In Case (b) of Region $\II$, the Gromov-Hausdorff map

(7.117)

$$F_{j}:(\mathcal{M},g_{j},\bm{x}_{j})\longrightarrow(\mathbb{R}^{3},g_{0},\bm{x}_{\infty})$$

can be given by the rescaled coordinate functions

(7.118)

$\displaystyle x_{j}\equiv\gamma_{j}\cdot x,\ y_{j}\equiv\gamma_{j}\cdot y,\ z_{j}\equiv\gamma_{j}\cdot z,$

where $\gamma_{j}>0$ is defined in the proof of Lemma 7.11. Moreover, $F\equiv(x_{j},y_{j},z_{j})$ satisfies

(7.119)

$$\Delta_{\tilde{g}_{j}}x_{j}=\Delta_{\tilde{g}_{j}}y_{j}=\Delta_{\tilde{g}_{j}}z_{j}=0$$

and satisfy

(7.120)

$$|\nabla_{\tilde{g}_{j}}x_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}y_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}z_{j}|_{\tilde{g}_{j}}\to 1$$

and away from the monopoles,

(7.121)

$$|\nabla_{\tilde{g}_{j}}^{2}x_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}^{2}y_{j}|_{\tilde{g}_{j}}=|\nabla_{\tilde{g}_{j}}^{2}z_{j}|_{\tilde{g}_{j}}\to 0.$$