Monotonicity of harmonic functions on $3$-manifolds with an asymptotically flat end

We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is also given.

💡 Research Summary

The paper investigates positive harmonic functions on three‑dimensional, complete, orientable Riemannian manifolds (M,g) that are asymptotically flat (AF) and have non‑negative scalar curvature. The boundary Σ of M is assumed to be connected and to satisfy H₂(M,Σ)=0. For such a manifold the authors consider the unique harmonic function u that vanishes on Σ and tends to 1 at infinity. For each regular value t∈