Hyperbolic manifolds with convex boundary

Abstract

Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature $K>-1$, and that the third fundamental forms of $\dr M$ are exactly the metrics with curvature $K<1$, for which contractible closed geodesics have length $L>2\pi$. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on $\dr M$ is a linear combination of the first, second and third fundamental forms.Comment: Check the updated version(s) on http://picard.ups-tlse.fr/~schlenker/ Version 2: an error corrected. Version 3: simpler main statement, small corrections, more details on one technical statement. Version 5: one error correcte

Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

“…In this case we obtain a Fuchsian manifold with convex polyhedral boundary, and all the Fuchsian manifolds with convex polyhedral boundary can be obtained in this way: the lifting to the universal cover of a component of the boundary of the Fuchsian manifold gives a convex Fuchsian polyhedron in the hyperbolic space. Then Theorem 1.1 says exactly that for a choice of the metric h on the boundary, there exists a unique metric on the manifold such that it is a Fuchsian manifold with convex polyhedral boundary and the induced metric on the boundary is isometric to h: The statement of Conjecture 1.4 in the case where the boundary is smooth and strictly convex as been proved in [21] (the existence part was found in [9]). Remark that A.D. Alexandrov's theorem is a part of Conjecture 1.4 for the case of the hyperbolic ball.…”

Hyperbolization of cusps with convex boundary

“…See also [Sch06], which contains a uniqueness result. Of course, one can take for metrics m in the statement of Theorem 1.12 hyperbolic metrics on S with conical singularities of positive curvature [Slu14].…”

Notes on a paper of Mess

“…In particular, this is the case for the analysis of [31], and also for [93] (for which however the proof has to be adapted and the dictionary between quasi-Fuchsian and anti-de Sitter manifolds has to be extended a little).…”