The positive mass conjecture for asymptotically Euclidean manifolds

Let $(M^n,g)$ be a smooth orientable asymptotically Euclidean (AE) manifold of dimension $n\geq 3$: outside a compact set it has finitely many ends, each identified with the complement of a Euclidean ball, with metric coefficients satisfying $g_{ij}=\delta_{ij}+O(r^{-\sigma})$ and corresponding derivative estimates. Suppose the order satisfies $\sigma>(n-2)/2$, and the scalar curvature is nonnegative and integrable. Its ADM mass is

$$m(g)=\lim_{r\to\infty}\int_{S_r}(\partial_i g_{ij}-\partial_j g_{ii})\,dA^j.$$

Positive mass conjecture. Then $m(g)\geq 0$, with equality if and only if $(M,g)=(\mathbb{R}^n,g_E)$.

The conjecture is the fundamental positivity statement for ADM mass in general relativity. The supplied source presents it as a general conjecture while noting that the paper proves nonnegativity under long-time existence of Ricci flow and gives an independent proof of the positive mass theorem in dimension three; the general statement is therefore not established by the supplied text.