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

• Tomasz Adamowicz ^[1] ; Jarmo Jääskeläinen ^[2] ; Aleksis Koski ^[2]
  1. [1] Polish Academy of Sciences

     Polish Academy of Sciences

     Warszawa, Polonia

     
  2. [2] University of Jyväskylä

     University of Jyväskylä

     Jyväskylä, Finlandia

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 6, 2020, págs. 1779-1834
• Idioma: inglés
• DOI: 10.4171/rmi/1183
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• Resumen

  • 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 p-harmonic mappings between Riemannian surfaces.

    In our proof of the injectivity criterion we approximate the p-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.