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.

48 pages, 1 figure; improved and shortened some arguments

Cited by in corpus (2)

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