arXiv subjects

Robin Timsit

Publications and source records attributed to Robin Timsit.

3 recordsLinked to original sources

arXiv2020

Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

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 \[ M_4 (\Gamma) = \int_\Omega \frac{|q|^2}{(l_\Gamma) ^4} \mathrm{d} L^3.\]

math.DG↗

arXiv2018

Quadratic differentials in spherical CR geometry

This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on quadratic differentials that lead to analogue of half-translation structures on spherical CR manifolds. Finally, we work on known examples of quasiconformal maps in the Heisenberg group with extremal properties and explicit how quadratic differentials are involved in those. In addition, on our way to quadratic differentials, we define a differential complex on strictly pseudoconvex CR manifolds with a finite dimensional cohomology space. It leads to a new CR invariant that we compute for compact manifolds endowed with a CR action of the circle.

math.DG↗

arXiv2016

Geometric construction of quasiconformal mappings in the Heisenberg group

In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group H that minimise a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincar\'e half-plane $\mathbb H$. The first map we construct is a quasiconformal map between two cylinders. We explain the method used to find it and prove its uniqueness up to rotations. Then, we give geometric conditions for the construction to be the only way to find such minimizers. Eventually, as a non trivial example of the generalisation, we manage to reconstruct the map from \cite{BFP} between two spherical annuli.

math.DG↗