C\left(\left|\left|H_{X}\overset{\circ}{X}\right|\right|^{\frac{1}{2}}_{L^{1}(\mathbb{R}^{3})}+\left|\left|H_{Y}\overset{\circ}{Y}\right|\right|^{\frac{1}{2}}_{L^{1}(\mathbb{R}^{3})}+\left|\left|(1-\xi_{N}-\xi_{S})H_{X}\right|\right|_{L^{2}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}+\right.$
$\displaystyle\left.\left|\left|(1-\xi_{N}-\xi_{S})(1-\xi_{A})H_{Y}\right|\right|_{L^{2}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}+\left|\left|(1-\xi_{N}-\xi_{S})\xi_{A}X^{-1}_{K}H_{Y}\right|\right|_{L^{2}(\mathbb{R}^{7}_{A})\cap
L^{\infty}(\mathbb{R}^{7}_{A})}\right).$

  6. 6.

Using similar arguments and the previous lemmas, we estimate the $C^{1}$ norm
of $\xi_{N}\overset{\circ}{X}$ and $\xi_{S}\overset{\circ}{X}$.

∎

Finally, we prove Proposition 29

###### Proof.

We prove that there exists $C=C(\alpha_{0})>0$ such that

$\left|\left|(\overset{\circ}{X},\overset{\circ}{Y})\right|\right|_{\mathcal{L}_{X}\times\mathcal{L}_{Y}}\leq
C\left|\left|(H_{X},H_{Y})\right|\right|_{\mathcal{N}_{X}\times\mathcal{N}_{Y}}.$

Recall the norms:

$\displaystyle||\overset{\circ}{X}||_{\mathcal{L}_{X}}$
$\displaystyle=||\overset{\circ}{X}||_{\dot{W}^{1,2}_{axi}(\overline{\mathscr{B}})}+||\overset{\circ}{X}||_{\hat{C}^{2,\alpha_{0}}(\overline{\mathscr{B}})}+||r\overset{\circ}{X}||_{L^{\infty}(\overline{\mathscr{B}})}+||r^{2}\hat{\partial}\overset{\circ}{X}||_{L^{\infty}(\overline{\mathscr{B}})}+||r^{3}\log^{-1}(4r)\hat{\partial}^{2}\overset{\circ}{X}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}})},$
$\displaystyle||\overset{\circ}{Y}||_{\mathcal{L}_{Y}}$
$\displaystyle=||\overset{\circ}{Y}||_{\dot{W}^{1,2}_{axi}(\overline{\mathscr{B}})}+|||\partial
h|\overset{\circ}{Y}||_{L^{2}(\mathbb{R}^{3})}+||\overset{\circ}{Y}||_{\hat{C}^{2,\alpha_{0}}(\overline{\mathscr{B}})}+||X_{K}^{-1}\overset{\circ}{Y}||_{\hat{C}^{2,\alpha_{0}}(\overline{\mathscr{B}})}+||r^{3}X_{K}^{-1}\overset{\circ}{Y}||_{L^{\infty}(\overline{\mathscr{B}})}$
$\displaystyle+||r^{4}\hat{\partial}(X^{-1}_{K}\overset{\circ}{Y})||_{L^{\infty}(\overline{\mathscr{B}})}+||r^{5}\log^{-1}(4r)\hat{\partial}^{2}(X^{-1}_{K}\overset{\circ}{Y})||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}})},$

$\displaystyle||H_{X}||_{\mathcal{N}_{X}}$
$\displaystyle=||r^{3}(1-\xi_{N}-\xi_{S})H_{X}||_{C^{0,\alpha_{0}}(\mathbb{R}^{3})}+||(\chi^{2}+s^{2})\xi_{N}H_{X}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}_{N}})}+||((\chi^{\prime})^{2}+(s^{\prime})^{2})\xi_{S}H_{Y}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}_{S}})},$
$\displaystyle||H_{Y}||_{\mathcal{N}_{Y}}$
$\displaystyle=||H_{Y}r^{5}X^{-1}_{K}||_{\hat{C}^{0,\alpha_{0}}\left(\left(\overline{\mathscr{B}_{H}}\cup\overline{\mathscr{B}_{A}}\right)\cap\left\\{\rho\leq
1\right\\}\right)}+||H_{Y}r^{4}||_{\hat{C}^{0,\alpha_{0}}(\overline{\mathscr{B}}\cap\left\\{\rho\geq
1\right\\})}+||(\chi^{2}+s^{2})\xi_{N}H_{X}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}_{N}})}$
$\displaystyle+||((\chi^{\prime})^{2}+(s^{\prime})^{2})\xi_{S}H_{Y}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}_{S}})}$

  1. 1.

First of all, by we have

$\displaystyle\left|\left|(1-\xi_{N}-\xi_{S})H_{X}\right|\right|_{L^{2}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}\leq
C\left|\left|(1-\xi_{N}-\xi_{S})H_{X}\right|\right|_{C^{0,\alpha}(\mathbb{R}^{3})}$
$\displaystyle\left|\left|(s^{2}+\chi^{2})\xi_{N}H_{X}\right|\right|_{L^{\infty}(\mathbb{R}^{4}_{N})}\leq
C\left|\left|(s^{2}+\chi^{2})\xi_{N}H_{X}\right|\right|_{C^{0,\alpha}(\overline{\mathscr{B}_{N}})}$
$\displaystyle\left|\left|((s^{\prime})^{2}+(\chi^{\prime})^{2})\xi_{S}H_{X}\right|\right|_{L^{\infty}(\mathbb{R}^{4}_{S})}\leq
C\left|\left|((s^{\prime})^{2}+(\chi^{\prime})^{2})\xi_{S}H_{X}\right|\right|_{C^{0,\alpha}(\overline{\mathscr{B}_{S}})}$

  2. 2.

Similarly, we show that

$\displaystyle\left|\left|(s^{2}+\chi^{2})\xi_{N}X_{K}^{-1}H_{Y}\right|\right|_{L^{\infty}(\mathbb{R}^{8}_{N})}\leq
C\left|\left|(s^{2}+\chi^{2})\xi_{N}X_{K}^{-1}H_{Y}\right|\right|_{C^{0,\alpha}(\overline{\mathscr{B}_{N}})}$
$\displaystyle\left|\left|((s^{\prime})^{2}+(\chi^{\prime})^{2})\xi_{S}X_{K}^{-1}H_{Y}\right|\right|_{L^{\infty}(\mathbb{R}^{8}_{S})}\leq
C\left|\left|((s^{\prime})^{2}+(\chi^{\prime})^{2})\xi_{S}X_{K}^{-1}H_{Y}\right|\right|_{C^{0,\alpha}(\overline{\mathscr{B}_{S}})}$

  3. 3.

By similar arguments, we have

$\displaystyle\left|\left|(1-\xi_{N}-\xi_{S})(1-\xi_{A})H_{Y}\right|\right|_{L^{2}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}+\left|\left|(1-\xi_{N}-\xi_{S})\xi_{A}X^{-1}_{K}H_{Y}\right|\right|_{L^{2}(\mathbb{R}^{7}_{A})\cap
L^{\infty}(\mathbb{R}^{7}_{A})}\leq C$

  4. 4.

