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

Diptaishik Choudhury

Published 2021-11-02, updated 2022-02-12Version 3

Measured foliations at infinity of a quasi-Fuchsian hyperbolic $3$-manifold are the horizontal measured foliations of the Schwarzian derivatives of the Uniformisation maps associated with the connected components of the boundary at infinity. Given a pair of measured foliations $(\mathsf{F}_{+},\mathsf{F}_{-})$ which fill a closed hyperbolic surface $S$, we show that 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 hyperbolic $3$-manifold homeomorphic to $S\times\mathbb{R}$, which is in a suitably small neighbourhood of the Fuchsian locus. This is parallel to a theorem of Bonahon which proves that a quasi-Fuchsian manifold close to being Fuchsian can be uniquely determined by the data of filling measured bending laminations on the boundary of its convex core. We also interpret our result in half-pipe geometry.