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

TL;DR

This paper investigates the behavior of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends, establishing monotonicity properties and characterizing the Schwarzschild manifolds with two ends.

Contribution

It introduces new monotonicity results for harmonic functions on such manifolds and provides a rigidity characterization of Schwarzschild manifolds with two ends.

Findings

Monotonicity properties of harmonic functions are established.

Rigidity results for Schwarzschild manifolds with two ends are proved.

Insights into the geometric structure of 3-manifolds with asymptotically flat ends.

Abstract

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.