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

2019/11/15 by Hubert L. Bray, Demetre Kazaras, Bray, Hubert L. +5 · 4 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1911.06754

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.

Cited by

• Monotonicity of harmonic functions on 3-manifolds with an asymptotically flat end
• A Conformal Positive Mass Theorem with Noncompact Boundary
• Mass-capacity inequality modeled on conformally flat manifolds
• Level sets of harmonic functions on three-dimensional manifolds with nonnegative scalar curvature

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