Date of Award
Spring 2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Minsky, Yair
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 $T\se M$. We say a sequence of $\pi_1$-injective maps $f_i:S_i\to M$ of surfaces $S_i$ is \emph{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 $\nu_T$.
Recommended Citation
Al Assal, Fernando, "Limits of Asymptotically Fuchsian Surfaces in a Closed Hyperbolic 3-Manifold" (2023). Yale Graduate School of Arts and Sciences Dissertations. 1019.
https://elischolar.library.yale.edu/gsas_dissertations/1019