We estimate the term
$\displaystyle\left|\left|H_{X}\overset{\circ}{X}\right|\right|^{\frac{1}{2}}_{L^{1}(\mathbb{R}^{3})}$:

$\displaystyle\left|\left|H_{X}\overset{\circ}{X}\right|\right|^{\frac{1}{2}}_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle\leq\left(\left|\left|H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}\left|\left|\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left|\left|H_{X}\right|\right|^{\frac{1}{2}}_{L^{\frac{6}{5}}(\mathbb{R}^{3})}\left(\left|\left|(1-\xi_{N}-\xi_{S})\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}+\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}+\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}\right)^{\frac{1}{2}}.$

     * •

$\xi_{N}\overset{\circ}{X}$ and $\xi_{S}\overset{\circ}{X}$ are compactly
supported on $\overline{\mathscr{B}_{N}}$ and $\overline{\mathscr{B}_{S}}$
respectively. Hence,

$\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}+\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}\leq
C\left(\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}+\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}\right).$

     * •

By (9.71),

$\left|\left|(1-\xi_{N}-\xi_{S})\overset{\circ}{X}\right|\right|_{L^{6}(\mathbb{R}^{3})}\leq
C\left|\left|(1-\xi_{N}-\xi_{S})\overset{\circ}{X}\right|\right|_{\dot{H}^{1}(\mathbb{R}^{3})}.$

     * •

Thus, $\forall q>0$, we have

$\displaystyle\left|\left|H_{X}\overset{\circ}{X}\right|\right|^{\frac{1}{2}}_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle\leq
C\left(q^{-1}\left|\left|H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}+q\left|\left|(1-\xi_{N}-\xi_{S})\overset{\circ}{X}\right|\right|_{\dot{H}^{1}(\mathbb{R}^{3})}+q\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}\right.$
$\displaystyle\left.+q\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}\right)$
$\displaystyle\leq
C\left(q^{-1}\left|\left|(1-\xi_{N}-\xi_{S})H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}+q\left|\left|(1-\xi_{N}-\xi_{S})\overset{\circ}{X}\right|\right|_{\dot{H}^{1}(\mathbb{R}^{3})}\right.$
$\displaystyle\left.+q^{-1}\left|\left|\xi_{N}H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}+q\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}+q^{-1}\left|\left|\xi_{S}H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}+q\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}\right),$

     * •

$\xi_{N}H_{X}$ and $\xi_{S}H_{X}$ are compactly supported on
$\overline{\mathscr{B}_{N}}$ and $\overline{\mathscr{B}_{S}}$ respectively.
Hence,

$\left|\left|\xi_{N}H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}\leq
C\left|\left|\xi_{N}H_{X}\right|\right|_{L^{1}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}$

and

$\left|\left|\xi_{S}H_{X}\right|\right|_{L^{\frac{6}{5}}(\mathbb{R}^{3})}\leq
C\left|\left|\xi_{S}H_{X}\right|\right|_{L^{1}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})}$

     * •

Furthermore,

$\displaystyle\left|\left|\xi_{N}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}+\left|\left|\xi_{S}\overset{\circ}{X}\right|\right|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\leq
C\left|\left|\overset{\circ}{X}\right|\right|_{\hat{C}^{1}(\overline{\mathscr{B}})}$

     * •

We obtain similar estimates for $H_{Y}\overset{\circ}{Y}$.

  5. 5.

We choose $q>0$ so that by Lemma 84, we obtain

$\displaystyle||\overset{\circ}{X}||_{\dot{W}^{1,2}_{axi}(\overline{\mathscr{B}})}+||\overset{\circ}{Y}||_{\dot{W}^{1,2}_{axi}(\overline{\mathscr{B}})}+\left|\left|\overset{\circ}{X}\right|\right|_{\hat{C}^{1}(\overline{\mathscr{B}})}$
$\displaystyle+\left|\left|\overset{\circ}{Y}\right|\right|_{\hat{C}^{1}(\overline{\mathscr{B}})}\leq
C(\alpha_{0})\left|\left|(H_{X},H_{Y})\right|\right|_{\mathcal{N}_{X}\times\mathcal{N}_{Y}}.$

  6. 6.

Now, we estimate the $C^{2,\alpha_{0}}$ part of
$(\overset{\circ}{X},\overset{\circ}{Y})$ in terms of
$\left|\left|(H_{X},H_{Y})\right|\right|_{\mathcal{N}_{X}\times\mathcal{N}_{Y}}$.

     * •

First, since $(H_{X},H_{Y})_{\mathbb{R}^{3}}$ are compactly supported in
$\mathbb{R}^{3}$, $(\overset{\circ}{X},\overset{\circ}{Y})$ are compactly
supported in $\overline{\mathscr{B}}$ and we have

$\operatorname{supp}(\overset{\circ}{X},\overset{\circ}{Y})_{\mathbb{R}^{3}}\subset\operatorname{supp}({H_{X}}_{\mathbb{R}^{3}})\cap\operatorname{supp}({H_{Y}}_{\mathbb{R}^{3}}).$

Denote by $K\subset\subset\overline{\mathscr{B}}$ the support of
$\overset{\circ}{X}$ and $\overset{\circ}{Y}$ and assume for simplicity that
$K=\overline{B}(x_{0},1)$ where $x_{0}\in\overline{\mathscr{B}}$. Depending on
the position of $x_{0}$, $K$ lies either in
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$,
$\overline{\mathscr{B}_{N}}$ or $\overline{\mathscr{B}_{S}}$. In the
following, we establish the $C^{2,\alpha_{0}}$ estimates for
$(\overset{\circ}{X},\overset{\circ}{Y})$ in the regions. To this end, we use
Theorem 9.
First of all, note that
$(\overset{\circ}{X},\overset{\circ}{Y})\in\hat{C}^{2,\alpha_{0}}(\overline{\mathscr{B}})$.
Now, we have

       1. (a)

if $K\subset\subset\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$,
the second order operator $(\overset{\circ}{X},\tilde{Y})\to
L(\overset{\circ}{X},\tilde{Y})$ defined by

$L(\overset{\circ}{X},\tilde{Y})=\left(\begin{aligned}
&\Delta_{\mathbb{R}^{3}}\overset{\circ}{X}+\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{Y}}{X_{K}}-\frac{2|\partial
Y_{K}|^{2}}{X_{K}^{2}}\overset{\circ}{X}+2\frac{\partial X_{K}\cdot\partial
Y_{K}}{X_{K}^{2}}\overset{\circ}{Y}\\\
&\Delta_{\mathbb{R}^{7}}\tilde{Y}_{\mathbb{R}^{7}}+\tilde{e}_{A}\cdot\partial\tilde{Y}-2\frac{|\partial
Y_{K}|^{2}}{X^{2}_{K}}\tilde{Y}_{\mathbb{R}^{7}}-2\tilde{d}_{A}\cdot\partial\overset{\circ}{X}-\frac{2\overline{d}_{A}}{\rho}\partial_{\rho}\overset{\circ}{X}.\end{aligned}\right)$

is uniformly elliptic on
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$. We apply Theorem 9
with $\Omega=\overset{\circ}{K}$, $L$ and $f=(H_{X},X_{K}^{-1}H_{Y})$

       2. (b)

The remaining estimates in $\overline{\mathscr{B}_{N}}$ and
$\overline{\mathscr{B}_{S}}$ follow in a similar manner.

  7. 7.

Now, we show that

$\displaystyle||r\overset{\circ}{X}||_{L^{\infty}(\overline{\mathscr{B}})}+||r^{2}\hat{\partial}\overset{\circ}{X}||_{L^{\infty}(\overline{\mathscr{B}})}+||r^{3}\log^{-1}(4r)\hat{\partial}^{2}\overset{\circ}{X}||_{C^{0,\alpha_{0}}(\overline{\mathscr{B}})}\leq
C\left|\left|(H_{X},H_{Y})\right|\right|_{\mathcal{N}_{X}\times\mathcal{N}_{Y}}.$

In order to obtain the latter estimate, we apply the newtonian estimates
provided by Lemma 63 and Lemma 64 on each region
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$,
$\overline{\mathscr{B}_{N}}$ and $\overline{\mathscr{B}_{S}}$:

    1. (a)

on the region $\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$, with
$F$ given by

$F_{\mathbb{R}^{3}}:=H_{X}-\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{Y}}{X_{K}}+\frac{2|\partial
Y_{K}|^{2}}{X_{K}^{2}}\overset{\circ}{X}-2\frac{\partial X_{K}\cdot\partial
Y_{K}}{X_{K}^{2}}\overset{\circ}{Y}.$

Therefore, there exists $C>0$ such that $\forall x\in\mathbb{R}^{3}$

$\left|F(x)\right|\leq Cr^{-4}.$

    2. (b)

