Measured foliations at infinity of quasi-Fuchsian manifolds close to the Fuchsian locus

Given a closed, oriented surface with genus greater that 2, we study quasi-Fuchsian hyperbolic 3-manifolds homeomorphic to this surface times the interval. Different properties of these manifolds have been carefully studied in previous important works on 3 manifold geometry and topology and some interesting questions about them still remain to be answered.

In this talk, we will focus on a new geometric invariant associated to them which we call the measured foliations at infinity. These are horizontal measured foliations of a holomorphic quadratic differential ( the Schwarzian derivative )  associated canonically with each of the two connected component of the boundary at infinity of a quasi-Fuchsian manifold. We ask whether given any pair of measured foliations (F,G) on a surface, is there a quasi-Fuchsian manifold with F and G as it measured foliations at infinity. The answer is affirmative under certain assumptions; first, (F,G) satisfy the property of being an « arational filling pair » and second, the quasi-Fuchsian manifold should be very close to being « Fuchsian » . The goal of this talk would be introducing the concepts and outlining the proof idea.