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In this work, we consider static manifolds $M$ with nonempty boundary$\partial M$. In this case, we suppose that the potential $V$ also satisfies anoverdetermined Robin type condition on $\partial M$. We prove a rigiditytheorem for the Euclidean closed unit ball $B^3$ in $\mathbb{R}^3$. Moreprecisely, 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 andintersects $\partial M$. We also consider the case where $\Sigma=V^{-1}(0)$does not intersect $\partial M$.

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