Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

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