Hyperbolic manifolds with convex boundary |
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Authors: | Jean-Marc Schlenker |
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Institution: | (1) Laboratoire Emile Picard, UMR CNRS 5580, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France |
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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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Keywords: | |
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