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?