Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

Robin Timsit

Vol. 314 (2021), No. 1, 233–251

DOI: 10.2140/pjm.2021.314.233

Abstract

Our aim 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 $\Omega$ such that every curve in $\Gamma$ is a horizontal trajectory for $q$. Let $l_{\Gamma }:\Omega \to$]$0,+\infty$[ 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

$$M_4\left(\Gamma \right)={\int \; <!--nolimits-->}_{\Omega }\frac{|q{|}^2}{{\left(l_{\Gamma }\right)}^4}d{L}^3.$$

Keywords

quadratic differentials, moduli of curve families, Legendrian foliations, Heisenberg group, spherical CR manifolds

Mathematical Subject Classification

Primary: 32G07, 32V05

Milestones

Received: 4 February 2021

Revised: 12 July 2021

Accepted: 16 July 2021

Published: 15 October 2021