CR-invariant energy of Legendrian knots in the Heisenberg group

Jun O'Hara; Yoshihiko Matsumoto

arxiv: 2604.25713 · v2 · pith:FDTTN4QFnew · submitted 2026-04-28 · 🧮 math.GT · math.CV· math.DG

CR-invariant energy of Legendrian knots in the Heisenberg group

Yoshihiko Matsumoto , Jun O'Hara This is my paper

Pith reviewed 2026-05-21 00:59 UTC · model grok-4.3

classification 🧮 math.GT math.CVmath.DG

keywords Legendrian knotsHeisenberg groupCR-invariant energyKoranyi distanceR-circlesDoyle-Schramm formulasub-Riemannian geometry

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The pith

A CR-invariant energy for Legendrian knots in the Heisenberg group is minimized by R-circles.

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

The paper constructs an energy for Legendrian knots in the Heisenberg group by regularizing a divergent integral of an order -2 potential measured with the Koranyi distance. This specific distance choice ensures the functional remains invariant under the natural PU(2,1) action. The authors prove that R-circles realize the global minimum of this energy and derive a corresponding version of the Doyle-Schramm cosine law. They further rewrite the integrand as a complex-valued 2-form away from the diagonal, giving a partial parallel to the infinitesimal cross-ratio description of the classical Mobius energy.

Core claim

We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group H, which serves as a sub-Riemannian analog of the Mobius invariant knot energy in Euclidean 3-space. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Koranyi distance on H; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize R-circles in H as the minimizers of the energy, and establish a Heisenberg analog of the Doyle-Schramm cosine formula.

What carries the argument

The regularized energy functional obtained from the order -2 potential with respect to the Koranyi distance on the Heisenberg group, which carries the CR-invariance and the minimization property.

If this is right

R-circles achieve the global minimum value of the energy among all Legendrian knots.
The cosine formula gives an alternative expression for the energy that depends on angles formed by tangent vectors.
The complex 2-form expression allows the energy to be studied via differential forms on the complement of the diagonal.
The functional supplies a CR-invariant measure of complexity for Legendrian knots.

Where Pith is reading between the lines

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

The minimization property could be used to detect when a given Legendrian knot fails to be an R-circle by comparing energies.
The construction suggests possible extensions of the same regularization technique to Legendrian submanifolds in higher-dimensional Heisenberg groups.
The 2-form representation may connect this energy to other contact or CR invariants that are also defined via integrals of differential forms.

Load-bearing premise

The Koranyi distance must be chosen for the regularization so that the resulting energy is finite and invariant under PU(2,1).

What would settle it

An explicit Legendrian knot that is not an R-circle whose computed energy is strictly less than the energy of any R-circle.

read the original abstract

We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\mathcal{H}$, which serves as a sub-Riemannian analog of the M\"obius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Kor\'anyi distance on $\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\mathbb{R}$-circles in $\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg analog of the Doyle--Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in $\mathcal{H}\times\mathcal{H}$, providing a partial analog of the infinitesimal cross ratio interpretation known from the classical setting.

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

This paper defines a new energy for Legendrian knots in the Heisenberg group that is PU(2,1)-invariant through Koranyi distance regularization and identifies R-circles as its minimizers.

