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

Cruz, Tiarlos; Nunes, Ivaldo

doi:10.1090/proc/16497

On static manifolds satisfying an overdetermined Robin type condition on the boundary
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by Tiarlos Cruz and Ivaldo Nunes;

Proc. Amer. Math. Soc. 151 (2023), 4971-4982

DOI: https://doi.org/10.1090/proc/16497

Published electronically: July 3, 2023

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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 $\partial M$.

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