Polyhedral approximation of metric surfaces and applications to uniformization
Dimitrios Ntalampekos, Matthew Romney
Abstract
We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general “one-sided” quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk–Kleiner theorem characterizing Ahlfors 2-regular quasispheres.
Topics & Concepts
Uniformization (probability theory)MathematicsUniformization theoremHausdorff measureHausdorff spaceHausdorff distanceMetric spaceLimitingPure mathematicsManifold (fluid mechanics)Boundary (topology)Metric (unit)Limit (mathematics)Surface (topology)Space (punctuation)Riemann surfaceMeasure (data warehouse)Mathematical analysisHausdorff dimensionGeometryGeometric function theoryRiemann–Hurwitz formulaMechanical engineeringMarkov chainEconomicsComputer scienceEngineeringMarkov modelOperations managementBalance equationStatisticsDatabasePhilosophyLinguisticsAnalytic and geometric function theoryGeometric Analysis and Curvature FlowsGeometric and Algebraic Topology