In the region $\overline{\mathscr{B}_{N}}$ with $F$ given by

$F_{\mathbb{R}^{4}}:=(s^{2}+\chi^{2})H_{X}-2\frac{\underline{\partial}Y_{K}\cdot\partial\overset{\circ}{Y}}{X_{K}}+2\frac{|\underline{\partial}Y_{K}|^{2}}{X_{K}^{2}}\overset{\circ}{X}-2\frac{\underline{\partial}X_{K}\cdot\underline{\partial}Y_{K}}{X_{K}^{2}}\overset{\circ}{Y}.$

∎

####  9.6.2 Non-linear estimates

We apply Theorem 6 in order to obtain

######  Proposition 30.

Let $\alpha_{0}\in(0,1)$ and let $\overline{\delta}_{0}>0$. Then, there exists
$0<\delta_{0}\leq\overline{\delta}_{0}$ such that
$\displaystyle\forall\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda},\delta\right)\in
B_{\delta_{0}}\left(\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\times[0,\infty[\right)$
there exists a unique one parameter family
$\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right)\right)$
depending on
$\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda},\delta\right)\in
B_{\delta_{0}}\left(\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\right)$
which solves (5.49), (9.53) and (5.51) and which satisfies

$\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right)\right)\left(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda},\delta\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}}\leq
C(\alpha_{0})\left(||\overset{\circ}{\Theta}||^{2}_{\mathcal{L}_{\Theta}}+||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right)$

and
$\displaystyle\forall\left(\overset{\circ}{\Theta}_{i},\overset{\circ}{\lambda}_{i},\delta_{i}\right)\in
B_{\delta_{0}}\left(\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\times[0,\infty[\right)$,

$\displaystyle\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right)\right)\left(\overset{\circ}{\Theta}_{1},\overset{\circ}{\lambda}_{1},\delta_{1}\right)-\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right)\right)\left(\overset{\circ}{\Theta}_{2},\overset{\circ}{\lambda}_{2},\delta_{2}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}}$
$\displaystyle\leq
C(\alpha_{0})\left(\left(\left|\left|\left(\overset{\circ}{X}_{1},\overset{\circ}{\Theta}_{1},\overset{\circ}{\lambda}_{1}\right)\right|\right|_{\mathcal{L}_{X}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}+\left|\left|\left(\overset{\circ}{X}_{2},\overset{\circ}{\Theta}_{2},\overset{\circ}{\lambda}_{2}\right)\right|\right|_{\mathcal{L}_{X}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}\right)\left|\left|\left(\overset{\circ}{\Theta}_{1},\overset{\circ}{\lambda}_{1}\right)-\left(\overset{\circ}{\Theta}_{2},\overset{\circ}{\lambda}_{2}\right)\right|\right|_{\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}\right.$
$\displaystyle\left.+\left|\delta_{1}-\delta_{2}\right|\right).$

Before we prove the above proposition, we introduce the following notations

  1. 1.

Define $H_{X}$ and $H_{Y}$ on $\overline{\mathscr{B}}$ to be the mappings

$\displaystyle H_{X}(\rho,z)$
$\displaystyle=N_{X}^{(1)}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)+N_{X}^{(2)}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),$
$\displaystyle H_{Y}(\rho,z)$
$\displaystyle=N_{Y}^{(1)}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)+N_{Y}^{(2)}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),$

where

$N^{(1)}_{X}(\overset{\circ}{X},\overset{\circ}{Y})(\rho,z):=\frac{X_{K}^{2}(|\partial\overset{\circ}{X}|^{2}-|\partial\overset{\circ}{Y}|^{2})+(\overset{\circ}{X}\partial
Y_{K}-\overset{\circ}{Y}\partial
X_{K})\cdot(2X_{K}\partial\overset{\circ}{Y}-\overset{\circ}{X}\partial
Y_{K}+\overset{\circ}{Y}\partial X_{K})}{X_{K}^{2}(1+\overset{\circ}{X})},$
$N^{(1)}_{Y}(\overset{\circ}{X},\overset{\circ}{Y})(\rho,z):=\frac{\partial\overset{\circ}{X}\cdot\partial\overset{\circ}{Y}+2X_{K}(\overset{\circ}{Y}\partial
X_{K}-\overset{\circ}{X}\partial
Y_{K})\cdot\partial\overset{\circ}{X}}{X_{K}^{2}(1+\overset{\circ}{X})},$
$\displaystyle
N^{(2)}_{X}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)(\rho,z)$
$\displaystyle:=(\rho^{-1}-\sigma^{-1}\partial_{\rho}\sigma)\frac{\partial_{\rho}(X_{K}(1+\overset{\circ}{X}))}{X_{K}}-\sigma^{-1}\partial_{z}\sigma\frac{\partial_{z}(X_{K}(1+\overset{\circ}{X}))}{X_{K}}$
$\displaystyle-\frac{2\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y})B_{\rho}}{X_{K}^{2}(1+\overset{\circ}{X})}-\frac{2\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y})B_{z}}{X_{K}^{2}(1+\overset{\circ}{X})}$
$\displaystyle-\frac{B_{\rho}^{2}+B_{z}^{2}}{X_{K}^{2}(1+\overset{\circ}{X})}+X^{-1}_{K}F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z),$
$\displaystyle
N^{(2)}_{Y}(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)(\rho,z)$
$\displaystyle:=(\rho^{-1}-\sigma^{-1}\partial_{\rho}\sigma)\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y})X_{K}^{-1}-\sigma^{-1}\partial_{\rho}\sigma
B_{\rho}X_{K}^{-1}$
$\displaystyle-\sigma^{-1}\partial_{z}\sigma(\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y})+B_{z})X_{K}^{-1}$
$\displaystyle+\frac{B_{\rho}\partial_{\rho}(X_{K}(1+\overset{\circ}{X}))+B_{z}\partial_{z}(X_{K}(1+\overset{\circ}{X}))}{X_{K}^{2}(1+\overset{\circ}{X})},$

  2. 2.

Let $\overline{\delta}_{0}>0$ be obtained by Proposition (28). Define the
nonlinear operator $N_{(X,Y)}$ on
$B_{\overline{\delta}_{0}}(\mathcal{L}_{X}\times\mathcal{L}_{Y})\times
B_{\overline{\delta}_{0}}\left(\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\times[0,\infty[\right)$
by

$N_{(X,Y)}\left(\left(\overset{\circ}{X},\overset{\circ}{Y}\right),(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)(\rho,z):=\left(\begin{aligned}
&(N^{(1)}_{X}+N^{(2)}_{X})(\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),B(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\\\
&(N^{(1)}_{Y}+N^{(2)}_{Y})(\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),B(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta),(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\end{aligned}\right).$

where $\overset{\circ}{\sigma}$ and $B$ are the mappings obtained by
Proposition 28.

Now, we state the following lemma

######  Lemma 85.

  1. 1.

$\displaystyle
N_{(X,Y)}\left(B_{\overline{\delta}_{0}}(\mathcal{L}_{X}\times\mathcal{L}_{Y})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{X}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\times[0,\infty[)\right)\subset\mathcal{N}_{X}\times\mathcal{N}_{Y}$.

  2. 2.

There exist $0<\delta_{0}\leq\overline{\delta}_{0}$ and $C(\alpha_{0})>0$ such
that
$\forall\left(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta\right)\in
B_{\overline{\delta}_{0}}(\mathcal{L}_{X}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\times[0,\infty[)$

$||(H_{X},H_{Y})||_{\mathcal{N}_{X}\times\mathcal{N}_{Y}}\leq
C(\alpha_{0})\left(\left|\left|\overset{\circ}{X}\right|\right|^{2}_{\mathcal{L}_{X}}+\left|\left|\overset{\circ}{Y}\right|\right|^{2}_{\mathcal{L}_{Y}}+\left|\left|(\overset{\circ}{\Theta},\overset{\circ}{\lambda})\right|\right|^{2}_{\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}+\delta\right).$
(9.72)

###### Proof.

  1. 1.

