Level sets of harmonic functions on three-dimensional manifolds with nonnegative scalar curvature

We investigate the level sets of harmonic functions on $(\mathbb{R}^{3}\setminus \{0\},g)$. Drawing inspiration from Miao, we adopt the method developed by Munteanu-Wang to derive a monotonic quantity associated with the level sets of harmonic functions on $(\mathbb{R}^{3}\setminus \{0\},g)$ with nonnegative scalar curvature, under certain conditions. Furthermore, we establish a rigidity result for this quantity. Additionally, we find an extra scalar-flat metric on $\mathbb{R}^{3}\setminus \{0\}$.