The positive mass conjecture for asymptotically flat manifolds with boundary

The simplest model is the closed half-space $\mathbb R^n_+$ with its standard flat metric $\delta$. Let $(M,g)$ be an asymptotically flat manifold with decay rate $\tau>(n-2)/2$, and let $R_g$ and $H_g$ denote its scalar curvature and boundary mean curvature. Its mass is denoted by $\mathfrak m_{(M,g)}$. Positive mass conjecture. If $(M,g)$ is asymptotically flat with decay rate $\tau>(n-2)/2$ as above and satisfies $R_g\geq 0$ and $H_g\geq 0$, then

$$\mathfrak m_{(M,g)}\geq 0.$$

Equality occurs if and only if $(M,g)$ is isometric to $(\mathbb R^n_+,\delta)$. This is the boundary analogue of the positive mass theorem and is expected to give rigidity of the flat half-space in the zero-mass case; the supplied source does not establish the conjecture's resolution.