JournalsrmiVol. 36, No. 6pp. 1779–1834

The Radó–Kneser–Choquet theorem for pp-harmonic mappings between Riemannian surfaces

  • Tomasz Adamowicz

    Polish Academy of Sciences, Warsaw, Poland
  • Jarmo Jääskeläinen

    University of Jyväskylä, Finland and University of Helsinki, Finland
  • Aleksis Koski

    University of Jyväskylä, Finland
The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces cover

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Abstract

In the planar setting, the Radó–Kneser–Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó–Kneser–Choquet for pp-harmonic mappings between Riemannian surfaces.

In our proof of the injectivity criterion we approximate the pp-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.

Cite this article

Tomasz Adamowicz, Jarmo Jääskeläinen, Aleksis Koski, The Radó–Kneser–Choquet theorem for pp-harmonic mappings between Riemannian surfaces. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1779–1834

DOI 10.4171/RMI/1183