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We study the Obata equation with Robin boundary condition on manifolds with boundary, where 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 plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both and . We also consider the Obata equation with non-vanishing Neumann condition .
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Xuezhang Chen, Mijia Lai, Fang Wang, The Obata equation with Robin boundary condition. Rev. Mat. Iberoam. 37 (2021), no. 2, pp. 643–670DOI 10.4171/RMI/1212