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arxiv: 2605.31397 · v1 · pith:L2VFOAFGnew · submitted 2026-05-29 · 🧮 math.DG · math.MG

Constant mean curvature surfaces in the sub-Lorentzian Heisenberg group

Samu\"el Borza , Andrea Pinamonti , Omar Zoghlami This is my paper

Pith reviewed 2026-06-28 20:52 UTC · model grok-4.3

classification 🧮 math.DG math.MG
keywords constant mean curvaturesub-Lorentzian Heisenberg groupisoperimetric problemboost-symmetric surfaceshorizontal mean curvaturePansu bubblescharacteristic set
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The pith

A family of boost-symmetric surfaces with constant horizontal mean curvature in the sub-Lorentzian Heisenberg group is conjectured to solve the isoperimetric problem.

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

The paper derives the first-variation formula for horizontal area under volume-preserving radial variations and concludes that smooth isoperimetric candidates must have constant horizontal mean curvature away from the characteristic set. It then classifies all smooth boost-symmetric constant mean curvature surfaces by their characteristic sets, causal behaviour, and ambient isometry classes. From the classification the authors isolate one family of smooth, acausal, nonzero-constant-mean-curvature surfaces that can be expressed as two-sheeted graphs over the exterior of a future hyperbola. These surfaces are presented as the direct sub-Lorentzian analogue of the Pansu bubbles. The authors conjecture that the family realises the isoperimetric maximum in the sub-Lorentzian Heisenberg group.

Core claim

From this classification, we single out a family of smooth, acausal, boost-symmetric surfaces with nonzero constant mean curvature. Written as a two-sheeted graph over the exterior of a future hyperbola, this family is a natural sub-Lorentzian analogue of the Pansu bubbles and leads us to conjecture that it gives the isoperimetric maximisers in the sub-Lorentzian Heisenberg group.

What carries the argument

The complete classification of smooth boost-symmetric constant mean curvature surfaces by characteristic sets, causal behaviour, and sub-Lorentzian isometry classes.

If this is right

  • Smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set.
  • The identified family consists of acausal surfaces that remain regular for the isoperimetric problem.
  • These surfaces supply explicit examples against which the isoperimetric conjecture in the sub-Lorentzian Heisenberg group can be tested.
  • The classification organises all boost-symmetric constant-mean-curvature surfaces into finitely many isometry classes.

Where Pith is reading between the lines

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

  • If isoperimetric maximisers need not be boost-symmetric, asymmetric competitors could exceed the ratio attained by the identified family.
  • The first-variation formula and classification method may apply directly to other sub-Lorentzian or sub-Riemannian three-dimensional geometries.
  • Numerical computation of the isoperimetric ratio attained by the two-sheeted graph family would provide a concrete test of the conjecture.

Load-bearing premise

The classification and conjecture rest on restricting attention to boost-symmetric surfaces.

What would settle it

Existence of a smooth isoperimetric surface that is not boost-symmetric and achieves a strictly larger isoperimetric ratio than the identified family.

Figures

Figures reproduced from arXiv: 2605.31397 by Andrea Pinamonti, Omar Zoghlami, Samu\"el Borza.

Figure 1
Figure 1. Figure 1: A timelike-parameter surface S(C,0) with C > 0. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Although no regularity issue arises in the classical sense, these surfaces still exhibit a form of “non-regularity” from the sub-Lorentzian point of view: their set of characteristic points is non-empty. This is described in the following proposition [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A surface S(c,d) with null parameter. 3.4. Surfaces of null parameters. We next consider the family of surfaces S(c,d) with null parameters, that is, those for which c 2 − d 2 = 0. Equivalently, apart from the degenerate case (c, d) = (0, 0), we may write c = ωd, where ω ∈ {−1, 1}. In this case the parametrisation (20) reduces to S(c,ωc) (t, u) = Bu  cosh(t) + c, sinh(t) + ωc, 1 2  t − ωce−ωt + ωc . (3… view at source ↗
Figure 4
Figure 4. Figure 4: A maximal surface. We first show that it is not restrictive to assume that ε = 1, since changing the sign of ε produces an isometric surface via a timed rotation. Proposition 3.39. For every α, β ∈ R, the maximal surfaces S(1,α,β) and S(−1,α,β) are isometric. More precisely, they are isometric both by a time-preserving rotation and by a vertical translation. Proof. By Theorem 2.1, the map A(x, y, z) = (x, … view at source ↗
read the original abstract

