2002/05/31 by Jean-Marc Schlenker · 32 citations
Mathematics · #Analytic and geometric function theory #Boundary (topology) #Contractible space #Curvature #Geodesic #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Hyperbolic 3-manifold #Hyperbolic manifold #Metric (unit) #Regular polygon #Uniqueness #math.DG #math.GT #msc:53C45 #msc:57M50 #msc:58J32
paper · pdf · doi:10.1007/s00222-005-0456-x
published in Inventiones mathematicae 163(1), 109-169 (Springer Science+Business Media) · Check the updated version(s) on http://picard.ups-tlse.fr/~schlenker/ Version 2: an error corrected. Version 3: simpler main statement, small corrections, more details on one technical statement. Version 5: one error corrected
arxiv created 2004/08/18 · openalex publication_date 2005/07/18 · arxiv updated 2015/06/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let (M, ∂ 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>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 \dr M is a linear combination of the first, second and third fundamental forms.