Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

2010/11/13 by François Fillastre, Fillastre, François
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.DG

paper · pdf · doi:10.48550/arxiv.1011.3123

Survey paper. No proof. 10 pages

arxiv created 2010/11/13 · openalex publication_date 2010/11/13 · arxiv updated 2010/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface S can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when S is the sphere are classical.

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