I am trying to understand and further formalize Witten's proof of the positive mass theorem. Dan Lee, in his book "Geometric relativity" did a wonderful job with formalizing and carrying out the details of Parker and Taubes' work, which was already a formalization of Witten's work.

The statement of the theorem in his book is more or less the following:

Theorem:Let $(N,g)$ be a complete asymptotically euclidean spin $n$-manifold with nonnegative scalar curvature and $n \geq 3$. Suppose further that $N$ has a well defined ADM mass. Then the ADM mass of each end is nonnegative. Moreover, if the mass of any end is zero, then $(N,g)$ is globally isometric to euclidean space.

I do not particularly like the completeness hypothesis as in most cases of interest in physics, the manifold is not complete. Therefore I am wondering why the completeness hypothesis is necessary. The only place I can find in the proof in his book where the completeness hypothesis is used explicitly is for positive mass rigidity, that is to say to prove that if the mass of any end is zero, then $(N,g)$ is globally isometric to euclidean space.

The completeness hypothesis is almost never stated in other surveys. Parker and Lee, in their survey on the Yamabe problem state the theorem as follows:

Theorem:Let $(N,g)$ be an asymptotically flat Riemannian manifold of dimension $n \geq 3$ such that the ADM mass is well defined, and with nonnegative scalar curvature. Then its mass $m(g)$ is nonnegative, with $m(g) = 0$ if and only if $(N, g)$ is isometric to $\mathbb{R}^n$ with its Euclidean metric.

The positive mass rigidity part in this theorem is plainly false, as $\mathbb{R}^n \setminus \{0\}$ satisfies all hypotheses but is not isometric to $\mathbb{R}^n$, so for this part the completeness is necessary. However, without completeness it is possible to prove that the manifold has to be flat. Hence I think the following theorem is also true:

Theorem:Let $(N,g)$ be an asymptotically euclidean spin $n$-manifold with nonnegative scalar curvature and $n \geq 3$. Suppose further that $N$ has a well defined ADM mass. Then the ADM mass of each end is nonnegative. Moreover, if the mass of any end is zero, then $(N,g)$ is flat.

Can someone confirm this?

asymptotically flat Riemannian manifoldis that $N$ is the union of a compact set $N_0$ together with a set $N_\infty$ that is diffeomorphic to the complement of closed ball in $\mathbb R^n$, with a metric that approaches the Euclidean metric at a certain rate. This implies that $N$ must be complete. It's not too hard to show that every Cauchy sequence converges, but this comment is too small to contain the proof. :-) $\endgroup$