Harmonic Functions and The Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

TL;DR

This paper establishes a new lower bound for the mass of 3D asymptotically flat Riemannian manifolds using harmonic functions, providing a novel proof of the positive mass theorem with parallels to existing methods.

Contribution

It introduces a new approach that replaces harmonic spinors with harmonic functions and level sets, offering an alternative proof of the positive mass theorem in three dimensions.

Findings

Derived explicit lower bounds for mass using harmonic functions.

Provided a new proof of the positive mass theorem in dimension three.

Connected harmonic functions with geometric and physical properties of manifolds.

Abstract

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.