A construction of trivial beltrami coefficients

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A measurable function μ on the unit disk D of the complex plane with ‖μ‖ < 1 is sometimes called a Beltrami coefficient. We say that μ is trivial if it is the complex dilatation (Formula Presented) of a quasiconformal automorphism f of D satisfying the trivial boundary condition f (z) = z, |z| = 1. Since it is not easy to solve the Beltrami equation explicitly, to detect triviality of a given Beltrami coefficient is a hard problem, in general. In the present article, we offer a sufficient condition for a Beltrami coefficient to be trivial. Our proof is based on Betker’s theorem on Löwner chains.

Original languageEnglish
Pages (from-to)629-635
Number of pages7
JournalProceedings of the American Mathematical Society
Issue number2
Publication statusPublished - 2019 Feb


  • Löwner chain
  • Quasiconformal mapping
  • Universal teichmüller space


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