The Obata Equation With Robin Boundary Condition

We study the Obata equation with Robin boundary condition partial derivative f/partial derivative v + 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 derivative f/partial derivative v = 1.