Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

TL;DR

This paper establishes a formula for the modulus of Legendrian curve families in the Heisenberg group using quadratic differentials and horizontal trajectories, linking geometric structures with analytical tools.

Contribution

It proves a new relation between the modulus of Legendrian foliations and quadratic differentials in the Heisenberg group, extending geometric analysis in sub-Riemannian settings.

Findings

Derived an explicit integral formula for the modulus of Legendrian curve families.

Connected quadratic differentials to the geometry of Legendrian foliations.

Provided a new analytical tool for studying curve families in the Heisenberg group.

Abstract

The aim of the paper is to prove the following result concerning moduli of curve families in the Heisenberg group. Let $\Omega$ be a domain in the Heisenberg group foliated by a family $\Gamma$ of legendrian curves. Assume that there is a quadratic differential $q$ on $\Omega$ in the kernel of an operator defined in \cite{Tim2} and every curve in $\Gamma$ is a horizontal trajectory for $q$. Let $l_\Gamma : \Omega \rightarrow ]0,+\infty[$ be the function that associates to a point $p\in \Omega$, the $q$-length of the leaf containing $p$. Then, the modulus of $\Gamma$ is \[ M_4 (\Gamma) = \int_\Omega \frac{|q|^2}{(l_\Gamma) ^4} \mathrm{d} L^3.\]