We study constant horizontal mean curvature surfaces in the sub-Lorentzian Heisenberg group. We derive the first-variation formula for horizontal area under volume-preserving radial variations and show that smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set. We then give a complete classification of smooth boost-symmetric constant mean curvature surfaces: their characteristic sets, causal behaviour, and ambient sub-Lorentzian isometry classes. From this classification, we single out a family of smooth, acausal, boost-symmetric surfaces with nonzero constant mean curvature. Written as a two-sheeted graph over the exterior of a future hyperbola, this family is a natural sub-Lorentzian analogue of the Pansu bubbles and leads us to conjecture that it gives the isoperimetric maximisers in the sub-Lorentzian Heisenberg group.

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.

Referee Report

1 major / 0 minor

Summary. The paper derives a first-variation formula for horizontal area under volume-preserving radial variations in the sub-Lorentzian Heisenberg group, shows that smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set, and gives a complete classification of smooth boost-symmetric constant mean curvature surfaces (their characteristic sets, causal behaviour, and isometry classes). It identifies a specific family of smooth, acausal, boost-symmetric nonzero-CMC surfaces realized as two-sheeted graphs over the exterior of a future hyperbola and conjectures that this family consists of the isoperimetric maximizers.

Significance. If the first-variation formula and the classification of boost-symmetric CMC surfaces are correct, the work supplies a direct sub-Lorentzian analogue of the Pansu bubbles together with an explicit first-variation tool that can be used for further isoperimetric analysis in this geometry.

major comments (1)
  1. [Abstract] Abstract: the conjecture that the identified two-sheeted boost-symmetric family 'gives the isoperimetric maximisers' is load-bearing for the paper's stated motivation, yet the classification and first-variation formula are derived only under the boost-symmetry assumption; no symmetry-forcing result, calibration argument, or comparison with non-symmetric competitors is supplied to justify that maximizers must lie in this class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this point concerning the abstract. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the conjecture that the identified two-sheeted boost-symmetric family 'gives the isoperimetric maximisers' is load-bearing for the paper's stated motivation, yet the classification and first-variation formula are derived only under the boost-symmetry assumption; no symmetry-forcing result, calibration argument, or comparison with non-symmetric competitors is supplied to justify that maximizers must lie in this class.

    Authors: The first-variation formula is derived without any symmetry assumption, and the classification is explicitly restricted to the boost-symmetric case; both results stand independently of the conjecture. The conjecture itself is framed as arising from the classification together with the known analogy to the rotationally symmetric Pansu bubbles in the sub-Riemannian Heisenberg group. The manuscript does not assert, nor does it attempt to prove, that every isoperimetric maximizer must be boost-symmetric; no symmetry-forcing theorem, calibration, or direct comparison with non-symmetric surfaces is claimed or supplied. We therefore view the conjecture as an open statement motivated by the explicit family we construct, rather than a proven characterization. Because the paper already presents the statement as a conjecture and does not rely on it for any of the proved results, we do not believe a change to the abstract is required. revision: no

Circularity Check

0 steps flagged

No circularity: classification derived from first-variation formula and explicit symmetry reduction; conjecture stated as such

full rationale

The paper derives the first-variation formula for horizontal area, shows CMC is necessary for isoperimetric candidates, and classifies boost-symmetric CMC surfaces by direct computation of their characteristic sets and causal properties. The family is singled out from this classification and the isoperimetric conjecture is explicitly labeled as a conjecture resting on the boost-symmetry restriction. No step equates a derived quantity to its input by construction, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present work. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the work relies on standard differential-geometric background.

axioms (1)
  • standard math Smoothness of surfaces away from the characteristic set and well-defined horizontal distribution on the sub-Lorentzian Heisenberg group.
    Implicit in the statement that smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set.

pith-pipeline@v0.9.1-grok · 5675 in / 1294 out tokens · 30671 ms · 2026-06-28T20:52:56.054964+00:00 · methodology

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Reference graph

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