Doubly connected minimal surfaces and extremal harmonic mappings

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8 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)726-762
Number of pages37
JournalJournal of Geometric Analysis
Volume22
Issue number3
DOIs
StatePublished - Jul 2012

Keywords

  • Björling problem
  • Harmonic mapping
  • Minimal surface
  • Quasiconformal mapping

ASJC Scopus subject areas

  • Geometry and Topology

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