Asian Journal of Mathematics

Volume 23 (2019)

Number 1

Convergence of discrete conformal geometry and computation of uniformization maps

Pages: 21 – 34



David Gu (Department of Computer Science, Stony Brook University, Stony Brook, New York, U.S.A.)

Feng Luo (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Tianqi Wu (Courant Institute of Mathematics, New York University, New York, N.Y., U.S.A.)


The classical uniformization theorem of Poincaré and Koebe states that any simply connected surface with a Riemannian metric is conformally diffeomorphic to the Riemann sphere, or the complex plane or the unit disk. Using the work by Gu–Luo–Sun–Wu on discrete conformal geometry for polyhedral surfaces, we show that the uniformization maps for simply connected Riemann surfaces are computable.


polyhedral surfaces, discrete conformal geometry, uniformizations and convergences

2010 Mathematics Subject Classification

Primary 57Q15. Secondary 30F10, 30G25, 52B70.

Received 25 April 2017

Accepted 27 June 2017

Published 3 May 2019