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

In this work, we consider static manifolds $M$ with nonempty boundary $\partial M$. In this case, we suppose that the potential $V$ also satisfies an overdetermined Robin type condition on $\partial M$. 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 $\Sigma=V^{-1}(0)$ of the potential $V$, when $\Sigma$ is connected and intersects $\partial M$. We also consider the case where $\Sigma=V^{-1}(0)$ does not intersect $\partial M$.