Hyperbolic manifolds with convex boundary

• 成果类型: Article
• 署名作者: Schlenker, JM
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0456-x
• 发表日期: 2006
• 页码: 109-169
• 关键词: holomorphic-curves; constant curvature; einstein manifolds; ideal polyhedra; SURFACES; 3-space; SPACE; RIGIDITY; laminations; dimension

摘要

Let (M, partial derivative 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 partial derivative M are exactly the metrics with curvature K < 1, for which the closed geodesics which are contractible in M 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 partial derivative M is a linear combination of the first, second and third fundamental forms.