A Green’s Function Proof of the Positive Mass Theorem
Virginia Agostiniani, Lorenzo Mazzieri, Francesca Oronzio
Abstract
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-manifolds. In the same context and for $$1<p<3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.