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

TL;DR

This paper investigates the limiting behavior of measures induced by sequences of nearly Fuchsian surface maps in a closed hyperbolic 3-manifold, showing they converge to convex combinations of uniform and totally geodesic surface measures.

Contribution

It characterizes all possible weak-* limit measures of asymptotically Fuchsian surface maps in hyperbolic 3-manifolds, extending understanding of surface geometry in these spaces.

Findings

Limits include all convex combinations of Haar measure and totally geodesic surface measures.

Asymptotically Fuchsian maps approximate Fuchsian representations as their quasifuchsian constant approaches 1.

The result describes the measure-theoretic boundary of surface embeddings in hyperbolic 3-manifolds.

Abstract

Let $M$ be a closed hyperbolic 3-manifold. Let $\nu_{Gr(M)}$ denote the probability volume (Haar) measure of the 2-plane Grassmann bundle $Gr(M)$ of $M$ and let $\nu_T$ denote the area measure on $Gr(M)$ of an immersed closed totally geodesic surface $T\subset M$. We say a sequence of $\pi_1$-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 $\nu_{Gr(M)}$ and the $\nu_T$.