We use the same arguments used to prove Lemma 86 and Lemma 85: the compact
support of
$F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma};\delta)$
and the differentiability of $F_{1}$ with respect to each variable.
In order to obtain (9.72), we write

$\displaystyle
F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta);\delta)$
$\displaystyle=D_{h}F_{1}(0,0,0;0)[(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))]+D_{\delta}F_{1}(0,0,0;0)\delta$
$\displaystyle+O(||(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))||^{2}+\delta^{2})$
$\displaystyle=D_{\delta}F_{1}(0,0,0;0)\delta+O(||(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))||^{2}+\delta^{2}).$

By Proposition 26, we have

$\left|\left|\overset{\circ}{\sigma}\left(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda},\delta\right)\right|\right|_{\mathcal{L}_{\sigma}}\leq
C(\alpha_{0})\left(||\overset{\circ}{X}||^{2}_{\mathcal{L}_{X}}+||\overset{\circ}{\Theta}||^{2}_{\mathcal{L}_{\Theta}}+||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right).$

Thus,

$||F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta);\delta)||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}})}\leq
C(\alpha_{0})\left(||\overset{\circ}{X}||^{2}_{\mathcal{L}_{X}}+||\overset{\circ}{\Theta}||^{2}_{\mathcal{L}_{\Theta}}+||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right).$

Now, we write

$X_{K}^{-1}F_{1}=(1-\xi_{N}-\xi_{S})X_{K}^{-1}F_{1}+\xi_{N}X_{K}^{-1}F_{1}+\xi_{S}X_{K}^{-1}F_{1}.$

By Proposition 24, $F_{1}$ is compactly supported in
$\overline{\mathscr{B}_{A}}$. Therefore,

$\displaystyle||X_{K}^{-1}F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta);\delta)||_{\mathcal{N}_{X}}$
$\displaystyle=||r^{3}(1-\xi_{N}-\xi_{S})X_{K}^{-1}F_{1}||_{C^{0,\alpha_{0}}(\mathbb{R}^{3})}$
$\displaystyle\leq
C||F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta);\delta)||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}})}$
$\displaystyle\leq
C(\alpha_{0})\left(||\overset{\circ}{X}||^{2}_{\mathcal{L}_{X}}+||\overset{\circ}{\Theta}||^{2}_{\mathcal{L}_{\Theta}}+||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right)$

  2. 2.

We refer to the proof of Proposition $9.2.1$ in [17] in order to estimate
$N_{X}^{(1)}$, $N_{Y}^{(1)}$, $N_{Y}^{(2)}$ and the remaining terms of
$N_{X}^{(2)}$.

∎

Now, we prove Proposition 28.

###### Proof.

  1. 1.

First of all, by Proposition 26 and Proposition 28, there exists a solution
map
$(\overset{\circ}{\sigma},B)(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)$
defined on $B_{\delta_{0}}$ which solves (5.49) and (5.50).

  2. 2.

We apply Theorem 6 with $\displaystyle L$ defined by

$L(\overset{\circ}{X},\overset{\circ}{Y}):=\left(\begin{aligned}
&\Delta_{\mathbb{R}^{3}}\overset{\circ}{X}+\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{Y}}{X_{K}}-\frac{2|\partial
Y_{K}|^{2}}{X_{K}^{2}}\overset{\circ}{X}+2\frac{\partial X_{K}\cdot\partial
Y_{K}}{X_{K}^{2}}\overset{\circ}{Y}\\\
&\Delta_{\mathbb{R}^{3}}\overset{\circ}{Y}-\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{X}}{X_{K}}-\frac{(|\partial
X_{K}|^{2}+|\partial
Y_{K}|^{2})}{X_{K}^{2}}\overset{\circ}{Y}\end{aligned}\right),$

$\displaystyle
N=N_{(X,Y)}\;,\;\mathcal{L}=\mathcal{L}_{X}\times\mathcal{L}_{Y}\;,\;\mathcal{Q}=\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}$
and $\displaystyle\mathcal{P}=[0,\delta_{0}[$. By the previous lemma, all the
assumptions are satisfied and we obtain the desired result.

∎

###  9.7 Solving for $\Theta$

We recall that $\overset{\circ}{\Theta}$ satisfies

$\displaystyle\partial_{\rho}\overset{\circ}{\Theta}$
$\displaystyle=-\frac{\sigma}{X^{2}}(\partial_{z}Y+B_{z})+\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K},$
$\displaystyle\partial_{z}\overset{\circ}{\Theta}$
$\displaystyle=\frac{\sigma}{X^{2}}(\partial_{\rho}Y+B_{\rho})-\frac{\rho}{X_{K}^{2}}\partial_{\rho}Y_{K}.$

####  9.7.1 Linear problem

We prove the following result

######  Proposition 31.

Let $H_{\Theta}\in\mathcal{N}_{\Theta}$. Then, there exists
$\overset{\circ}{\Theta}\in\mathcal{L}_{\Theta}$ which solves the equation

$d\overset{\circ}{\Theta}=H_{\Theta}.$ (9.73)

It is given by

$\overset{\circ}{\Theta}(\rho,z):=-\int_{\rho}^{\infty}\,\left(H_{\Theta}\right)_{\rho}(\tilde{\rho},z)\,d\tilde{\rho}.$
(9.74)

Moreover, there exists $C(\alpha_{0})>0$ such that

$||\overset{\circ}{\sigma}||_{\mathcal{L}_{\sigma}}\leq
C(\alpha_{0})||H_{\sigma}||_{\mathcal{N}_{\sigma}}.$

Recall that $\mathcal{N}_{\Theta}$ is the completion of smooth compactly
supported closed $1-$forms under the norm

$\displaystyle||F||_{\mathcal{N}_{\Theta}}$
$\displaystyle:=||r^{3}(1+\rho^{-1})F_{\rho}||_{\hat{C}^{1,\alpha_{0}}\left(\overline{\mathscr{B}_{H}}\cup\overline{\mathscr{B}_{A}}\right)}+||r^{3}F_{z}||_{\hat{C}^{1,\alpha_{0}}\left(\overline{\mathscr{B}_{H}}\cup\overline{\mathscr{B}_{A}}\right)}+||s^{-1}F_{s}||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{N}})}$
$\displaystyle+||F_{\chi}||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{N}})}+||(s^{\prime})^{-1}F_{s^{\prime}}||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{S}})}+||F_{\chi^{\prime}}||_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{S}})}.$

###### Proof.

  1. 1.

First of all, we show that $\overset{\circ}{\Theta}$ given by (9.74) is well-
defined .
When $\rho\to\infty$, we have

$\left(H_{\Theta}\right)_{\rho}=O_{\rho\to\infty}(\rho^{-3}).$

Therefore, $\overset{\circ}{\Theta}$ is well defined.

  2. 2.

Now, we show that $\overset{\circ}{\Theta}$ solves (9.73).
Since $H_{\theta}$ is closed, we have

$d(H_{\rho}d\rho+H_{z}dz)=dH_{\Theta}=0.$

Therefore,

$\partial_{z}\left(H_{\Theta}\right)_{\rho}=\partial_{\rho}\left(H_{\Theta}\right)_{z}.$

Now, we compute

$\partial_{\rho}\overset{\circ}{\Theta}=\left(H_{\Theta}\right)_{\rho}(\rho,z).$

We apply the dominated convergence theorem to obtain

$\displaystyle\partial_{z}\overset{\circ}{\Theta}$
$\displaystyle=-\int_{\rho}^{\infty}\,\partial_{z}\left(H_{\Theta}\right)_{\rho}(\tilde{\rho},z)\,d\tilde{\rho}$
$\displaystyle=-\int_{\rho}^{\infty}\,\partial_{\rho}\left(H_{\Theta}\right)_{z}(\tilde{\rho},z)\,d\tilde{\rho}$
$\displaystyle=\left(H_{\Theta}\right)_{z}.$

  3. 3.

