I'm currently reading Schoen & Yau's 1979 proof of the positive-mass theorem and arrived at the very last sentence on the very last page (modulo the appendix) where they say:
Hence we conclude that $\mathop{Ric} = 0$ and because we are working in dimension three, $ds²$ is flat. This completes the proof of Theorem 2.
Theorem 2, however, made the claim that the manifold-with-boundary $N$ is (globally) isometric to $\mathbb{R}^3$ and I haven't been able to figure out in detail why local isometry to $\mathbb{R}^3$ (away from the boundary, of course) implies global isometry in this case. (My gut tells me that I'm missing something very elementary about what 3-manifolds of the given kind must look like, so I hope you'll excuse if this is a very simple question.)
For definiteness, let me restate the situation and the claim in detail:
Let $N$ be a connected¹, complete², asymptotically flat Riemannian manifold-with-boundary. Here, asymptotic flatness means that (1) there is a compact set $K \subset N$ such that the open set $N \setminus K$ consists of finitely many connected components ("ends") $N_k$ each of which is diffeomorphic to some $\mathbb{R^3} \setminus (\text{closed ball})$, and (2) the boundary of $N$ has mean curvature $H < 0$ with respect to the outward-pointing normal $n$. (Here, $H$ is defined as $H := \mathop{tr}_g II$ and $II(v, w) := \langle \nabla_v w, n \rangle$ is the 2nd fundamental form.)
Claim: Suppose $N$ is flat and only has one end (called $N_k$ in the theorem). Then $N$ is isometrically isomorphic to $\mathbb{R}^3$.
¹, ²: These requirements are not explicitly mentioned by Schoen & Yau but seem natural and, in fact, are necessary for some of the proofs in the paper to work. In particular, without them the claim would be false right away. I hope I'm not missing any further implicit assumptions on $N$.
My intuition is that the requirement on the boundary's mean curvature prevents it from bounding any "holes" in $N$. (In particular, K cannot merely be the boundary of $N_k \subset \mathbb{R}^3$, i.e. a sphere.) Moreover, the completeness of $N$ prevents us from choosing $K$ to be e.g. the empty set.
Finally, the fact that $N = K \cup N_k$ and that $N$ is flat everywhere should prevent $K$ from including something like a flat 3-torus. After all, topologically, there would be nothing preventing me from e.g. gluing $N_k$ to $T^3$. So I suspect that it must be the flatness forbidding this, in the sense that I cannot actually choose the throat between $N_k$ and $T^3$ to be flat. But I'm having trouble making this precise and, in particular, generalizing this argument to any other $K$ that is not a ball.
This is only a partial answer as of now but in the case of a manifold $N$ without boundary, a comparison result like the Bishop-Gromov inequality (see this PDF for a good introduction) can be used together with asymptotic flatness to show that every geodesic ball in $N$ is isometric to a ball in $\mathbb{R}³$ and these isometries can be glued together to an isometry between $N$ and $\mathbb{R}³$.