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

Abstract

Measured foliations at infinity of quasi-Fuchsian manifolds are a natural analog at infinity to the measured bending laminations on the boundary of its convex core. We show that given a pair of arational measured foliations $({\mathsf{F}}_{+},{\mathsf{F}}_{-})$ which fill a closed hyperbolic surface $S$, for $t>0$ sufficiently small, $t{\mathsf{F}}_{+}$ and $t{\mathsf{F}}_{-}$ can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian manifold homeomorphic to $S\times \mathbb{R}$, which is sufficiently close to the Fuchsian locus. Here arationality means that the corresponding measured laminations are maximal. The proof is inspired by Bonahon's in\cite{bonahon05} which shows that a quasi-Fuchsian manifold close to the Fuchsian locus can be uniquely determined by the data of filling measured bending laminations on the boundary of its convex core. We also interpret the result in half-pipe geometry

arXiv commentRevised version with some modifications