2001/11/12 by Jean-Marc Schlenker, Jean‐Marc Schlenker, Schlenker, Jean-Marc · 2 citations
Mathematics · Physics and Astronomy · #53C45 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum chaos and dynamical systems #math.DG #math.GT #msc:53C45
paper · pdf · doi:10.48550/arxiv.math/0111136
Updated version on http://picard.ups-tlse.fr/~schlenker/texts/papers.html New version: several typos corrected, a few remarks added
openalex publication_date 2001/11/12 · arxiv created 2002/09/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M, ∂ M) be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric g on M such that \dr M is smooth and strictly convex, the induced metric on \dr M has curvature K>-1, and each such metric on \dr M is obtained for a unique choice of g. A dual statement is that, for each g as above, the third fundamental form of \dr M has curvature K<1, and its closed geodesics which are contractible in M have length L>2π. Conversely, any such metric on \dr M is obtained for a unique choice of g. We are interested here in the similar situation where ∂ M is not smooth, but rather looks locally like an ideal polyhedron in H3. We can give a fairly complete answer to the question on the third fundamental form -- which in this case concerns the dihedral angles -- and some partial results about the induced metric. This has some by-products, like an affine piecewise flat structure on the Teichmueller space of a surface with some marked points, or an extension of the Koebe circle packing theorem to many 3-manifolds with boundary.