TY - JOUR
T1 - POLYHEDRAL APPROXIMATION OF METRIC SURFACES AND APPLICATIONS TO UNIFORMIZATION
AU - Ntalampekos, Dimitrios
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2023 Duke University Press. All rights reserved.
PY - 2023/6/15
Y1 - 2023/6/15
N2 - 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.
AB - 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.
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U2 - 10.1215/00127094-2022-0061
DO - 10.1215/00127094-2022-0061
M3 - Article
AN - SCOPUS:85174292036
SN - 0012-7094
VL - 172
SP - 1673
EP - 1734
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 9
ER -