We will fix $\epsilon({\rm v})>0$ later. For the moment let any $\epsilon_{1}>0$ be arbitrary with $\delta_{1}({\rm v},\epsilon_{1})>0$ the corresponding number from Theorem 8.3. If $T^{\delta_{1}}_{\alpha_{1}}(x)=0$ then there exists a diffeomorphism

$\displaystyle\Phi_{\alpha_{1}}:A_{r_{\alpha_{1}}/2,2r_{\alpha_{1}}}(0)\to U_{\alpha_{1}}\,,$

(8.31)

where $0\in\mathds{R}^{4}/\Gamma_{\alpha_{1}}$ and $A_{r_{\alpha_{1}}/2,2r_{\alpha_{1}}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha_{1}}/2,(1+\epsilon)r_{\alpha_{1}}}(x)$, such that

$\displaystyle||\Phi_{\alpha_{1}}^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha_{1}}/2},2r_{\alpha_{1}})}+r_{\alpha_{1}}||\partial_{k}\Phi_{\alpha_{1}}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha_{1}}/2},2r_{\alpha_{1}})}<\epsilon_{1}\,.$

(8.32)

In particular, if $\epsilon>0$ is fixed and $2\delta(n,\epsilon)$ is the corresponding number from Theorem 8.3, then we can choose $\epsilon_{1}=\epsilon_{1}(\epsilon,{\rm v})$ sufficiently small so that

$\displaystyle\mathcal{V}^{\delta}_{r_{\alpha_{1}}}(x)<\ln|\Gamma_{\alpha_{1}}|+\delta\,.$

(8.33)

Thus, if $\alpha_{2}$ is such that

$\displaystyle\mathcal{V}^{\delta}_{r_{\alpha_{2}}/2}(x)\geq\ln|\Gamma_{\alpha_{1}}|-\delta\,,$

(8.34)

then for all $\alpha_{1}\leq\alpha\leq\alpha_{2}$ we have $T^{2\delta}_{\alpha}(x)=0$.

By Theorem 8.3, there exists for each $\alpha_{1}\leq\alpha\leq\alpha_{2}$, a diffeomorphism

$\displaystyle\Phi_{\alpha}:A_{r_{\alpha}/2,2r_{\alpha}}(0)\to U_{\alpha}\,,$

(8.35)

where $0\in\mathds{R}^{4}/\Gamma_{\alpha}$ and $A_{r_{\alpha}/2,2r_{\alpha}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha}/2,2(1+\epsilon)r_{\alpha}}(x)$, such that

$\displaystyle||\Phi_{\alpha}^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}+r_{\alpha}||\partial_{k}\Phi_{\alpha}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}<\epsilon\,.$

(8.36)

In particular this implies that $\Gamma_{\alpha}=\Gamma$ is independent of $\alpha$.

Next we focus on the inverse maps

$\displaystyle\Phi_{\alpha}^{-1}:U_{\alpha}\to A_{r_{\alpha}/2,2r_{\alpha}}(0)\,.$

(8.37)

Observe that by (8.36), after possibly composing $\Phi_{\alpha}$ with a rotation of $\mathds{R}^{4}/\Gamma$ we can assume for $x\in U_{\alpha}\cap U_{\beta}$ that

$\displaystyle|\Phi_{\alpha}^{-1}(x)-\Phi_{\beta}^{-1}(x)|<\epsilon r_{\alpha}\,.$

(8.38)

Now let $\epsilon<\epsilon({\rm v})$ be sufficiently small, so that if $x\in\mathds{R}^{4}/\Gamma$, then $B_{\epsilon|x|}(x)\subseteq\mathds{R}^{4}/\Gamma$ is isometric to the standard Euclidean ball $B_{\epsilon|x|}(0^{4})\subseteq\mathds{R}^{4}$. Note in particular that if $\{x_{i}\}\in B_{\epsilon|x|}(x)$ is a collection of points, then any convex combination is well defined.

For each $\alpha$ let $\varphi^{\prime}_{\alpha}:U_{\alpha}\to\mathds{R}$ be a smooth cutoff function such that

$\displaystyle\varphi^{\prime}_{\alpha}(x)=\begin{cases}&1\text{ if }x\in A_{3r_{\alpha}/8,15r_{\alpha}/8}(x)\,,\\ &0\text{ if }x\not\in A_{r_{\alpha}/2,2r_{\alpha}}(x)\,,\end{cases}$

and such that $|\nabla\varphi^{\prime}_{\alpha}|\leq 10r_{\alpha}^{-1}$. If we set $\varphi^{\prime}(x)=\sum_{\alpha}\varphi^{\prime}_{\alpha}(x)$ then $1\leq\varphi^{\prime}(x)\leq 4$. In particular,

$\displaystyle\varphi_{\alpha}=\frac{\varphi^{\prime}_{\alpha}(x)}{\varphi^{\prime}(x)}:U_{\alpha}\to\mathds{R}\,,$

(8.39)

sarisfies $\sum\varphi_{\alpha}(x)=1$, and so, is a partition of unity, with $|\nabla\varphi_{\alpha}|\leq 40r_{\alpha}^{-1}$.

Define the map

$\displaystyle\Phi^{-1}:U=\bigcup_{\alpha}U_{\alpha}\to A_{r_{\alpha_{2}}/2,2r_{\alpha_{1}}}(0)\,,$

(8.40)

given by

$\displaystyle\Phi^{-1}(x)=\sum_{\alpha}\varphi_{\alpha}(x)\Phi^{-1}_{\alpha}(x)\,.$

(8.41)

(As previously noted, the convex combination is well defined since the $\Phi^{-1}_{\alpha}(x)$ all live in a ball which is isometric to a Euclidean ball.) On each domain, $U_{\alpha}$, we have by (8.32) and (8.38) that $\Phi^{-1}$ and $\Phi^{-1}_{\alpha}$ are $C^{1}$-close. Hence, $\Phi^{-1}$ is a diffeomorphism, and a quick computation using (8.32) and (8.38) verifies the desired estimates:

$\displaystyle||\Phi^{*}g_{ij}-\delta_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}+r_{\alpha}||\partial_{k}\Phi_{\alpha}^{*}g_{ij}||_{C^{0}(A_{r_{\alpha}/2},2r_{\alpha})}<C\epsilon\,.$

(8.42)

By choosing $\epsilon$ appropriately small, we complete the proof. ∎