# A step in the proof on the uniqueness of mass

I am reading the survey paper "The Yamebe Problem" 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" 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.

## 1 Answer

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.