Hyperbolic manifolds with convex boundary

Details Hyperbolic manifolds with convex boundary Hyperbolic manifolds with convex boundary

2002-05-29

arxiv.org/abs/math/0205305v5

Inventiones math. 163(2006):1, 109-169.

Mathematics

Differential Geometry

Check the updated version(s) on http://picard.ups-tlse.fr/~schlenker/ Version 2: an error corrected. Version 3: simpler main s

Scientific paper

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.

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