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

Abstract

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 ${\mathbb{R}}^3\setminus \{0\}$.

12 pages; Comments are welcome