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Rigidity of circle packings with crosscuts

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Abstract

Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply connected domain. We introduce the concept of crosscuts and investigate the rigidity of circle packings with respect to maximal crosscuts. The main result is a discrete version of an indentity theorem for analytic functions (in the spirit of Schwarz’ Lemma), which has implications to uniqueness statements for discrete conformal mappings.

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Acknowledgments

We would like to thank the referee for reading this long manuscript with great care and making valuable comments and suggestions. We also thank Beate Uhl and the Springer Correction Team for their constructive collaboration in resolving some issues concerning page referencing.

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Authors and Affiliations

  1. Institute of Applied Analysis, TU Bergakademie Freiberg, Akademiestr. 6, 09596, Freiberg, Germany

    David Krieg & Elias Wegert

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  1. David Krieg
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  2. Elias Wegert
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Correspondence to Elias Wegert.

Additional information

D. Krieg was supported by Sächsisches Landesgraduiertenstipendium. E. Wegert was supported by the Deutsche Foschungsgemeinschaft, Grant We 1704/8-2.

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Krieg, D., Wegert, E. Rigidity of circle packings with crosscuts. Beitr Algebra Geom 57, 1–36 (2016). https://doi.org/10.1007/s13366-015-0245-7

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  • DOI: https://doi.org/10.1007/s13366-015-0245-7

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