I am reading the survey paper "[The Yamebe Problem](https://projecteuclid.org/euclid.bams/1183553962)" by Lee and Parker. In section 9, Theorem 9.6 in P.78, it was proved that the mass is well defined in the sense that $m(g)$ depends only on the metric $g$. But there is one step in the proof which I cannot understand: in the proof of Theorem 9.6, they said that "The first key observation is that the radial distance functions $\rho=|z|$ and $\tilde{\rho}=|\tilde{z}|$ are related by $C^{-1}\tilde{\rho}\leq \rho\leq C\tilde{\rho}$ for some positive constant." I wonder how one can conclude this. I guess it follows from the previous sentence, which says
"$\tilde{z}^i=z^i+\varphi^i$, where $\varphi^i\in C^{2,\alpha}\_{-\tau+1}(N\_\infty)$". But I am not exactly sure. I also tried to read the original paper "[The mass of an asymptotically flat manifold](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160390505/abstract)" by Bartnik. But it seems to me that it was not mentioned explicitly. I guess there are many people in this site who have read these papers. I would appreciate if I can get help from these people.



A step in the proof on the uniqueness of mass
Yes it follows from the previous sentence. Since $\tau > (n-2)/2 \geq 0$ by assumption you have that $|\varphi^i| \leq C \rho$ from the definition of the norm. This implies that $\tilde{\rho} \leq C \rho$ for some possibly different $C$ by triangle inequality. The condition is symmetric between $\rho$ and $\tilde{\rho}$, so the other inequality follows also immediately. 


