A Green's function proof of the Positive Mass Theorem

Abstract

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat $3$-manifold. In the same context and for $1<p<3$, a Geroch-type calculation is performed along the level sets of $p$-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.

arXiv commentThis version contains a new session regarding the small sphere limits

References in corpus (7)

• On the mass of static metrics with positive cosmological constant - I
• On the $1/H$-flow by $p$-Laplace approximation: new estimates via fake distances under Ricci lower bounds
• On the uniqueness of Schwarzschild-de Sitter spacetime
• Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature
• Spacetime Harmonic Functions and Applications to Mass
• Comparison theorems for three-dimensional manifolds with scalar curvature bound
• Anisotropic $p$-capacity and anisotropic Minkowski inequality

Cited by in corpus (3)

• Stable minimal hypersurfaces in ${\mathbf{R}}^5$
• Optimal growth bounds for eigenfunctions
• Symmetry results for Serrin-type problems in doubly connected domains