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In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf

Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same result that did not rely on the Riemann mapping theorem.

Start with a simply connected domain, fill it with coins making an hexagonal pattern (or any other packing). Add an extra vertex to their intersection graph, that will correspond to the boundary. Now apply the Koebe-Andreev-Thurston theorem to this planar graph. This defines a map from the centres of the coins to the centres of some circle packing, filling in each triangle by affine maps gives a map that converges to a conformal one.

See https://en.wikipedia.org/wiki/Circle_packing_theorem for details.

Can this approach be generalised to to obtain the full uniformization theorem? https://en.wikipedia.org/wiki/Uniformization_theorem

Is there some discouraging reason why this might be false or hard?

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    $\begingroup$ "Is there some discouraging reason why this might be false or hard?" Well, Dennis didn't prove it. $\endgroup$
    – Silvia Ghinassi
    Commented Nov 11, 2015 at 21:50
  • $\begingroup$ One problem is this: what is a "circle" on an arbitrary Riemann surface? $\endgroup$
    – Alexandre Eremenko
    Commented Nov 11, 2015 at 22:25

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Firstly, the Rodin-Sullivan argument can not, in principle, be used to give a proof of uniformization, since it uses uniformization of domains of infinite type (due to Marden, if I recall) as an ingredient.

Secondly, there are many circle-packing approaches to uniformization, the first of them given by G. Leibon, using variational ideas of Y. Colin de Verdiere and of I. Rivin, see

MR1903777 (2003d:57038) Reviewed 
Leibon, Gregory(1-DTM)
Random Delaunay triangulations and metric uniformization. 
Int. Math. Res. Not. 2002, no. 25, 1331–1345

Leibon's argument is philosophically (and a bit more than that) very similar to the argument of Osgood-Phillips-Sarnak (who do things by maximizing log det of the laplacian).

Since then, there has been a mini-industry in approximating conformal maps by circle packing maps (based on the same variational ideas), where the actors are G. Brock Williams, A. Bobenko, B. Springborn, F. Luo, B. Chow, and others. A mathscinet search for these names will reveal much.

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  • $\begingroup$ Thanks. My question was not what I wrote :) $\endgroup$
    – Alfredo Hubard
    Commented Nov 12, 2015 at 8:23
  • $\begingroup$ I corrected the implicit mistake, the answer still looks adequate. $\endgroup$
    – Alfredo Hubard
    Commented Nov 12, 2015 at 8:40

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