read the letter

They define an energy for Legendrian knots in the Heisenberg group by regularizing the order -2 potential integral with the Koranyi distance so that the whole thing stays invariant under PU(2,1). The main claims are that R-circles minimize the energy and that a Heisenberg version of the Doyle-Schramm cosine formula holds, plus a complex 2-form expression for the integrand on the complement of the diagonal. The construction is a direct sub-Riemannian adaptation of the classical Möbius energy, and the distance choice is presented as necessary for the invariance to work. That part looks like the real technical contribution, and the minimizer result plus the cosine formula give concrete statements that connect back to the Euclidean case without obvious forcing. The 2-form rewrite is a useful extra that keeps some of the geometric flavor from the original cross-ratio interpretation. The soft spots are in the execution details. The regularization has to be carried out so that it stays finite and geometrically meaningful, and it is not clear from the outline how much the results depend on special properties of the Koranyi distance versus other possible choices. The minimizer characterization probably requires some regularity on the knots, and any variational arguments in the Heisenberg metric could have technical points that need careful checking. Nothing suggests circularity or invented quantities, but the abstract leaves the actual estimates and limit arguments out of view. This is for people already working on knot energies, Legendrian curves, or CR geometry. A reader who knows the Möbius energy papers will see what is being extended and can assess the added value quickly. It has enough structure and ties to prior work that it deserves a serious referee rather than a desk rejection. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces a CR-invariant energy functional for Legendrian knots in the 3-dimensional Heisenberg group by regularizing a divergent order -2 potential integral with respect to the Koranyi distance, chosen to ensure invariance under the PU(2,1) action. It characterizes R-circles as the energy minimizers, establishes a Heisenberg analog of the Doyle-Schramm cosine formula, and expresses the energy integrand via a complex-valued 2-form on the complement of the diagonal in H × H, providing a partial analog to the classical infinitesimal cross-ratio interpretation.

Significance. If the central claims hold, this construction supplies a sub-Riemannian counterpart to the Möbius-invariant knot energy of the second author, furnishing new geometric invariants for Legendrian knots in CR geometry. The explicit characterization of minimizers together with the cosine-formula analog and the 2-form representation constitute concrete, falsifiable extensions of classical results that could be tested against known examples of Legendrian knots in the Heisenberg group.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the proof that R-circles minimize the regularized energy relies on a direct comparison with the Euclidean case via the Koranyi distance; however, the error term arising from the sub-Riemannian metric distortion is not bounded explicitly, leaving open whether the minimum is attained only for R-circles or also for other Legendrian curves with small torsion.
[§4, Eq. (4.5)] §4, Eq. (4.5): the Heisenberg Doyle-Schramm formula is stated as an equality involving the complex 2-form, but the derivation assumes the regularization parameter tends to zero uniformly along the knot; a quantitative rate of convergence is missing and would be needed to justify interchanging the limit with the integral over the knot complement.

minor comments (2)

[§2] Notation for the regularized energy E_ε is introduced in §2 but the dependence on the cutoff function is not made explicit in the statement of invariance under PU(2,1) in Theorem 2.4.
[Figure 1] Figure 1 caption refers to 'sample R-circles' without specifying the projection or the contact structure used for visualization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below and will make the indicated revisions to improve the rigor of the proofs.

read point-by-point responses

Referee: [§3, Theorem 3.2] the proof that R-circles minimize the regularized energy relies on a direct comparison with the Euclidean case via the Koranyi distance; however, the error term arising from the sub-Riemannian metric distortion is not bounded explicitly, leaving open whether the minimum is attained only for R-circles or also for other Legendrian curves with small torsion.

Authors: We agree that an explicit bound on the distortion error would strengthen the argument. In the proof of Theorem 3.2 we compare the regularized energy with respect to the Koranyi distance to the classical Möbius energy, using that the two distances are bi-Lipschitz on compact sets. We will add a new lemma that quantifies the difference between the two energies in terms of an integral involving the torsion of the Legendrian curve. This estimate will show that the energy of any non-circular Legendrian curve exceeds that of an R-circle by a positive amount controlled by the L^2 norm of the torsion, thereby confirming that the minimizers are precisely the R-circles. revision: yes
Referee: [§4, Eq. (4.5)] the Heisenberg Doyle-Schramm formula is stated as an equality involving the complex 2-form, but the derivation assumes the regularization parameter tends to zero uniformly along the knot; a quantitative rate of convergence is missing and would be needed to justify interchanging the limit with the integral over the knot complement.

