Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.08521 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916867467116544 |
|---|---|
| author | Mesbah, Abderrahim |
| author_facet | Mesbah, Abderrahim |
| contents | Let $S$ be an oriented closed surface of genus at least two, and let $M = S \times (0,1)$. Suppose that $h$ is a Riemannian metric on $S$ with curvature strictly greater than $-1$, $h^{*}$ is a Riemannian metric on $S$ with curvature strictly less than $1$, and every contractible closed geodesic with respect to $h^{*}$ has length strictly greater than $2π$. Let $μ$ be a measured lamination on $S$ such that every closed leaf has weight strictly less than $π$. Then, we prove the existence of a convex hyperbolic metric $g$ on the interior of $M$ that induces the Riemannian metric $h$ (respectively $h^{*}$) as the first (respectively third) fundamental form on $S \times \left\{ 0\right\}$ and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $μ$. This statement remains valid even in limiting cases where the curvature of $h$ is constant and equal to $-1$. Additionally, when considering a conformal class $c$ on $S$, we show that there exists a convex hyperbolic metric $g$ on the interior of $M$ that induces $c$ on $S \times \left\{ 0\right\}$, which is viewed as one component of the ideal boundary at infinity of $(M,g)$, and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $μ$. Our proof differs from previous work by Lecuire for these two last cases. Moreover, when we consider a lamination which is small enough, in a sense that we will define, and a hyperbolic metric, we show that the metric on the interior of $M$ that realizes these data is unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08521 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The induced metric and bending lamination on the boundary of convex hyperbolic 3-manifolds Mesbah, Abderrahim Geometric Topology Let $S$ be an oriented closed surface of genus at least two, and let $M = S \times (0,1)$. Suppose that $h$ is a Riemannian metric on $S$ with curvature strictly greater than $-1$, $h^{*}$ is a Riemannian metric on $S$ with curvature strictly less than $1$, and every contractible closed geodesic with respect to $h^{*}$ has length strictly greater than $2π$. Let $μ$ be a measured lamination on $S$ such that every closed leaf has weight strictly less than $π$. Then, we prove the existence of a convex hyperbolic metric $g$ on the interior of $M$ that induces the Riemannian metric $h$ (respectively $h^{*}$) as the first (respectively third) fundamental form on $S \times \left\{ 0\right\}$ and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $μ$. This statement remains valid even in limiting cases where the curvature of $h$ is constant and equal to $-1$. Additionally, when considering a conformal class $c$ on $S$, we show that there exists a convex hyperbolic metric $g$ on the interior of $M$ that induces $c$ on $S \times \left\{ 0\right\}$, which is viewed as one component of the ideal boundary at infinity of $(M,g)$, and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $μ$. Our proof differs from previous work by Lecuire for these two last cases. Moreover, when we consider a lamination which is small enough, in a sense that we will define, and a hyperbolic metric, we show that the metric on the interior of $M$ that realizes these data is unique. |
| title | The induced metric and bending lamination on the boundary of convex hyperbolic 3-manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2306.08521 |