Polyhedral approximation of metric surfaces and applications to uniformization

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.

arXiv comment48 pages, 1 figure; improved and shortened some arguments

References in corpus (8)

• Quasisymmetric parametrizations of two-dimensional metric spheres
• Quasisymmetric parameterizations of two-dimensional metric planes
• Area minimizing discs in metric spaces
• Canonical parametrizations of metric discs
• Rigidity theorems for circle domains
• On the inverse absolute continuity of quasiconformal mappings on hypersurfaces
• Local characterization of polyhedral spaces
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Cited by in corpus (2)

• Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces
• Conformal uniformization of planar packings by disk packings