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On Riemann-Hilbert Problems in Circle Packing

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Abstract

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.

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References

  1. D. Bauer, K. Stephenson and E. Wegert, Circle packings as differentiable manifolds, in preparation.

  2. Ch. Glader and E. Wegert, Nonlinear Riemann-Hilbert problems with circular target curves, Math. Nachr. 281 No.9 (2008), 1221–1239.

    Article  MathSciNet  MATH  Google Scholar 

  3. Z.-X. He and O. Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), 285–305.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse 88 (1936), 141–164.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  8. K. Stephenson, Introduction to Circle Packing — The Theory of Discrete Analytic functions, Cambridge Univ. Press, Cambridge 2005.

    MATH  Google Scholar 

  9. K. Stephenson, A probabilistic proof of Thurston’s conjecture on circle packings. Rend. Semin. Mat. Fis. Milano 66 (1998), 201–291.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Wegert, Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations, Akademie Verlag, Berlin 1992.

    MATH  Google Scholar 

  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.

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Authors and Affiliations

  1. Institute of Applied Analysis, Tech Univ Bergakademie Freiberg, D-09596, Freiberg, Germany

    Elias Wegert

  2. MPI for Mathematics in the Sciences, Inselstrae 22, D-04103, Leipzig, Germany

    David Bauer

Authors
  1. Elias Wegert
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  2. David Bauer
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Correspondence to Elias Wegert.

Additional information

This research was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant WE 1704/8-1.

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Wegert, E., Bauer, D. On Riemann-Hilbert Problems in Circle Packing. Comput. Methods Funct. Theory 9, 609–632 (2009). https://doi.org/10.1007/BF03321748

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  • DOI: https://doi.org/10.1007/BF03321748

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