CR-invariant energy of Legendrian knots in the Heisenberg group

TL;DR

This paper introduces a new energy functional for Legendrian knots in the Heisenberg group, invariant under PU(2,1), with minimizers characterized as R-circles and connections to classical knot energies.

Contribution

It defines a PU(2,1)-invariant energy for Legendrian knots in the Heisenberg group, characterizes minimizers, and relates the energy to complex 2-forms and classical invariants.

Findings

R-circles minimize the energy functional

Established a Heisenberg analog of the Doyle--Schramm cosine formula

Expressed the energy integrand via a complex 2-form

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.