A Green's function proof of the Positive Mass Theorem

Abstract

In this short note, 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, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality in some case studies.

Geometry of three-dimensional manifolds with scalar curvature lower bound

“…We point out that in [2] this type of result was applied to reprove the positive mass theorem [33]. Recently, using a variant of the above theorem, Chodosh and Li [8] have affirmed a conjecture of Schoen that a stable minimal hypersurface in Euclidean space R 4 must be flat.…”

“…2 − |∇ |∇u|| 2 via a classical regularization procedure[26] from the above identity by noticing that it is nonnegative due to the Kato inequality.Using the co-area formula to rewrite the last term Since u is harmonic, we know that ´l(t) |∇u| is constant in t . Together So, for every t > t 0 there exists t i ∈ (t, t + 1)dw dt (t i ) ≤ Υ 2 .…”

Symmetry results for Serrin-type problems in doubly connected domains

“…In fact, the most part of our arguments (with the notable exception of the Pohozaev identity) do not rely on the structure of the Euclidean space at all. As a couple of examples of further applications, this method has proven to be quite successful to characterize static spacetimes in General Relativity [10,11,12,13] and somewhat similar techniques have been employed for other problems in General Relativity [4,21] and for p-harmonic functions in manifolds with nonnegative Ricci curvature in [3,20].…”

Monotonicity of the $p$-Green functions

“…The method is far-reaching and has been proven to be successful in view of the recent progresses in [5,6,10,35,36]. In particular, Bray-Kazaras-Khuri-Stern [6] introduced a new formula relating certain integral of the harmonic functions to the ADM mass of an asymptotically flat manifold and give an alternative proof to the positive mass theorem in dimension 3 (see also [5] for a proof using monotonicity of harmonic function). Motivated by the results in [11,13], Munteanu-Wang [35,36] investigated the quantity…”

“…Later, Chodosh-Li [10] applied the method to study Green functions on minimal hyper-surfaces in R 4 and proved the conjecture that any complete, two-sided, stable minimal hyper-surface in R 4 must be a hyper-plane. In [5], Agostiniani-Mazzieri-Oronzio also studied an analogous monotone quantity involving the Green function and gave an alternative proof to the positive mass theorem in dimension 3.…”