Inventiones mathematicae

, Volume 163, Issue 1, pp 109–169

Hyperbolic manifolds with convex boundary


DOI: 10.1007/s00222-005-0456-x

Cite this article as:
Schlenker, JM. Invent. math. (2006) 163: 109. doi:10.1007/s00222-005-0456-x


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.

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Laboratoire Emile Picard, UMR CNRS 5580, UFR MIGUniversité Paul SabatierToulouse Cedex 4France

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