Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations

Argam Ohanyan; Gregory J. Galloway; Robert J. McCann

arxiv: 2606.21997 · v1 · pith:4NHSPFEInew · submitted 2026-06-20 · 🧮 math.DG · gr-qc

Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations

Gregory J. Galloway , Robert J. McCann , Argam Ohanyan This is my paper

Pith reviewed 2026-06-26 11:28 UTC · model grok-4.3

classification 🧮 math.DG gr-qc

keywords Lorentzian distance functionsCauchy temporal functionequi-Lipschitz continuityLorentzian splitting theoremstimelike lineCauchy surface foliation

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The pith

Lorentzian distances to level sets of Cauchy temporal functions are not locally equi-Lipschitz in general.

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

The paper establishes that while families of Lorentzian distance functions to points on a complete timelike line are locally equi-Lipschitz, this property does not hold when the distances are taken instead to the level sets of a Cauchy temporal function. This matters because the equi-Lipschitz property is essential in proofs of classical Lorentzian splitting theorems. The failure is shown to occur even in spacetimes that meet the conditions for the line case. The work also introduces conjectures linking the equi-Lipschitz property for such foliations to a splitting conjecture.

Core claim

We show that the local equi-Lipschitz continuity of Lorentzian distance functions fails in general for families of distances to and from the level sets of a given Cauchy temporal function, unlike the case for complete timelike lines.

What carries the argument

The family of Lorentz distances to and from level sets of a Cauchy temporal function, shown not to be locally equi-Lipschitz.

If this is right

Classical proofs of Lorentzian splitting theorems that rely on equi-Lipschitz continuity along timelike lines cannot be straightforwardly adapted to Cauchy surface foliations.
The existence of equi-Lipschitz Cauchy temporal functions in certain spacetimes is conjectured to be equivalent to a splitting conjecture.
Additional assumptions beyond the existence of a Cauchy temporal function are needed to guarantee equi-Lipschitz behavior for distances to its level sets.

Where Pith is reading between the lines

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

This highlights a special geometric role for complete timelike lines that may not be shared by general Cauchy foliations.
Approaches to splitting theorems in cosmological spacetimes may need to focus on lines rather than arbitrary temporal functions.
Concrete examples of spacetimes violating the equi-Lipschitz property for foliations could be constructed to test the boundary of the failure.

Load-bearing premise

The spacetime has a Cauchy temporal function allowing definition of the distance families, even when it has a complete timelike line where the property holds for lines.

What would settle it

A specific spacetime example where Lorentz distances to every Cauchy temporal function level sets are locally equi-Lipschitz would show the failure does not occur in general.

read the original abstract

The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.

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 shows equi-Lipschitzness fails for Cauchy foliation distances while holding for timelike lines, then links the positive case to Bartnik's conjecture via two new conjectures.

read the letter

The key point is that the equi-Lipschitz property for Lorentz distances, already known to hold along complete timelike lines, does not hold in general when the distances are taken to level sets of a Cauchy temporal function.

The paper supplies an explicit counterexample for the negative claim and then states two conjectures under which the foliation distances would be equi-Lipschitz in cosmological or timelike geodesically complete spacetimes; these conjectures are presented as equivalent to Bartnik's splitting conjecture. That equivalence is the main new contribution.

The work is useful because it isolates a distinction that the splitting-theorem literature had not addressed and because it converts an open question into a pair of concrete statements that can be checked. The abstract is clear on both the negative result and the conjectural positive direction.

The main soft spot is the one raised in the stress-test: the abstract does not state whether the counterexample spacetime still contains a complete timelike line. If the constructed example drops that hypothesis, the failure occurs outside the regime where the line result applies, so the contrast with the existing splitting literature is weaker than claimed. Without the construction details it is impossible to tell whether this is handled.

This is the sort of paper that belongs in a reading group on Lorentzian geometry or global hyperbolicity. It is worth refereeing because it directly engages the splitting literature and offers falsifiable reformulations of Bartnik's conjecture, even if the counterexample needs closer inspection on the line condition.

