Abstract: In this talk, I will discuss some recent developments on static manifolds. We also consider static manifolds  with nonempty boundary. In this case, we suppose that the potential  satisfies an overdetermined Robin type condition on the boundary. We prove a rigidity theorem for the Euclidean closed unit ball  in the Euclidean space. More precisely, we give a sharp upper bound for the area of the zero set of the potential. This is a joint work with Ivaldo Nunes.