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.
Citation
Dimitrios Ntalampekos. Matthew Romney. "Polyhedral approximation of metric surfaces and applications to uniformization." Duke Math. J. 172 (9) 1673 - 1734, 15 June 2023. https://doi.org/10.1215/00127094-2022-0061
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