Confusion with conclusion to positive mass theorem I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050
I am fundamentally confused by the structure of their argument. Under consideration is the Einstein constraint equations with fundamental form for a spacelike hypersurface $M$ equal to $0$, so the constraint equations take the simple form 
$$ R(g)=0. $$
For metrics conformal to the euclidean metric, $g_{ij}(x)=u^4(x) \delta_{ij}$, for a distinguished "harmonic coordinate" $x$, we have that having vanishing scalar curvature is equivalent to $u$ being harmonic, $\Delta u=0$.
But if I simply define $u(x)=1+\frac{m}{2|x|}$ for a negative constant $m<0$, this is a harmonic function and so I have a solution to the constraint equations in this simple form. So it must be that this solution is incompatible with something other that the constraint equations $R(g)=0$. But the whole proof seems to deal only with this simple form of the equations!
Can someone offer a reason why this solution is incompatible?
 A: I recently had this question answered for me by someone knowledgeable so will post the answer here, although there are gaps in my understanding so apologies that this might not be a fully satisfactory transcription of their answer. I will leave the question open in case someone can provide a more informative response.
The resolution is that the positive mass theorem ONLY applies to a certain class of solutions to the field equations, namely those space-times for which a space-like hypersurface is a complete Riemannian manifold, with $k$ asymptotically-flat "ends". The Schwarzschild solution with $m>0$ falls under this class as it may be analytically extended in a radial coordinate $r$ to a "mirror solution" for $r<0$. This makes the space-like hypersurfaces for this stationary solution complete with two asymptotically flat ends. 
For the Schwarzschild solution with $m<0$, spacelike hypersurfaces are incomplete, with a coordinate singularity at $|x|=\frac{-m}{2}$ and no clear way to extend these coordinates to give a solution to the constraint equations on a complete Riemannian manifold.
