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

TL;DR

This paper investigates static manifolds with boundary under Robin boundary conditions on the potential, establishing a rigidity theorem and bounds on the zero set of the potential in Euclidean space.

Contribution

It provides a new rigidity result and sharp bounds for the zero set of the potential in static manifolds with Robin boundary conditions.

Findings

Rigidity theorem for the Euclidean unit ball in D.

Sharp upper bound for the area of the zero set C0=V^{-1}(0).

Results for cases where the zero set does or does not intersect 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 $\partial M$.