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

Abstract

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