Finally, we prove that $\overset{\circ}{\Theta}$ lies in
$\mathcal{L}_{\Theta}$, that is

$\left((1-\xi_{N}-\xi_{S})r^{2}\overset{\circ}{\Theta}\right)_{\mathbb{R}^{3}}\in
C^{2,\alpha_{0}}(\mathbb{R}^{3})\;,\;\left(\xi_{N}r^{2}\overset{\circ}{\Theta}\right)_{\mathbb{R}^{4}}\in
C^{2,\alpha_{0}}(\mathbb{R}^{4})\;\text{and}\;\left(\xi_{N}r^{2}\overset{\circ}{\Theta}\right)_{\mathbb{R}^{4}}\in
C^{2,\alpha_{0}}(\mathbb{R}^{4}).$

To this end, we prove the estimates on the different region: away from the
boundary of $\overline{\mathscr{B}}$, near the axis, the horizon and near the
poles.
First of all, recall the following change of variables

$\displaystyle x$ $\displaystyle=\rho\cos\vartheta$ $\displaystyle y$
$\displaystyle=\rho\sin\vartheta.$

We have

$\partial_{x}=\frac{x}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\;,\;\partial_{y}=\frac{y}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}.$

    1. (a)

We control the $L^{\infty}$ norm of $r^{2}\overset{\circ}{\Theta}$:

      1. i.

Away from the region $\partial\overline{\mathscr{B}}$, say
$[\rho_{0},\infty[\times\mathbb{R}$ for some $\rho_{0}>0$, there exists $C>0$
such that

$\forall(\rho,z)\in[\rho_{0},\infty[\times\mathbb{R}\;:\;\left|\overset{\circ}{\Theta}(\rho,z)\right|\leq
Cr^{-2}$

Indeed,

         * •

if $(\rho,z)\in[\rho_{0},\infty[\times\mathbb{R}\cap B((0,0),1)$, then the
estimates are straightforward.

         * •

Otherwise, we have $\rho^{2}+z^{2}>1.$ Therefore,

$\displaystyle\left|\overset{\circ}{\Theta}(\rho,z)\right|$ $\displaystyle\leq
C\int_{\rho}^{\infty}\;r^{-3}(\tilde{\rho},z)\,d\tilde{\rho}$
$\displaystyle\leq
C\int_{\rho}^{\infty}\;\frac{1}{(\tilde{\rho}^{2}+z^{2})^{\frac{3}{2}}}\,d\tilde{\rho}$
$\displaystyle=C\int_{0}^{\infty}\frac{1}{((\tau+\rho)^{2}+z^{2})^{\frac{3}{2}}}\,d\tau$
$\displaystyle=C\int_{0}^{\infty}\frac{1}{(\rho^{2}+z^{2}+2\rho\tau+\tau^{2})^{\frac{3}{2}}}\,d\tau$
$\displaystyle\leq
C(\rho^{2}+z^{2})^{-\frac{3}{2}}\int_{0}^{\infty}\frac{1}{\left(1+\left(\frac{\tau}{\sqrt{\rho^{2}+z^{2}}}\right)^{2}\right)^{\frac{3}{2}}}\,d\tau$
$\displaystyle=C(\rho^{2}+z^{2})\int_{0}^{\infty}\,\frac{1}{(1+u^{2})^{\frac{3}{2}}}\,du$
$\displaystyle\leq Cr^{-2}(\rho,z).$

Therefore, we obtain the estimate for $r^{2}\overset{\circ}{\Theta}$.

      2. ii.

In $\tilde{\mathscr{A}}\subset\overline{\mathscr{B}_{A}}$, a neighbourhood of
the axis or in $\tilde{\mathscr{H}}\subset\overline{\mathscr{B}_{H}}$, a
neighbourhood of the horizon, we have

$\displaystyle\left|\overset{\circ}{\Theta}(\rho,z)\right|$
$\displaystyle=\left|-\int_{\rho}^{\infty}\,\left(H_{\Theta}\right)_{\rho}(\tilde{\rho},z)\,d\tilde{\rho}\right|$
$\displaystyle\leq
C\int_{\rho}^{\infty}\;\frac{\tilde{\rho}}{1+\tilde{\rho}}r^{-3}(\tilde{\rho},z)\,d\tilde{\rho}$
$\displaystyle\leq
C\int_{\rho}^{\infty}\;2\tilde{\rho}\left(1+\tilde{\rho}^{2}+z^{2}\right)^{-\frac{3}{2}}\,d\tilde{\rho}$

      3. iii.

In $\overline{\mathscr{B}_{N}}$, we write (9.73) in the $(s,\chi)$ coordinate
system. We have

$\partial_{s}\overset{\circ}{\Theta}=(H_{\Theta})_{s}\quad\text{;}\quad\partial_{\chi}\overset{\circ}{\Theta}=(H_{\Theta})_{\chi}.$

and

$\overset{\circ}{\Theta}(s,\chi)=\int_{0}^{s}\;(H_{\Theta})_{s}(\tilde{s},\chi)\,d\tilde{s}.$

We have

$\displaystyle\left|(r^{2}\xi_{N}\overset{\circ}{\Theta})_{\mathbb{R}^{4}}\right|\leq
C\int_{0}^{s}\tilde{s}d\tilde{s}\leq C.$

Here, we used the estimate of $s^{-1}(H_{\Theta})_{s}$ in the region
$\overline{\mathscr{B}_{N}}$.

    2. (b)

We control the $L^{\infty}$ norm of
$\nabla_{\mathbb{R}^{3}}(r^{2}(1-\xi_{N}-\xi_{S})\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$,
$\nabla_{\mathbb{R}^{4}}(r^{2}\xi_{N}\overset{\circ}{\Theta})_{\mathbb{R}^{4}}$
and
$\nabla_{\mathbb{R}^{4}}(r^{2}\xi_{S}\overset{\circ}{\Theta})_{\mathbb{R}^{4}}$
We have

$\displaystyle\partial_{x}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$
$\displaystyle=\frac{x}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\left(r^{2}(\rho,z)\overset{\circ}{\Theta}(\rho,z)\right),$
$\displaystyle\partial_{y}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$
$\displaystyle=\frac{y}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\left(r^{2}(\rho,z)\overset{\circ}{\Theta}(\rho,z)\right),$
$\displaystyle\partial_{z}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$
$\displaystyle=\partial_{z}\left(r^{2}(\rho,z)\overset{\circ}{\Theta}(\rho,z)\right).$

      1. i.

Away from the boundary $\left\\{0\right\\}\times\mathbb{R}$, the estimates are
straightforward since $H_{\Theta}\in\mathcal{N}_{\Theta}$.

      2. ii.

In $\tilde{\mathscr{A}}\cup\tilde{\mathscr{H}},$ we have

$\displaystyle\partial_{x}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$
$\displaystyle=\frac{x}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\left(r^{2}(\rho,z)\overset{\circ}{\Theta}(\rho,z)\right)$
$\displaystyle=x\overset{\circ}{\Theta}(\rho,z)+r^{2}(\rho,z)\frac{x}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\overset{\circ}{\Theta}.$

The first term is easily bounded by $r^{-2}$ thanks to (a). As for the second
term, we have

$\displaystyle\left|r^{2}(\rho,z)\frac{x}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\overset{\circ}{\Theta}\right|$
$\displaystyle=r^{2}(\rho,z)\frac{x}{\sqrt{x^{2}+y^{2}}}\left|(H_{\Theta})_{\rho}\right|$
$\displaystyle\leq Cr^{-1}\frac{\rho}{1+\rho}\frac{x}{\sqrt{x^{2}+y^{2}}}$
$\displaystyle\leq C.$

We control $\partial_{y}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$ in a
similar way. As for
$\partial_{z}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{3}}$, it is
straightforward.

      3. iii.

In $\overline{\mathscr{B}_{N}}$, we have

$\partial_{s}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{4}}=\partial_{s}r^{2}\overset{\circ}{\Theta}(s,\chi)+r^{2}(H_{\Theta})_{s}$

The previous point and the estimate for $H_{\Theta}$ yield the result.

