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Hyperbolic manifolds with convex boundary

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Abstract

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. 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 ∂M are exactly the metrics with curvature K<1, for which the closed geodesics which are contractible in M have length L>2π. Each is obtained exactly once.

Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on ∂M is a linear combination of the first, second and third fundamental forms.

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  1. Laboratoire Emile Picard, UMR CNRS 5580, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 4, France

    Jean-Marc Schlenker

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Schlenker, JM. Hyperbolic manifolds with convex boundary. Invent. math. 163, 109–169 (2006). https://doi.org/10.1007/s00222-005-0456-x

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