Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911387366719488 |
|---|---|
| author | Assal, Fernando Al |
| author_facet | Assal, Fernando Al |
| contents | 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 $ν_T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02164 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold Assal, Fernando Al Geometric Topology Differential Geometry Dynamical Systems 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 $ν_T$. |
| title | Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold |
| topic | Geometric Topology Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2309.02164 |