#### Vol. 314, No. 1, 2021

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Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

### Robin Timsit

Vol. 314 (2021), No. 1, 233–251
##### 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

${M}_{4}\left(\Gamma \right)={\int }_{\Omega }\frac{|q{|}^{2}}{{\left({l}_{\Gamma }\right)}^{4}}\phantom{\rule{0.3em}{0ex}}d{L}^{3}.$

##### Keywords
quadratic differentials, moduli of curve families, Legendrian foliations, Heisenberg group, spherical CR manifolds
##### Mathematical Subject Classification
Primary: 32G07, 32V05