    3. (c)

Now, we control the $L^{\infty}$ norm of
$\left(\nabla^{2}_{\mathbb{R}^{3}}r^{2}\overset{\circ}{\Theta}\right)$.

      1. i.

Away from the boundary $\left\\{0\right\\}\times\mathbb{R}$, the estimates are
straightforward since $H_{\Theta}\in\mathcal{N}_{\Theta}$.

      2. ii.

In $\tilde{\mathscr{A}}\cup\tilde{\mathscr{H}},$ we have

$\displaystyle\partial_{xx}\left(r^{2}\overset{\circ}{\Theta}\right)_{\mathbb{R}^{3}}$
$\displaystyle=\overset{\circ}{\Theta}+\frac{x^{2}}{\sqrt{x^{2}+y^{2}}}\partial_{\rho}\overset{\circ}{\Theta}+x\partial_{x}\left(\frac{r^{2}}{\rho}(H_{\Theta})_{\rho}\right)+\frac{r^{2}}{\rho}(H_{\Theta})_{\rho}.$

The above terms are controlled using the previous estimates and the
$L^{\infty}$ estimate of
$\displaystyle\nabla_{\mathbb{R}^{3}}\left(r^{3}\left(1+\frac{1}{\rho}\right)H_{\Theta}\right)$.

      3. iii.

In $\overline{\mathscr{B}_{N}}$, we have

$\partial_{ss}(r^{2}\overset{\circ}{\Theta})_{\mathbb{R}^{4}}=\partial_{ss}r^{2}\overset{\circ}{\Theta}(s,\chi)+r^{2}\partial_{s}(H_{\Theta})+2\partial_{s}r^{2}(H_{\Theta})_{s}.$

The estimate is straigthforward.

    4. (d)

Finally, we control

$\sup_{X_{1}\neq
X_{2}}\frac{\left|\left|\left(\nabla^{2}_{\mathbb{R}^{3}}r^{2}\overset{\circ}{\Theta}\right)(X_{1})-\left(\nabla^{2}_{\mathbb{R}^{3}}r^{2}\overset{\circ}{\Theta}\right)(X_{2})\right|\right|}{\left|X_{1}-X_{2}\right|^{\alpha_{0}}}\leq
Cr^{-2}.$

The estimates are straightforward thanks to the control of $H_{\Theta}$ and
the regularity of the different terms.

∎

####  9.7.2 Non-linear estimates

We apply Theorem 6 in order to obtain

######  Proposition 32.

Let $\alpha_{0}\in(0,1)$ and let $\overline{\delta}_{0}>0$. Then, there exists
$0<\delta_{0}\leq\overline{\delta}_{0}$ such that
$\displaystyle\forall\left(\overset{\circ}{\lambda},\delta\right)\in
B_{\delta_{0}}\left(\mathcal{L}_{\lambda}\times[0,\infty[\right)$ there exists
a unique one parameter family
$\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\overset{\circ}{\lambda},\delta\right)\in
B_{\delta_{0}}$
$\left(\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}\right)$
which solves (5.49), (9.53), (5.51) and (5.52) and which satisfies

$\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\overset{\circ}{\lambda},\delta\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}}\leq
C(\alpha_{0})\left(||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right)$

and
$\displaystyle\forall\left(\overset{\circ}{\lambda}_{i},\delta_{i}\right)\in
B_{\delta_{0}}\left(\mathcal{L}_{\lambda}\times[0,\infty[\right)$,

$\displaystyle\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\overset{\circ}{\lambda}_{1},\delta_{1}\right)-\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\overset{\circ}{\lambda}_{2},\delta_{2}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}}$
$\displaystyle\leq
C(\alpha_{0})\left(\left(\left|\left|\overset{\circ}{\lambda}_{1}\right|\right|_{\mathcal{L}_{\lambda}}+\left|\left|\overset{\circ}{\lambda}_{2}\right|\right|_{\mathcal{L}_{\lambda}}\right)\left|\left|\overset{\circ}{\lambda}_{1}-\overset{\circ}{\lambda}_{2}\right|\right|_{\mathcal{L}_{\lambda}}+\left|\delta_{1}-\delta_{2}\right|\right).$

Before we prove the above proposition, we introduce the following notations

  1. 1.

Define $H_{\Theta}$ on $\overline{\mathscr{B}}$ to be the one form which
expression in the $(\rho,z)$ coordinates is

$\displaystyle(H_{\Theta})_{\rho}$
$\displaystyle=-\frac{\rho}{X_{K}^{2}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}(\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y})+B_{z})+\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K},$
$\displaystyle(H_{\Theta})_{z}$
$\displaystyle=\frac{\rho}{X_{K}^{2}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}(\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y})+B_{\rho})-\frac{\rho}{X_{K}^{2}}\partial_{\rho}Y_{K}.$

  2. 2.

Let $\overline{\delta}_{0}>0$ be obtained by Proposition 30. Define the
nonlinear operator $N_{\Theta}$ on
$B_{\overline{\delta}_{0}}(\mathcal{L}_{\Theta})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{\lambda}\times[0,\infty[)$ by

$\displaystyle\left[N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)\right]_{\rho}(\rho,z):=-\frac{\rho}{X_{K}^{2}}\frac{(1+\overset{\circ}{\sigma}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))}{(1+\overset{\circ}{X}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))^{2}}(\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))+B_{z}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))+\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K}$
$\displaystyle\left[N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)\right]_{z}(\rho,z):=\frac{\rho}{X_{K}^{2}}\frac{(1+\overset{\circ}{\sigma}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))}{(1+\overset{\circ}{X}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))^{2}}(\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))+B_{\rho}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta))-\frac{\rho}{X_{K}^{2}}\partial_{\rho}Y_{K}$

and set

$\displaystyle\left[N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)\right]$
$\displaystyle=\left[N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)\right]_{\rho}d\rho+\left[N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)\right]_{z}dz,$

where $(\overset{\circ}{\sigma},B,\overset{\circ}{X},\overset{\circ}{Y})$ are
the solution mappings obtained by Proposition 30.

In order to prove the above proposition, we will need the following lemma

######  Lemma 86.

  1. 1.

There exists $\overline{\delta}_{0}>0$ such that $\displaystyle
N_{\Theta}\left(B_{\overline{\delta}_{0}}(\mathcal{L}_{\Theta})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{\lambda}\times[0,\infty[)\right)\subset\mathcal{N}_{\Theta}$.

  2. 2.

There exist $0<\delta_{0}\leq\overline{\delta}_{0}$ and $C(\alpha_{0})>0$ such
that
$\forall\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta\right)\in
B_{\overline{\delta}_{0}}(\mathcal{L}_{\Theta})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{\lambda}\times[0,\infty[)$

$||H_{\Theta}||_{\mathcal{N}_{\sigma}}\leq
C(\alpha_{0})\left(\left|\left|\overset{\circ}{\Theta}\right|\right|^{2}_{\mathcal{L}_{\Theta}}+||\overset{\circ}{\lambda}||^{2}_{\mathcal{L}_{\lambda}}+\delta\right).$

###### Proof.

  1. 1.

Let $\overline{\delta}_{0}>0$ be obtained by Proposition 30 and let
$(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\in(B_{\overline{\delta}_{0}}(\mathcal{L}_{\Theta})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{\lambda}\times[0,\infty[).$ First of
all, we show that $\displaystyle
N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta\right)$
is a differentiable closed one-form on $\overline{\mathscr{B}}$.

     * •

For the differentiability, the only terms that we need to analyse are

$\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K}\quad\text{,}\quad\frac{\rho}{X_{K}^{2}}\partial_{\rho}Y_{K}\quad\text{and}\quad\frac{\rho}{X_{K}^{2}}.$

The differentiability of the remaining terms as well as the differentiability
away from the $\partial\overline{\mathscr{B}}$ follow because
$(\overset{\circ}{\sigma},B,\overset{\circ}{X},\overset{\circ}{Y})(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\lambda)$
is differentiable.

