Analysis Seminar -- Discrete uniformization via hyper-ideal circle patterns

11/20/2015 - 13:30
11/20/2015 - 14:30
Nikolay Dimitrov (Berlin)
Burnside Hall, Rm. 920 (McGill University)

In this talk I will present a discrete version of the classical uniformization theorem based on the theory of hyper-ideal circle patterns. It applies to surfaces represented as finite branched covers over the Riemann sphere as well as to compact polyhedral surfaces with non-positive cone singularities. The former include all Riemann surfaces realized as algebraic curves, and more generally, any closed Riemann surface with a choice of a meromorphic function on it. The latter include any closed Riemann surface with a choice of a quadratic differential on it. We show that for such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique (up to isometry). This kind of discrete uniformization is the result of an interplay between realization theorems for ideal (Rivin) and hyper-ideal (Bao and Bonahon) polyhedra in hyperbolic three-space, and their generalization to hyper-ideal circle patterns on surfaces with cone-singularities (Schlenker). We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.

Last edited by on Wed, 11/11/2015 - 10:16