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Circle packings and conformal approximations. (Empilements de cercles et approximations conformes.) (French) Zbl 0932.52004

Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 253-272 (1996).
In the beginning of the article, the authors – starting from a holomorphic immersion \(f\) of a compact region of the plane and using the standard hexagonal circle packing, deforming each border circle \(C(z_{k},\frac{1}{2N})\) by an amount of \(|f'(z_{k})|\) – construct new circle packings.
Using this packing they build for every positive integer a piecewise affine map \(f_N\). The main result is that the sequence \((f_{N})_{N\geq 1}\) is converging in \(C^1\)-topology to \(f\). Also is given an estimate of packing circles radii \[ r_{N}(z)=\frac{f'(z_{k})}{2N}+O\biggl(\frac{1}{N^2}\biggr) \] uniformly when \(N\rightarrow \infty\).
The proof is based on the variational approach previously developed by the first author [Y. Colin de Verdière, Invent. Math. 104, No. 3, 655-669 (1991; Zbl 0745.52010)].
In the special case of \(f(z)=\exp(az)\) there is an exact formula for the radii.
For the entire collection see [Zbl 0859.00016].

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
30E10 Approximation in the complex plane

Citations:

Zbl 0745.52010
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