POLYHEDRAL APPROXIMATION OF METRIC SURFACES AND APPLICATIONS TO UNIFORMIZATION

Dimitrios Ntalampekos, Matthew Romney

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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 "onesided" 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.

Original languageEnglish
Pages (from-to)1673-1734
Number of pages62
JournalDuke Mathematical Journal
Volume172
Issue number9
DOIs
StatePublished - 15 Jun 2023

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