The positive mass conjecture with arbitrary ends

Let $N^n$ be a complete manifold diffeomorphic to $\mathbb{R}^n$, let $X^n$ be a complete connected non-compact manifold, and let $g$ be a smooth complete metric on

$$M^n=N^n\mathbin{\#}X^n$$

with nonnegative scalar curvature $R\geq 0$. Assume that $g$ is asymptotically flat on $N^n$ with suitable fall-off. Positive mass conjecture with arbitrary ends. The ADM mass of $(M^n,g)$ measured in the asymptotically flat end $N^n$ is nonnegative. This extends the positive mass principle to an asymptotically flat end connected to an otherwise arbitrary complete non-compact manifold. The source presents it as a conjectural implication from which the LCF Liouville conjecture follows; no resolution is supplied in the given text.