We have verified the weighted Sobolev inequality (cf. section 5.2), the existence of the distance-like function (cf. section 4.7), and the existence of $C^{k,\alpha}$ quasi-atlas (cf. section 4.1, 4.2).

The volume form error for $dd^{c}\phi_{glue}$ decays like $O(\tilde{\rho}^{-\frac{2n(2n-1)}{(n-1)(n+2)}})$ (cf. (36)). On the other hand, the volume growth rate is $O(\tilde{\rho}^{p^{\prime}})$ with $2<p^{\prime}=\frac{4n}{n+2}<\frac{2n(2n-1)}{(n-1)(n+2)}$. In particular the volume error is bounded by $O(\tilde{\rho}^{-p^{\prime}})$. Applying Hein’s package, we can find a $C^{k,\alpha}$ bounded solution $\phi_{rel}$, such that

with decay estimate $|\phi_{rel}|=O(\tilde{\rho}^{2-p_{0}+\epsilon})=O(\tilde{\rho}^{-q})$. Since the charts on the local universal covers have harmonic radius scale $O(\rho)$, elliptic regularity improves the decay estimate to local $C^{k,\alpha}$-norms. ∎