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The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing.

However I have seen a comment that circle packing on surfaces and Ricci flow on surfaces are related. The circle packing here is an extension of circle packing to include intersection angles between the circles with a particular choice for these angles. My initial question is to ask for an explanation of this.

My real question should now be apparent. There is an extension of Ricci flow to three dimensions: so is there some version of circle packing in three dimensions which can be interpreted as a combinatorial version of Ricci flow?

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This is a big open problem. See for example:… – Ryan Budney May 22 '11 at 5:19
D. Glickenstein has done some work that might interest you, "Combinatorial Yamabe flow in three dimensions" – Ramsay May 22 '11 at 15:45
up vote 14 down vote accepted

Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popular variational approach (though the two approaches intersect nontrivially).

MR0127372 (23 #B418) Regge, T. General relativity without coordinates. (Italian summary) Nuovo Cimento (10) 19 1961 558–571.

MR1393382 (97k:52022) Cooper, Daryl(1-UCSB); Rivin, Igor(4-WARW-MI) Combinatorial scalar curvature and rigidity of ball packings. Math. Res. Lett. 3 (1996), no. 1, 51–60.

MR2136536 (2006a:53081) Glickenstein, David A maximum principle for combinatorial Yamabe flow. Topology 44 (2005), no. 4, 809–825. (Reviewer: Igor Rivin), 53C44 (52C15)

MR2136535 (2005k:53108) Glickenstein, David A combinatorial Yamabe flow in three dimensions. Topology 44 (2005), no. 4, 791–808. (Reviewer: Igor Rivin), 53C44 (52C15)

MR2100762 (2005m:53122) Luo, Feng Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6 (2004), no. 5, 765–780. (Reviewer: Igor Rivin), 53C44 (53C21)

MR2015261 (2005a:53106) Chow, Bennett; Luo, Feng Combinatorial Ricci flows on surfaces. J. Differential Geom. 63 (2003), no. 1, 97–129. (Reviewer: Igor Rivin), 53C44

arXiv:1010.4070 [pdf, ps, other] Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric Xianfeng David Gu, Ren Guo, Feng Luo, Wei Zeng

arXiv:1005.4648 [pdf, other] Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow W. Zeng, L.M. Lui, F. Luo, J.S. Liu T.F. Chan, S.T. Yau, X.F. Gu

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A considerable inspiration for the 2003 paper by Feng Luo and myself was in fact the 1996 paper by Daryl Cooper and Igor Rivin. – Bennett Chow Oct 15 '13 at 5:51

Not yet mentioned is the interesting definition of Ricci curvature by Yann Ollivier, a definition especially suited to discrete spaces, such as graphs. His definition "can be used to define a notion of 'curvature at a given scale' for metric spaces." For example, he shows how the discrete cube $\{ 0,1 \}^n$ behaves like $\mathbb{S}^n$ in having constant positive curvature, and possessing an analog of the Lévy "concentration of measure" (the mass of $\mathbb{S}^n$ is concentrated about its equator).

His definition is used in the recent (April, 2011) paper by Jürgen Jost and Shiping Liu: "Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs."

Here are two primary sources:

Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), No. 3, 810-864.

Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains, in Probabilistic approach to geometry, 343-381, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010.

Update (7Feb13). Noticed this recent posting to the arXiv:

Warner A. Miller, Jonathan R. McDonald, Paul M. Alsing, David Gu, Shing-Tung Yau, "Simplicial Ricci Flow," arXiv:1302.0804 [math.DG].

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This last paper by Miller et al. appears to be dimension-agnostic. In particular, a discrete curvature can be defined at any $(n-2)$-cell in a piecewise-flat simplicial manifold simply by considering the area of its dual cell. – Vidit Nanda Feb 8 '13 at 19:17

The following paper

F. Luo, A combinatorial curvature flow for compact 3-manifolds with boundary, (now published in electronic research announcements, AMS, Volume 11, Pages 12--20)

provides a combinatorial flow for 3-manifolds with boundary consisting of surfaces with negative Euler characteristic. It deals with the convergence of an initial "piecewise-hyperbolic" metric to an actual hyperbolic metric with geodesic boundary. The analogy with Ricci flow is very very mild, but I hope you may be interested in this reference anyway.

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