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

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 ${\nu }_T$.

51 pages, 19 figures. Second version has many small corrections and improvements. Accepted to Geometry and Topology