The method is to set up a Banach iteration scheme to correct the volume form error. The generalised Gibbons-Hawking equation can be rewritten in the linearised form

where the linearised operator

The key point below is that in $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ the quadratic term is small while $\mathcal{L}$ is approximately $\Delta_{a}$.

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  Start with the initial volume form error $\tilde{E}^{(1)}$ on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell\geq A^{1/2}\}$, where $\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}$ according to Lemma 3.16. We will only need the precise value of $\tilde{E}^{(1)}$ in the shrinked region $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$.

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  Define the extension norm for a function $f$ on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ as the infimum of the $C^{k,\alpha}_{-1,0}$-norms for all functions $f^{\prime}$ extending $f$ with compact support inside $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$. The extension norm of $\tilde{E}^{(1)}$ is bounded by $CA^{-3/4}\nu^{2}$, since we can find an appropriate cutoff function $\chi$ such that $\chi\tilde{E}^{(1)}$ provides a required extension.

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  Apply Proposition 3.21 to produce $u_{1}=\Delta_{a}^{-1}(\chi\tilde{E}^{(1)}),$ with second derivative bound on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$,

  $$\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{3}A^{-3/4}.$$

  In particular on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$,

  $$|\nabla^{2}_{g_{a}}u_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2},$$

  which in fact holds on the entire $\mathcal{B}^{+}_{\nu+1}$ using $\Delta_{a}$-harmonicity in $\{\ell<A^{1/2}\}$. Whence the quadratic term is bounded on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ by

  $$\begin{split}\left\lVert\frac{1}{\tilde{W}_{(1)}}\det(\frac{\partial^{2}u_{1}}{\partial\mu_{i}\partial\mu_{j}})\right\rVert_{C^{k,\alpha}_{-1,0}}\leq&C\nu^{3}A^{-3/2}\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \leq&C\nu^{4}A^{-3/2}\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \ll&\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}.\end{split}$$

  The last inequality uses the condition $\nu\ll A^{3/8}$.

  The linearised equation is approximately satisfied on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$:

  $$\begin{split}&\left\lVert\mathcal{L}u_{1}-\chi\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}=\left\lVert\mathcal{L}u_{1}-\Delta_{a}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \leq&A^{-3/4}\nu\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\ll\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}.\end{split}$$

  where we used the metric deviation estimate in Lemma 3.12.

  Elementary algebra shows that inside $\mathcal{B}^{+}_{\nu}\cap\{\ell>A^{1/2}\}$, the volume form error is improved:

  $$\tilde{E}^{(1)}_{1}=\det(W^{p\bar{q}}_{(1)}-4\frac{\partial^{2}u_{1}}{\partial\eta_{p}\partial\bar{\eta}_{q}})(V_{(1)}+\frac{\partial^{2}u_{1}}{\partial\mu\partial\mu})^{-1}-1,$$

  $$\left\lVert\tilde{E}^{(1)}_{1}\right\rVert_{C^{k,\alpha}_{-1}(\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\})}\ll\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\})}.$$

  More formally the extension norm of $E^{(1)}$ is far smaller than that of $E^{(1)}$, after taking into account the cutoff procedures.

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  Iterate this procedure to produce $u_{1},u_{2},\ldots$, each time improving the extension norm by a factor say $10^{-1}$. The second derivative estimate

  $$\left\lVert\nabla^{2}_{g_{a}}u_{j}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C10^{-j}\nu^{3}A^{-3/4}$$

  implies that the series $\sum_{j}\nabla^{2}_{g_{a}}u_{j}$ converges. The series $\varphi_{1}=\sum_{j}u_{j}$ also converges after possibly adjusting $u_{j}$ by some affine linear functions, and satisfies the Hessian estimate $\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{3}A^{-3/4}.$ By construction the generalised Gibbons-Hawking equation holds on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$.

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