The positive mass conjecture

Let $(M^n,g)$, $n\geq 3$, be an asymptotically flat Riemannian manifold of order $\tau>(n-2)/2$ whose scalar curvature is nonnegative and integrable. The ADM mass is defined by

$$\mathrm{m}_{\mathrm{ADM}}=\frac{1}{2(n-1)\omega_{n-1}}\lim_{r\to\infty}\int_{S_r}(g_{ij,i}-g_{ii,j})\nu^j\,dS_r.$$

Positive mass conjecture. The ADM mass of $(M,g)$ is nonnegative. Moreover, if the mass is zero, then $(M,g)$ is isometric to Euclidean space $(\mathbb{R}^n,\delta)$.

The conjecture is known in dimensions $n\leq 7$, for conformally flat manifolds, for spin manifolds, and for Euclidean graphs under the conditions described in the source. The general statement is therefore not open in all cases, but the supplied text does not specify a complete resolution status for the unrestricted formulation.