The Obata equation with Robin boundary condition

• Xuezhang Chen ^[1] ; Mijia Lai ^[2] ; Fang Wang ^[2]
  1. [1] Nanjing University

     Nanjing University

     China

     
  2. [2] Shanghai Jiao Tong University

     Shanghai Jiao Tong University

     China

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 2, 2021, págs. 643-670
• Idioma: inglés
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• Resumen

  • We study the Obata equation with Robin boundary condition ∂f/∂ν+af=0 on manifolds with boundary, where a is a non-zero constant. Dirichlet and Neumann boundary conditions were previously studied by Reilly, Escobar and Xia. Compared with their results, the sign of a plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both a>0 and a<0. We also consider the Obata equation with non-vanishing Neumann condition ∂f/∂ν=1.