Volume 26, issue 3 (2022)

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Discrete conformal geometry of polyhedral surfaces and its convergence

Feng Luo, Jian Sun and Tianqi Wu

Geometry & Topology 26 (2022) 937–987
Abstract

We prove a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin and Sullivan’s theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin and Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.

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Keywords
polyhedral metrics, discrete conformal map, Riemann mapping, triangulation, Delaunay triangulation, convex polyhedra, discrete harmonic functions
Mathematical Subject Classification 2010
Primary: 52C26, 53C44, 58E30
References
Publication
Received: 8 April 2018
Revised: 28 December 2020
Accepted: 8 February 2021
Published: 3 August 2022
Proposed: David Gabai
Seconded: Tobias H Colding, Dmitri Burago
Authors
Feng Luo
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Jian Sun
Mathematical Science Center
Tsinghua University
Beijing
China
Tianqi Wu
Center of Mathematical Sciences and Applications
Harvard University
Cambridge, MA
United States