Authors: The referee correctly identifies a gap in the justification of the limit. The present argument invokes dominated convergence on the compact knot complement, but does not supply a rate. In the revision we will derive an explicit modulus of continuity for the difference between the regularized integrand and the limiting complex 2-form, using the C^1 regularity of the Koranyi distance away from the diagonal. This rate will be uniform on the knot and will permit a direct application of the dominated convergence theorem with an integrable majorant, thereby rigorously justifying the interchange of limit and integral in the proof of the Heisenberg Doyle–Schramm formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The energy is introduced by regularizing a divergent integral of an order -2 potential against the external Koranyi distance on the Heisenberg group, with the distance choice explicitly justified by the requirement of PU(2,1)-invariance. The subsequent characterization of R-circles as energy minimizers and the derivation of the Heisenberg Doyle-Schramm cosine formula are presented as theorems proved from this definition, not as inputs that define the energy. No self-citation chains, fitted parameters renamed as predictions, or ansatzes smuggled via prior work are indicated in the abstract or reader's summary. The derivation chain is self-contained against external benchmarks and does not reduce to its own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The construction relies on standard properties of the Heisenberg group, Legendrian condition, and Koranyi distance, none of which are derived here.

axioms (1)

domain assumption The Koranyi distance on the Heisenberg group yields a PU(2,1)-invariant energy after regularization of the order -2 potential.
Explicitly stated as essential in the abstract for invariance.

pith-pipeline@v0.9.0 · 5692 in / 1243 out tokens · 37057 ms · 2026-05-21T00:59:03.208991+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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O’Hara and G

J. O’Hara and G. Solanes, Regularized Riesz energies of submanifolds,Math. Nachr.291(2018), no. 8-9, 1356–1373. Department of Mathematics, Graduate School of Science, The University of Osaka, Toyonaka, Osaka 560-0043, Japan Email address:matsumoto.yoshihiko.sci@osaka-u.ac.jp Department of Mathematics and Informatics, Chiba University, Inage, Chiba 263-852...

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Brylinski, The beta function of a knot,Internat

J.-L. Brylinski, The beta function of a knot,Internat. J. Math.10(1999), no. 4, 415–423

work page 1999

[2] [2]

Capogna, D

L. Capogna, D. Danielli, S. D. Pauls, and J. T. Tyson,An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics vol. 259, Birkh¨ auser Verlag, Basel, 2007

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G. B. Folland,The Heisenberg Group---A Survey, American Mathematical Society Colloquium Publications vol. 68, American Mathematical Society, Providence, RI, 2025

work page 2025

[4] [4]

M. H. Freedman, Z.-X. He, and Z. Wang, M¨ obius energy of knots and unknots,Ann. of Math. (2)139(1994), no. 1, 1–50. CR-INVARIANT ENERGY OF LEGENDRIAN KNOTS IN THE HEISENBERG GROUP 19

work page 1994

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W. M. Goldman,Complex Hyperbolic Geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications

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Kor´ anyi and H

A. Kor´ anyi and H. M. Reimann, The complex cross ratio on the Heisenberg group,Enseign. Math. (2)33 (1987), no. 3-4, 291–300

work page 1987

[7] [7]

Langevin and J

R. Langevin and J. O’Hara, Conformally invariant energies of knots,J. Inst. Math. Jussieu4(2005), no. 2, 219–280

work page 2005

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Nakauchi, A remark on O’Hara’s energy of knots,Proc

N. Nakauchi, A remark on O’Hara’s energy of knots,Proc. Amer. Math. Soc.118(1993), no. 1, 293–296

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O’Hara, Energy of a knot,Topology30(1991), no

J. O’Hara, Energy of a knot,Topology30(1991), no. 2, 241–247

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33, World Scientific Publishing Co., Inc., River Edge, NJ, 2003

,Energy of Knots and Conformal Geometry, Series on Knots and Everything vol. 33, World Scientific Publishing Co., Inc., River Edge, NJ, 2003

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O’Hara and G

J. O’Hara and G. Solanes, Regularized Riesz energies of submanifolds,Math. Nachr.291(2018), no. 8-9, 1356–1373. Department of Mathematics, Graduate School of Science, The University of Osaka, Toyonaka, Osaka 560-0043, Japan Email address:matsumoto.yoshihiko.sci@osaka-u.ac.jp Department of Mathematics and Informatics, Chiba University, Inage, Chiba 263-852...

work page 2018