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.