2024), no. 2, Paper No. 38, 24 pp.; MR4703452

work page 2024

[32] [32]

R. J. McCann and C. S¨ amann.A Lorentzian analog for Hausdorff dimension and measure. Pure Appl. Anal.4(2022), no. 2, 367–400; MR4496090

work page 2022

[33] [33]

Minguzzi and S

E. Minguzzi and S. Suhr.Lorentzian metric spaces and their Gromov–Hausdorff convergence. Lett. Math. Phys.114(2024), no. 3, Paper No. 73, 63 pp.; MR4752400

work page 2024

[34] [34]

Mondino and C

A. Mondino and C. S¨ amann.Lorentzian Gromov-Hausdorff convergence and pre-compactness. arXiv:2504.10380

work page arXiv

[35] [35]

Mondino and S

A. Mondino and S. Suhr.An optimal transport formulation of the Einstein equations of general relativity. J. Eur. Math. Soc. (JEMS)25(2023), no. 3, 933–994; MR4577957

work page 2023

[36] [36]

M¨ uller.Lorentzian Gromov–Hausdorff theory and finiteness results

O. M¨ uller.Lorentzian Gromov–Hausdorff theory and finiteness results. Gen. Relativity Grav- itation54(2022), no. 10, Paper No. 117, 17 pp.; MR4493631

work page 2022

[37] [37]

Noldus.A Lorentzian Gromov-Hausdorff notion of distance

J. Noldus.A Lorentzian Gromov-Hausdorff notion of distance. Classical Quantum Gravity 21(2004), no. 4, 839–850; MR2036128

work page 2004

[38] [38]

S¨ amann.Global hyperbolicity for spacetimes with continuous metrics

C. S¨ amann.Global hyperbolicity for spacetimes with continuous metrics. Ann. Henri Poincar´ e 17(2016), no. 6, 1429–1455; MR3500220

work page 2016

[39] [39]

Sormani and C

C. Sormani and C. Vega.Null distance on a spacetime. Classical Quantum Gravity33(2016), no. 8, 085001, 29 pp.; MR3476515. 24

work page 2016

[40] [40]

K. T. SturmOn the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65–131

work page 2006

[41] [41]

K. T. SturmOn the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133–177. 25

work page 2006