Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold

Al Assal, Fernando

doi:10.2140/gt.2026.30.23

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ISSN (electronic): 1364-0380

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Limits of asymptotically Fuchsian surfaces in a closed hyperbolic $3$-manifold

Fernando Al Assal

Geometry & Topology 30 (2026) 23–70

DOI: 10.2140/gt.2026.30.23

Abstract

Let $M$ be a closed hyperbolic 3-manifold. Let $ν_{Gr M}$ denote the probability volume (Haar) measure of the 2-plane Grassmann bundle $Gr M$ of $M$ and let $ν_{T}$ denote the area measure on $Gr M$ of an immersed closed totally geodesic surface $T \subset M$ . We say a sequence of $π_{1}$ -injective maps $f_{i} : S_{i} \to M$ of surfaces $S_{i}$ is asymptotically Fuchsian if $f_{i}$ is $K_{i}$ -quasifuchsian with $K_{i} \to 1$ as $i \to \infty$ . We show that the set of weak- $*$ limits of the probability area measures induced on $Gr M$ by asymptotically Fuchsian minimal or pleated maps $f_{i} : S_{i} \to M$ of closed connected surfaces $S_{i}$ consists of all convex combinations of $ν_{Gr M}$ and the $ν_{T}$ .

Keywords

quasifuchsian surfaces, hyperbolic 3-manifolds, minimal surfaces, pleated surfaces, equidistribution, nonequidistribution, surface subgroup theorem, good pants, invariant measures, Ratner measure classification theorem

Mathematical Subject Classification

Primary: 22E40, 53A10, 57K32

References

Publication

Received: 9 November 2023

Revised: 11 September 2024

Accepted: 31 October 2024

Published: 19 January 2026

Proposed: Ian Agol

Seconded: David Fisher, Mladen Bestvina

Authors

Fernando Al Assal
Department of Mathematics University of Wisconsin-Madison Madison, WI United States

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Volume 30, issue 1 (2026)