Combinatorial Ricci flows for ideal circle patterns
In his well-known lecture notes, \textit{W. Thurston} [Geometry and topology of 3-manifolds. Princeton: Princeton University Press (1976)] describes a method to show existence and uniqueness theorems about circle packings. Roughly speaking, given a cellular decomposition \(\mathcal{D}\) of a surface, the question is to find a constant curvature metric for that surface, and a set of circles any two of whose are either tangent or disjoint, such that the circles are in one-to-one correspondence with the vertices of \(\mathcal{D}\) and two of them are tangent if and only if the corresponding vertices of \(\mathcal{D}\) are adjacent. It has been shown by \textit{B. Chow} and \textit{F. Luo} [J. Differ. Geom. 63, 97--129 (2003; Zbl 1070.53040)] that Thurston's circle packing metrics can be recovered as limits of the solutions to a combinatorial analogue of Hamilton's Ricci flow. If the circles are allowed to intersect in two points (in which case they form a circle pattern instead of a circle packing), the way two circles intersect can be set by assigning their intersection angle to the corresponding edge of \(\mathcal{D}\). The same question as with circle packings can then be asked in this case, where the edges of \(\mathcal{D}\) are weighted with a prescribed set of angles. Existence and uniqueness theorems for circle patterns have been provided by \textit{A. Bobenko} and \textit{B. Springborn} [Trans. Am. Math. Soc. 356, 659--689 (2004; Zbl 1044.52009)] for a wide range of surfaces, extending a result by \textit{I. Rivin} [Ann. Math. 143, 51--70 (1996; Zbl 0874.52006)] in the spherical case. Here, the authors use the approach of Chow and Luo in the case of circle patterns and give necessary and sufficient conditions for the convergence of the solutions to the combinatorial Ricci flow. In particular, they recover the results by Rivin and by Bobenko and Springborn from the limit of the Ricci flow.
scientific article; zbMATH DE number 7337387
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