The induced metric and bending lamination of the boundary of convex hyperbolic 3-manifolds with boundary
| Autor: | Mesbah, Abderrahim |
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| Rok vydání: | 2023 |
| Předmět: | |
| DOI: | 10.48550/arxiv.2306.08521 |
| Popis: | Let $S$ be a closed hyperbolic surface and $M = S \times (0,1)$. Suppose $h$ 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 $2\pi$. Let $L$ be a measured lamination on $S$ such that every closed leaf has weight strictly less than $\pi$. 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 $L$. 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 one component of the boundary at infinity of $(M,g)$, and induces a pleated surface structure on $S \times \left\{1\right\}$ with bending lamination $L$. Our proof differs from previous work by Lecuire for these two last cases. Comment: 22 pages, 4 figures. Comments are welcome! |
| Databáze: | OpenAIRE |
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