On Riemann-Hilbert problems in circle packing. (English) Zbl 1179.30038
Summary: We propose a discrete counterpart of non-linear boundary value problems for holomorphic functions (Riemann-Hilbert problems) in the framework of circle packing. For packings with simple combinatorial structure and circular target curves appropriate solvability conditions are given and the set of all solutions is described. We compare the discrete and the continuous setting and discuss several discretization effects. In the last section we indicate promising directions for further research and report on the results of some test calculations which show that solutions of the circle packing problem approximate the classical solutions surprisingly well.
MSC:
| 30E25 | Boundary value problems in the complex plane |
| 52C26 | Circle packings and discrete conformal geometry |
| 30C35 | General theory of conformal mappings |
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
Software:
CirclePackReferences:
| [1] | D. Bauer, K. Stephenson and E. Wegert, Circle packings as differentiable manifolds, in preparation. · Zbl 1258.52011 |
| [2] | Ch. Glader and E. Wegert, Nonlinear Riemann-Hilbert problems with circular target curves, Math. Nachr. 281 No.9 (2008), 1221–1239. · Zbl 1230.30027 · doi:10.1002/mana.200710673 |
| [3] | Z.-X. He and O. Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), 285–305. · Zbl 0868.30010 · doi:10.1007/s002220050076 |
| [4] | P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse 88 (1936), 141–164. · Zbl 0017.21701 |
| [5] | B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, in: R. Dedekind and H. Weber (eds.), Gesammelte mathematische Werke und wissenschaftlicher Nachlaß, Leipzig 1876, 3–47. |
| [6] | B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geometry 26 (1987), 349–360. · Zbl 0694.30006 |
| [7] | A. Shnirel’man, The degree of a quasi-linearlike mapping and the nonlinear Hilbert problem, (in Russian), Mat. Sb. 89 (1972) 3, 366–389; English transl.: Math. USSR Sbornik 18 (1974), 373–396. |
| [8] | K. Stephenson, Introduction to Circle Packing – The Theory of Discrete Analytic functions, Cambridge Univ. Press, Cambridge 2005. · Zbl 1074.52008 |
| [9] | K. Stephenson, A probabilistic proof of Thurston’s conjecture on circle packings. Rend. Semin. Mat. Fis. Milano 66 (1998), 201–291. · Zbl 0997.60504 · doi:10.1007/BF02925361 |
| [10] | K. Stephenson and J. Ashe, Fractal branching, personal communication, University of Tennessee at Knoxville 2007. |
| [11] | E. Wegert, Topological methods for strongly nonlinear Riemann-Hilbert problems for holomorphic functions, Math. Nachr. 134 (1987), 201–230. · Zbl 0638.30039 · doi:10.1002/mana.19871340114 |
| [12] | E. Wegert, Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations, Akademie Verlag, Berlin 1992. · Zbl 0745.30040 |
| [13] | E. Wegert, Nonlinear Riemann-Hilbert problems – history and perspectives, in: N. Papamichael et al. (eds.), Proc. 3rd CMFT Conf. 1997 World Scientific, Ser. Approx. Decompos. 11 (1999), 583–615. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
