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

TL;DR

This paper demonstrates that small measured foliations at infinity on quasi-Fuchsian manifolds near the Fuchsian locus can be uniquely realized, extending the analogy with measured bending laminations on convex core boundaries.

Contribution

It establishes a unique realization of scaled arational measured foliations at infinity for quasi-Fuchsian manifolds close to the Fuchsian locus, linking boundary data to geometric structures.

Findings

Unique realization of scaled measured foliations at infinity.

Extension of Bonahon's results to foliations at infinity.

Interpretation within half-pipe geometry.

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