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

Abstract

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 $\partial M$.

arXiv commentSome typos corrected. Adicional references and a new result (Theorem 3) were included. Comments are welcome!