     * •

Now, we check that $\displaystyle
N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta\right)$
is closed.

       * –

$(\overset{\circ}{Y},B)(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)$
solve (5.51) and (5.50). We set

$\theta:=dY+B\quad\text{and}\quad\theta_{K}:=dY_{K}.$

       * –

Therefore,

$N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)=-\frac{\sigma}{X^{2}}\left(\theta_{z}d\rho-\theta_{\rho}dz\right)+\frac{\rho}{X_{K}^{2}}\left(\theta_{z}^{K}-\theta_{\rho}^{K}dz\right).$

       * –

Moreover, $\theta$ verifies

$\sigma^{-1}\partial_{\rho}(\sigma\theta_{\rho})+\sigma^{-1}\partial_{z}(\sigma\theta_{z})=\frac{2\theta_{\rho}\partial_{\rho}X+2\theta_{z}\partial_{z}X}{X}$

       * –

Straightforward computations and the used of the above equation imply that the
one forms
$\displaystyle\frac{\sigma}{X^{2}}\left(\theta_{z}d\rho-\theta_{\rho}dz\right)$
and
$\displaystyle\frac{\rho}{X_{K}^{2}}\left(\theta_{z}^{K}-\theta_{\rho}^{K}dz\right)$
are closed.

       * –

Hence, $\displaystyle
N_{\Theta}\left(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\right)$
is closed.

  2. 2.

Let
$\displaystyle(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)\in(B_{\overline{\delta}_{0}}(\mathcal{L}_{\Theta})\times
B_{\overline{\delta}_{0}}(\mathcal{L}_{\lambda}\times[0,\infty[).$ We show
that $N_{\Theta}(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)$
provided $\delta$ sufficiently small. To this end, we show the estimates in
the region $\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$, then in
$\overline{\mathscr{B}_{N}}$ and $\overline{\mathscr{B}_{S}}$. In order to
lighten the expressions, we omit the dependence of
$\overset{\circ}{\sigma},\overset{\circ}{X},\overset{\circ}{Y},B$ on the
quantities
$\displaystyle(\overset{\circ}{\Theta},\overset{\circ}{\lambda};\delta)$. We
re-write $\left(N_{\Theta}\right)_{\rho}$ and $\left(N_{\Theta}\right)_{z}$ on
the form

$\displaystyle\left(N_{\Theta}\right)_{\rho}$
$\displaystyle=-\frac{\rho}{X_{K}^{2}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}(\partial_{z}(X_{K}\overset{\circ}{Y}+Y_{K}))+B_{z})+\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K}$
$\displaystyle=\frac{\rho}{X_{K}^{2}}\frac{-1-\overset{\circ}{\sigma}+(1+\overset{\circ}{X})^{2}}{(1+\overset{\circ}{X}^{2})^{2}}\partial_{z}Y_{K}-\frac{\rho}{X_{K}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}B_{z}-\frac{\sigma}{X_{K}^{2}(1+\overset{\circ}{X})^{2}}\partial_{z}(X_{K}^{2}(X_{K}^{-1}\overset{\circ}{Y}))$
$\displaystyle=\frac{\rho}{X_{K}^{2}}\frac{-\overset{\circ}{\sigma}+\overset{\circ}{X}^{2}+2\overset{\circ}{X}}{(1+\overset{\circ}{X}^{2})^{2}}\partial_{z}Y_{K}-\frac{\rho}{X_{K}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}B_{z}-\frac{\rho(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}\partial_{z}(X_{K}^{-1}\overset{\circ}{Y})$
$\displaystyle-2\frac{\rho(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}X_{K}^{-1}\overset{\circ}{Y}\partial_{z}\log
X_{K}.$ $\displaystyle\left(N_{\Theta}\right)_{z}$
$\displaystyle=\frac{\rho}{X_{K}^{2}}\frac{-\overset{\circ}{\sigma}+\overset{\circ}{X}^{2}+2\overset{\circ}{X}}{(1+\overset{\circ}{X}^{2})^{2}}\partial_{\rho}Y_{K}-\frac{\rho}{X_{K}}\frac{(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}B_{\rho}-\frac{\rho(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}\partial_{\rho}(X_{K}^{-1}\overset{\circ}{Y})$
$\displaystyle-2\frac{\rho(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}X_{K}^{-1}\overset{\circ}{Y}\partial_{\rho}\log
X_{K}.$

     * •

In $\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$, we prove that
$\displaystyle r^{3}(1+\rho^{-1})\left(N_{\Theta}\right)_{\rho}$ and
$r^{3}(N_{\Theta})_{z}$ are bounded in $\hat{C}^{1,\alpha_{0}}$. We have

       * –

By Lemma 54,

$\frac{1}{X^{2}_{K}}|\partial_{z}Y_{K}|\leq
Cr^{-5},\quad|\partial\left(X^{-2}_{K}\partial Y_{K}\right)|\leq
Cr^{-6},\quad|\partial^{2}\left(X^{-2}_{K}\partial Y_{K}\right)|\leq Cr^{-7}$

and

$\frac{\rho}{X^{2}_{K}}|\partial Y_{K}|\leq
Cr^{-4},\quad|\partial\left(X^{-1}_{K}\partial Y_{K}\right)|\leq
Cr^{-4},\quad|\partial^{2}\left(X^{-1}_{K}\partial Y_{K}\right)|\leq Cr^{-5}.$

Thus, the term $\displaystyle\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K}$ is
bounded by $\displaystyle
r^{3}(1+\rho^{-1})\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K}$ is bounded in
$\hat{C}^{1,\alpha_{0}}$.

       * –

By Lemma 56, the term $\partial_{z}(\log X_{K})\in
C^{\infty}(\overline{\mathscr{B}})$ and satisfies

$\left|\partial_{z}\log X_{K}\right|=O_{r\to\infty}(r^{-2}).$

       * –

Moreover, the quantities $\displaystyle r^{3}X_{K}^{-1}\overset{\circ}{Y}$,
$\displaystyle r^{4}X_{K}^{-1}\overset{\circ}{Y}$ and $\displaystyle
r^{5}X_{K}^{-1}\overset{\circ}{Y}$ are bounded in
$L^{\infty}(\overline{\mathscr{B}})$, which implies that the term
$\displaystyle r^{3}(1+\rho^{-1})X_{K}^{-1}\overset{\circ}{Y}$ is bounded in
$\hat{C}^{1,\alpha_{0}}$

       * –

Finally, recall that we control
$\displaystyle\frac{(1+\rho^{10})(1+r^{10})}{\rho^{10}}B_{z}$ in
$\hat{C}^{1,\alpha_{0}}$.

       * –

This yields to estimates in the region
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$.

     * •

In the region $\overline{\mathscr{B}_{N}}$, we write $N_{\Theta}$ in
$(s,\chi)$ coordinates

       * –

We have

$\displaystyle N_{\Theta}$
$\displaystyle=(N_{\Theta})_{\rho}d\rho+(N_{\Theta})_{z}dz$
$\displaystyle=(\chi(N_{\Theta})_{\rho}(s,\chi)-s(N_{\Theta})_{z}(s,\chi))ds+(s(N_{\Theta})_{\rho}(s,\chi)+\chi(N_{\Theta})_{z}(s,\chi))d\chi.$

       * –

Moreover,

$\partial_{\rho}=\frac{\chi}{\chi^{2}+s^{2}}\partial_{s}+\frac{s}{\chi^{2}+s^{2}}\partial_{\chi},\quad\quad\partial_{z}=\frac{-s}{\chi^{2}+s^{2}}\partial_{s}+\frac{\chi}{\chi^{2}+s^{2}}\partial_{\chi}.$

