Measured foliations at infinity of quasi-Fuchsian manifolds near the Fuchsian locus

Abstract

Given a pair of measured foliations (F + , F − ) which fill a closed hyperbolic surface S, we show that for t > 0 sufficiently small, tF + and tF − can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian hyperbolic 3-manifold homeomorphic to S × R, which is in a suitably small neighbourhood of the Fuchsian locus. This is parallel to a theorem of Bonahon that partially answers a conjecture of Thurston by proving that a quasi-Fuchsian manifold close to being Fuchsian can be uniquely determined by the data of measured bending laminations on the boundary of its convex core. We also give an interpretation of our result in half-pipe geometry using intermediate steps in the proof.