CR-invariant energy of Legendrian knots in the Heisenberg group

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 Möbius 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 Korányi 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.

• The paper introduces a CR-invariant energy for Legendrian knots in the Heisenberg group, preserving Möbius symmetry via the Korányi metric.
• It employs analytic regularization through the beta function to rigorously define the energy despite divergence issues on the diagonal.
• The study shows that ℝ-circles are the unique minimizers, revealing a geometric rigidity through a Heisenberg analogue of the cosine formula.

CR-invariant Energy Functional for Legendrian Knots in the Heisenberg Group

Introduction and Motivation

This paper develops a Möbius-invariant energy for Legendrian knots embedded in the three-dimensional Heisenberg group $\mathcal{H} = \mathbb{C} \times \mathbb{R}$, analogously to Möbius-invariant knot energies in $\mathbb{R}^3$. In the classical setting, O’Hara’s energy functional and its regularized $r^{-2}$-potential serve as a canonically minimizing geometric functional, with invariance under Möbius transformations—a property fundamental to geometric knot theory and its applications for canonical knot representatives. The present work extends this paradigm to the sub-Riemannian geometric context of the Heisenberg group, with a functional respecting the underlying CR geometry, and focuses on Legendrian knots (horizontal curves with respect to the standard contact structure).

Korányi Metric and CR Invariance

Central to this construction is the Korányi metric on $\mathcal{H}$. Unlike the Carnot–Carathéodory metric, the Korányi metric is essential for Möbius (CR) invariance: the Korányi distance $(p,q)$ ensures that the energy integrand $dp\,dq/(p,q)^2$ is invariant under the full CR automorphism group $\PU(2,1)$, which plays the role analogous to the Möbius transformations $\PO(3,1)$ in the classical setting.

Legendrian knots are understood as closed, smooth, horizontally immersed curves in $\mathcal{H}$, and the relevant notion of length, as well as all analytic and geometric properties, are based on the sub-Riemannian (contact) structure.

Definition and Analytic Regularization of the Energy

A direct integral of $dp\,dq/(p,q)^2$ over $\mathbb{R}^3$0 diverges due to singularities on the diagonal, paralleling the divergence in classical energy definitions. The authors employ a regularization schema utilizing the beta function

$\mathbb{R}^3$1

valid initially for $\mathbb{R}^3$2 and meromorphically continued with simple poles at negative odd integers. The energy is then defined as the value at $\mathbb{R}^3$3:

$\mathbb{R}^3$4

This matches the critical exponent for self-repulsiveness (diverges at double points) and Möbius invariance, established through a careful asymptotic expansion near the diagonal and residue computation. The residue at $\mathbb{R}^3$5 gives twice the length, at $\mathbb{R}^3$6 involves the integral of curvature squared of the vertical projection, matching the geometric hierarchy observed in classical knot energies.

Möbius Invariance and Properties

The functional $\mathbb{R}^3$7 is rigorously established to be $\mathbb{R}^3$8-invariant. The proof utilizes inversion symmetry in the Heisenberg group, showing that the potential function at the image point $\mathbb{R}^3$9 under inversion scales appropriately such that the integral of the energy over the transformed knot equals that of the original. This is combined with invariance under Heisenberg translations, dilations, and rotations, thereby confirming Möbius invariance within the CR-geometry framework.

The self-repulsiveness property is preserved; $r^{-2}$0 as the knot approaches a self-intersection, with lower bound zero.

Minimizers: $r^{-2}$1-circles

The energy attains its minimum ($r^{-2}$2) exactly for $r^{-2}$3-circles, which serve as the analogs of circles in $r^{-2}$4. $r^{-2}$5-circles are the boundaries of totally geodesic, totally real planes within the complex hyperbolic ball model and are explicit in Heisenberg coordinates as affine lines with prescribed lifts. Any Legendrian knot can be mapped via the CR automorphism group to a $r^{-2}$6-circle, reflecting the transitive action of $r^{-2}$7 on these minimizers.

Heisenberg Analogue of the Cosine Formula

The energy $r^{-2}$8 also admits a geometric interpretation via an analog of the Doyle–Schramm cosine formula. For each pair $r^{-2}$9, one associates an angle $\mathcal{H}$0 between specified $\mathcal{H}$1-circles tangent at these points. The energy is then expressed as

$\mathcal{H}$2

a strictly geometric/canonical form reinforcing the Möbius invariance and further revealing the underlying geometric structure of the functional.

If this angle vanishes for all $\mathcal{H}$3, the knot must be a $\mathcal{H}$4-circle, providing a geometric rigidity characterization of energy minimizers.

Geometric and Complex-Analytic Interpretation of the Integrand

Extending the interpretation of $\mathcal{H}$5 as the absolute value of the infinitesimal cross ratio in the classical setting, the paper constructs a (partial) analog in the Heisenberg group. The complex 2-form

$\mathcal{H}$6

(with $\mathcal{H}$7) is shown to be Möbius invariant (under $\mathcal{H}$8), and the integrand for energy is directly related to its modulus on the product of the contact planes. The argument of this 2-form recovers the angle $\mathcal{H}$9, analogous to the symplectic interpretation known for the Euclidean theory.

Moreover, a connection to the Kähler structure on the Siegel domain (complex hyperbolic space) is observed, with the constructed 2-form being related to the Kähler form functorially restricted to the boundary ($(p,q)$0).

Implications and Future Directions

This construction firmly grounds knot energies in sub-Riemannian CR geometry, opening pathways for analytic and topological applications akin to those in the Euclidean context. The energy’s minimizers and invariance properties suggest a geometric flow theory and variational perspective for Legendrian knots in $(p,q)$1 and more broadly for CR-structures, with potential implications in mathematical physics, geometric analysis, and several complex variables.

Further extension to higher-dimensional CR manifolds and more general sub-Riemannian settings, study of the functional’s gradient flow, and ramifications for Legendrian isotopy classification in the Heisenberg group are immediate areas for development.

Conclusion

The paper realizes a natural generalization of Möbius-invariant knot energy to Legendrian knots in the Heisenberg group, ensuring invariance under CR automorphisms through the Korányi metric. The minimization characterization, analytic regularization, Möbius invariance, and geometric interpretations position the energy as a fundamental functional in CR and sub-Riemannian knot theory, bridging classical geometric knot theory and CR/complex hyperbolic geometry.