JournalsrmiVol. 37, No. 2pp. 643–670

The Obata equation with Robin boundary condition

  • Xuezhang Chen

    Nanjing University, China
  • Mijia Lai

    Shanghai Jiao Tong University, China
  • Fang Wang

    Shanghai Jiao Tong University, China
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Abstract

We study the Obata equation with Robin boundary condition f/ν+af=0\partial f/\partial \nu + af = 0 on manifolds with boundary, where aa 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 aa plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both a>0a > 0 and a<0a < 0. We also consider the Obata equation with non-vanishing Neumann condition f/ν=1\partial f/\partial \nu=1.

Cite this article

Xuezhang Chen, Mijia Lai, Fang Wang, The Obata equation with Robin boundary condition. Rev. Mat. Iberoam. 37 (2021), no. 2, pp. 643–670

DOI 10.4171/RMI/1212