On static manifolds satisfying an overdetermined Robin type condition on the boundary

In this work, we consider static manifolds $M$ with nonempty boundary $\mathrm{\partial }M$. In this case, we suppose that the potential $V$ also satisfies an overdetermined Robin type condition on $\mathrm{\partial }M$. We prove a rigidity theorem for the Euclidean closed unit ball $B^3$ in ${\mathbb{R}}^3$. More precisely, we give a sharp upper bound for the area of the zero set $\mathrm{\Sigma }=V^{-1}(0)$ of the potential $V$, when $\mathrm{\Sigma }$ is connected and intersects $\mathrm{\partial }M$. We also consider the case where $\mathrm{\Sigma }=V^{-1}(0)$ does not intersect $\mathrm{\partial }M$.