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Manifolds of quasiconformal map**s and the nonlinear Beltrami equation

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Abstract

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation \(\partial_{\bar{z}}f=\mathcal{H}(z,\partial_{z}f)\) generate a two-dimensional manifold of quasiconformal map**s \(\mathcal{F}_\mathcal{H} \subset {W_{\rm{loc}}^{1,2}(\mathbb{C})}\). Moreover, we show that under regularity assumptions on \(\mathcal{H}\), the manifold \(\mathcal{F}_\mathcal{H}\) defines the structure function \(\mathcal{H}\) uniquely.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076, Aalto, Finland

    Kari Astala

  2. LE STUDIUM Loire Valley Institute for Advanced Studies, Orléans & Tours, France

    Kari Astala

  3. Laboratoire de Mathématiques, Analyse, Probabilités, Modélisation, Université d’Orléans, 45067, Orléans cedex 2, France

    Kari Astala

  4. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 64, Pietari Kalmin Katu 5, FI-00014, Helsinki, Finland

    Kari Astala & Jarmo Jääskeläinen

  5. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Catalonia, Spain

    Albert Clop

  6. Department of Mathematics, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Daniel Faraco & Jarmo Jääskeläinen

  7. ICMAT CSIC-UAM-UCM-UC3M, 28049, Madrid, Spain

    Daniel Faraco

  8. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Jyväskylä, Finland

    Jarmo Jääskeläinen

Authors
  1. Kari Astala
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  2. Albert Clop
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  3. Daniel Faraco
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  4. Jarmo Jääskeläinen
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Corresponding author

Correspondence to Daniel Faraco.

Additional information

K. A. was supported by Academy of Finland project SA-307333.

A. C. was supported by research grants 2014SGR75 (Generalitat de Catalunya), MTM2016-75390-P and MTM2016-81703-ERC (Gobierno de Espana) and FP7-607647 (European Union).

D. F. was supported by research grant MTM2011-28198 from the Ministerio de Ciencia e Innovación (MCINN), by MINECO: ICMAT Severo Ochoa project SEV-2011-0087, and by the ERC 307179.

J. J. was supported by the ERC 307179 and Academy of Finland (no. 276233).

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Astala, K., Clop, A., Faraco, D. et al. Manifolds of quasiconformal map**s and the nonlinear Beltrami equation. JAMA 139, 207–238 (2019). https://doi.org/10.1007/s11854-019-0059-x

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  • DOI: https://doi.org/10.1007/s11854-019-0059-x

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