Referee Report

1 major / 2 minor

Summary. The manuscript shows that the local equi-Lipschitz continuity of Lorentzian distance functions, which holds for families associated to points along a complete timelike line, fails in general for families associated to the level sets of a Cauchy temporal function. It constructs a counterexample in a globally hyperbolic spacetime and formulates conjectures equating equi-Lipschitz behavior for such foliations with Bartnik's splitting conjecture in cosmological and timelike geodesically complete settings.

Significance. If the counterexample is shown to retain a complete timelike line, the negative result would be significant: it isolates the special role of lines in the equi-Lipschitz property used for classical splitting theorems and supplies concrete conjectures that could be tested against known positive results in the literature.

major comments (1)

[counterexample construction] Counterexample construction: the manuscript must explicitly verify that the constructed spacetime contains a complete timelike line (under the same global hyperbolicity and other background hypotheses used for the positive line result). Without this verification the claimed failure does not occur inside the regime where the line result applies, leaving the contrast with the splitting-theorem literature unsecured.

minor comments (2)

[introduction] Notation for the family of distance functions d_Σ_t should be introduced with a single displayed equation early in the text rather than piecemeal.
[conjectures] The conjectures in the final section should be numbered and stated with the precise hypotheses (e.g., existence of Cauchy temporal function plus timelike geodesic completeness) under which they are claimed to be equivalent to Bartnik's conjecture.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting an important point about the counterexample. We address the major comment below and will incorporate the requested verification into the revised manuscript to strengthen the connection to the splitting theorem literature.

read point-by-point responses

Referee: [counterexample construction] Counterexample construction: the manuscript must explicitly verify that the constructed spacetime contains a complete timelike line (under the same global hyperbolicity and other background hypotheses used for the positive line result). Without this verification the claimed failure does not occur inside the regime where the line result applies, leaving the contrast with the splitting-theorem literature unsecured.

Authors: We agree that explicit verification is necessary to place the counterexample firmly within the setting of the positive equi-Lipschitz result for complete timelike lines. In the revised version we will add a dedicated subsection verifying that the constructed globally hyperbolic spacetime admits a complete timelike line satisfying the same hypotheses (global hyperbolicity, smoothness, etc.) used in the line-based positive result. This verification will consist of exhibiting an explicit timelike curve, confirming its completeness with respect to the Lorentzian metric, and checking that the spacetime remains globally hyperbolic. The addition will secure the desired contrast with the splitting-theorem literature without altering the main negative result. revision: yes

Circularity Check

0 steps flagged

No circularity; counterexample contrasts external positive result for lines with new negative result for foliations

full rationale

The paper's central claim is a negative result: it constructs (or exhibits) a spacetime where the equi-Lipschitz property fails for distance families to level sets of a Cauchy temporal function. This is explicitly contrasted with a known positive result for complete timelike lines, which the abstract attributes to prior literature on Lorentzian splitting theorems rather than any self-citation chain. No equations or steps reduce a prediction to a fitted input, no ansatz is smuggled via self-citation, and the formulated conjectures are stated as equivalent to the external Bartnik splitting conjecture. The derivation is therefore self-contained as a counterexample paper; the load-bearing verification of the counterexample's hypotheses is a matter of correctness, not circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the claim rests on standard background results in Lorentzian geometry that are not detailed here.

pith-pipeline@v0.9.1-grok · 5647 in / 1028 out tokens · 22265 ms · 2026-06-26T11:28:32.375639+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

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1988
[2]

Beran, M

T. Beran, M. Braun, M. Calisti, N. Gigli, R. J. McCann, A. Ohanyan, F. Rott, and C. S¨ amann. A nonlinear d’Alembert comparison theorem and causal differential calculus on metric measure spacetimes.arXiv:2408.15968, 2024

arXiv 2024
[3]

Bernard and S

P. Bernard and S. Suhr. Lyapounov functions of closed cone fields: from Conley theory to time functions.Comm. Math. Phys., 359(2):467–498, 2018

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[4]

Braun, N

M. Braun, N. Gigli, R. J. McCann, A. Ohanyan, and C. S¨ amann. An elliptic proof of the splitting theorems from Lorentzian geometry.Comm. Amer. Math. Soc., to appear
[5]

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, 2025

arXiv 2025
[6]

