Topology Seminar for 3/27/2003

Jean-Marc Schlenker
Université Paul Sabatier

Hyperbolic Manifolds with Convex Boundary

Let M be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. One can describe the hyperbolic metrics on M for which the boundary is smooth and strictly convex.

Theorem A: the induced metrics have curvature K>-1, and each is obtained for a unique hyperbolic metric on M.

Theorem B: the third fundamental forms of the boundary have curvature K<1, and their closed geodesics which are contractible in M have length L>2\pi. Each is obtained for a unique hyperbolic metric on M.

Theorem B has analogs when the boundary is supposed to look locally like an ideal or a hyperideal polyhedron. As a consequence, we find an extension of the Koebe circle packing theorem when the sphere is replaced by the boundary of M.