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Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

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Abstract

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Grötzsch and Johannes C.C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Björling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.

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Authors and Affiliations

  1. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Tadeusz Iwaniec, Leonid V. Kovalev & Jani Onninen

  2. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Tadeusz Iwaniec

Authors
  1. Tadeusz Iwaniec
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  2. Leonid V. Kovalev
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  3. Jani Onninen
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Corresponding author

Correspondence to Leonid V. Kovalev.

Additional information

Communicated by Michael Wolf.

T. Iwaniec was supported by the NSF grant DMS-0800416 and Academy of Finland grant 1128331. L.V. Kovalev was supported by the NSF grant DMS-0913474. J. Onninen was supported by the NSF grant DMS-0701059.

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Iwaniec, T., Kovalev, L.V. & Onninen, J. Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings. J Geom Anal 22, 726–762 (2012). https://doi.org/10.1007/s12220-010-9212-6

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  • DOI: https://doi.org/10.1007/s12220-010-9212-6

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