Caponio, A

E. Caponio, A. Ohanyan, and S.-i. Ohta. Splitting theorems for weighted Finsler space- times via thep-d’Alembertian: beyond the Berwald case.J. Reine Angew. Math., to appear
[7]

P. E. Ehrlich and G. J. Galloway. Timelike lines.Classical Quantum Gravity, 7(3):297– 307, 1990

1990
[8]

Eschenburg

J.-H. Eschenburg. The splitting theorem for space-times with strong energy condition. J. Differential Geom., 27(3):477–491, 1988

1988
[9]

G. J. Galloway. The Lorentzian splitting theorem without the completeness assumption. J. Differential Geom., 29(2):373–387, 1989

1989
[10]

G. J. Galloway and A. Horta. Regularity of Lorentzian Busemann functions.Trans. Amer. Math. Soc., 348(5):2063–2084, 1996

2063
[11]

G. J. Galloway and C. Vega. Achronal limits, Lorentzian spheres, and splitting.Ann. Henri Poincar´ e, 15(11):2241–2279, 2014

2014
[12]

G. J. Galloway and C. Vega. Hausdorff closed limits and rigidity in Lorentzian geometry. Ann. Henri Poincar´ e, 18(10):3399–3426, 2017. 8

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[13]

R. P. A. C. Newman. A proof of the splitting conjecture of S.-T. Yau.J. Differential Geom., 31(1):163–184, 1990

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S. Zhu, H. Wu, and X. Cui. Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times.J. Differential Equations, 435:Paper No. 113323, 30, 2025. 9

2025

[1] [1]

R. Bartnik. Remarks on cosmological spacetimes and constant mean curvature surfaces. Comm. Math. Phys., 117(4):615–624, 1988

1988

[2] [2]

Beran, M

T. Beran, M. Braun, M. Calisti, N. Gigli, R. J. McCann, A. Ohanyan, F. Rott, and C. S¨ amann. A nonlinear d’Alembert comparison theorem and causal differential calculus on metric measure spacetimes.arXiv:2408.15968, 2024

arXiv 2024

[3] [3]

Bernard and S

P. Bernard and S. Suhr. Lyapounov functions of closed cone fields: from Conley theory to time functions.Comm. Math. Phys., 359(2):467–498, 2018

2018

[4] [4]

Braun, N

M. Braun, N. Gigli, R. J. McCann, A. Ohanyan, and C. S¨ amann. An elliptic proof of the splitting theorems from Lorentzian geometry.Comm. Amer. Math. Soc., to appear

[5] [5]

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, 2025

arXiv 2025

[6] [6]

Caponio, A

E. Caponio, A. Ohanyan, and S.-i. Ohta. Splitting theorems for weighted Finsler space- times via thep-d’Alembertian: beyond the Berwald case.J. Reine Angew. Math., to appear

[7] [7]

P. E. Ehrlich and G. J. Galloway. Timelike lines.Classical Quantum Gravity, 7(3):297– 307, 1990

1990

[8] [8]

Eschenburg

J.-H. Eschenburg. The splitting theorem for space-times with strong energy condition. J. Differential Geom., 27(3):477–491, 1988

1988

[9] [9]

G. J. Galloway. The Lorentzian splitting theorem without the completeness assumption. J. Differential Geom., 29(2):373–387, 1989

1989

[10] [10]

G. J. Galloway and A. Horta. Regularity of Lorentzian Busemann functions.Trans. Amer. Math. Soc., 348(5):2063–2084, 1996

2063

[11] [11]

G. J. Galloway and C. Vega. Achronal limits, Lorentzian spheres, and splitting.Ann. Henri Poincar´ e, 15(11):2241–2279, 2014

2014

[12] [12]

G. J. Galloway and C. Vega. Hausdorff closed limits and rigidity in Lorentzian geometry. Ann. Henri Poincar´ e, 18(10):3399–3426, 2017. 8

2017

[13] [13]

R. P. A. C. Newman. A proof of the splitting conjecture of S.-T. Yau.J. Differential Geom., 31(1):163–184, 1990

1990

[14] [14]

S. Zhu, H. Wu, and X. Cui. Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times.J. Differential Equations, 435:Paper No. 113323, 30, 2025. 9

2025