The positive mass conjecture for asymptotically flat manifolds

Let $N^n$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ and order $p>\frac{n-2}{2}$, with integrable scalar curvature $R_g$ and, when $\partial N$ is noncompact, integrable mean curvature $H_g$. Suppose that $\partial N$ is compact or empty, and let $m_{ADM}(g)$ denote its ADM mass. Positive mass conjecture. If $R_g,H_g\geq 0$, then

$$m_{ADM}(g)\geq 0,$$

and equality holds if and only if $N$ is isometric to $\mathbb{R}^n$; in particular, $\partial N=\emptyset$ in the equality case. This is the positive mass theorem for asymptotically flat manifolds. It is known when $n\leq 7$ or when $N$ is spin, while the general case remains open.