Measured foliations at infinity of quasi-Fuchsian manifolds

TL;DR

The paper analyzes the measured foliations at infinity $\lambda(M)=(\lambda^+(M),\lambda^-(M))$ of quasi-Fuchsian manifolds, proving that near the Fuchsian locus these pairs are filling and that every small multiple of any filling pair can be realized as $\lambda(M)$ for some $M$. It develops a precise analytic framework using harmonic Beltrami differentials, Bers embeddings, and a carefully constructed differential $\Phi(X,Y)$ to compare geometric data along Teichmüller geodesics, ultimately building a continuous deformation map $\mathbf{F}: \operatorname{L} \to \mathcal{MF}^2$ whose $t=0$ fiber is a homeomorphism onto the filling pair space. The results answer Schlenker’s questions near the Fuchsian locus and provide a concrete deformation theory tying conformal boundary data to boundary foliations at infinity via the Bers framework. The construction hinges on controlling the asymptotics of Bers data, Gardiner–Masur type parametrizations, and a degree-theoretic argument to extend local realizability to a neighborhood, with potential implications for renormalized volume variational formulas and the broader geometry of $\mathcal{QF}$.

Abstract

Let $(λ^+(M),λ^-(M))$ denote the pair of measured foliations at the boundary at infinity $\partial_\infty$ of a quasi-Fuchsian manifold $M$. We prove that $(λ^+(M),λ^-(M))$ is filling if $M$ is close to being Fuchsian. We also show that given any filling pair $(α_1,α_2)$ of measured foliations, and every small enough $t>0$, the pair $(tα_1,tα_2)$ is realised as the pair of measured foliations at infinity of some quasi-Fuchsian manifold $M$. This answers questions of Schlenker near the Fuchsian locus.

Theorems & Definitions (6)

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