The Obata equation with Robin boundary condition

Abstract

We study the Obata equation with Robin boundary condition $\partial f/\partial \nu +af=0$ 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 $\partial f/\partial \nu =1$.