       * –

Therefore,

$\displaystyle(N_{\Theta})_{s}$
$\displaystyle=\chi\left(-\frac{\sigma}{X^{2}}\left(\frac{-s}{\chi^{2}+s^{2}}\partial_{s}Y+\frac{\chi}{\chi^{2}+s^{2}}\partial_{\chi}Y\right)\right)-s\left(\frac{\sigma}{X^{2}}\left(\frac{\chi}{\chi^{2}+s^{2}}\partial_{s}Y+\frac{s}{\chi^{2}+s^{2}}\partial_{\chi}Y\right)\right)$
$\displaystyle-\frac{\sigma}{X^{2}}B_{\chi}$
$\displaystyle=\frac{s\chi}{X_{K}^{2}}\frac{-\overset{\circ}{\sigma}+\overset{\circ}{X}^{2}+2\overset{\circ}{X}}{(1+\overset{\circ}{X}^{2})^{2}}\partial_{\chi}Y_{K}-\frac{s\chi(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}\partial_{\chi}(X_{K}^{-1}\overset{\circ}{Y})-2\frac{s\chi(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}X_{K}^{-1}\overset{\circ}{Y}\partial_{\chi}\log
X_{K}$ $\displaystyle-\frac{\sigma}{X^{2}}B_{\chi}.$

In the same manner, we obtain

$(N_{\Theta})_{\chi}=\frac{s\chi}{X_{K}^{2}}\frac{-\overset{\circ}{\sigma}+\overset{\circ}{X}^{2}+2\overset{\circ}{X}}{(1+\overset{\circ}{X}^{2})^{2}}\partial_{s}Y_{K}-\frac{s\chi(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}\partial_{s}(X_{K}^{-1}\overset{\circ}{Y})-2\frac{s\chi(1+\overset{\circ}{\sigma})}{(1+\overset{\circ}{X})^{2}}X_{K}^{-1}\overset{\circ}{Y}\partial_{s}\log
X_{K}-\frac{\sigma}{X^{2}}B_{s}.$

     * •

Now, we estimate $s^{-1}(N_{\Theta})_{s}$ in
$\displaystyle\hat{C}^{1,\alpha_{0}}$:

       * –

We start with $\displaystyle\frac{s\chi\partial_{\chi}Y_{K}}{X_{K}^{2}}$. We
recall that $Y_{K}$

$dY_{K}=\theta_{K}$

which takes the following form in the $(s,\chi)$ coordinates

$\partial_{s}(X_{K}^{-1}W_{K})ds+\partial_{\chi}(X_{K}^{-1}W_{K})d\chi=\frac{s\chi}{X_{K}^{2}}\left(\partial_{\chi}Y_{K}-\partial_{s}Y_{K}\right).$

Moreover, we recall the $(x,y,u,v)$ coordinates defined by (5.59) so that any
function $f(s,\chi)$ defined on $\overline{\mathscr{B}_{N}}$ can be see as a
function $f_{\mathbb{R}^{4}}(x,y,u,v)$ defined on $\mathbb{R}^{4}$ Therefore,

$\displaystyle\frac{s\chi}{X_{K}^{2}}\partial_{\chi}Y_{K}$
$\displaystyle=\partial_{s}(X_{K}^{-1}W_{K})$
$\displaystyle=s\left.\partial_{xx}\left(\frac{W_{K}}{X_{K}}\right)\right|_{(0,\sqrt{x^{2}+y^{2}},u,v)}.$

Now, we recall that $\displaystyle\frac{W_{K}}{X_{K}}$ is smooth on
$\overline{\mathscr{B}}$. This yields the $\hat{C}^{1,\alpha_{0}}$ estimates
of $\displaystyle\frac{\chi}{X_{K}^{2}}\partial_{\chi}Y_{K}$.

       * –

The remaining terms are easily controlled in $\hat{C}^{1,\alpha_{0}}$ due to
the estimates for $X_{K}^{-1}\overset{\circ}{Y}$ and $B$.

       * –

Finally, the estimates for $(N_{\Theta})_{\chi}$ follow similarly.

     * •

In the region $\overline{\mathscr{B}_{S}}$, the estimates are obtained as in
$\overline{\mathscr{B}_{N}}$.

∎

###  9.8 Solving for $\lambda$

In this section, we prove the following result

######  Proposition 33.

Let $\alpha_{0}\in(0,1)$ and let $\overline{\delta}_{0}>0$. Then, there exists
$0<\delta_{0}\leq\overline{\delta}_{0}$ such that
$\displaystyle\forall\delta\in[0,\delta_{0}[$ there exists a unique one
parameter family
$\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)\left(\delta\right)\in
B_{\delta_{0}}$
$\left(\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}\right)$
which solves (5.49), (9.53), (5.51), (5.52), (5.53) and which satisfies

$\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)\left(\delta\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}\leq
C(\alpha_{0})\delta$

and $\displaystyle\delta_{i}\in[0,\delta_{0}[$,

$\displaystyle\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)\left(\delta_{1}\right)-\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)\left(\delta_{2}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}\times\mathcal{L}_{\lambda}}$
$\displaystyle\leq C(\alpha_{0})\left|\delta_{1}-\delta_{2}\right|.$

First of all, we recall the following Theorem 1.2 from [16]

######  Theorem 7 (O.CHODOSH, Y.SHLAPENTOKH-ROTHMAN).

Assume that the metric data $(X,W,\theta,\sigma,\lambda)$ is chosen. Assume
the energy momentum tensor $\mathbb{T}$ is chosen so that it satisfies (5.16).
Suppose that

  1. 1.

$\tensor{{T}}{{}_{\alpha\beta}}\in C^{1,\alpha}_{loc}(\mathscr{B})$, $X\in
C^{2,\alpha}_{loc}(\mathscr{B})$, $W\in C^{1,\alpha}_{loc}(\mathscr{B})$,
$\theta\in C^{1,\alpha}_{loc}(\mathscr{B})$, $\sigma\in
C^{3,\alpha}_{loc}(\mathscr{B})$ and $\lambda\in
C^{1,\alpha}_{loc}(\mathscr{B})$,

  2. 2.

$X,W,\theta,\sigma$ satisfy their respective equations on $\mathscr{B}$, as
listed in Theorem 2,

  3. 3.

$|d\sigma|\neq 0$ on $\mathscr{B}$.

  4. 4.

If we form the metric $g$ from (5.1), then $\mathbb{T}$ is divergence free
with respect to $g$.

Then, the $1-$form arising in Equation (5.22) satisfies the following
compatibility condition

$d\alpha=(\beta_{2})\wedge\left(d\lambda-\alpha-\frac{1}{2}d\log X\right),$
(9.75)

where $\beta=(\beta_{2})_{\rho}d\rho+(\beta_{2})_{z}dz$ is a one-form which
depends on my on $\sigma$ through the equations:

$\displaystyle(\beta_{2})_{z}$
$\displaystyle=\frac{1}{2}\frac{(\partial_{\rho}\sigma)(\partial^{3}_{\rho\rho
z}\sigma-\partial^{3}_{z}\sigma)+(\partial_{z}\sigma)(\partial_{\rho}^{3}\sigma-\partial_{\rho
zz}^{3}\sigma+2\partial^{2}_{\rho}\sigma+2\partial_{z}^{2}\sigma)}{(\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2}}$
(9.76) $\displaystyle(\beta_{2})_{\rho}$
$\displaystyle=\frac{1}{2}\frac{(\partial_{z}\sigma)(\partial^{3}_{\rho
zz}\sigma-\partial^{3}_{\rho}\sigma)+(\partial_{\rho}\sigma)(\partial_{z}^{3}\sigma-\partial_{\rho\rho
z}^{3}\sigma+2\partial^{2}_{\rho}\sigma+2\partial_{z}^{2}\sigma)}{(\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2}}$

Furthermore, under the above hypothesis, if $\lambda$ satisfies its first
order equation (5.22), then we automatically have $\lambda\in
C_{loc}^{2,\alpha}(\mathscr{B})$ and that $\lambda$ satisfies the second order
equation given in Theorem 2.

Now we recall that $\overset{\circ}{\lambda}$ satisfies the following
equations

$\displaystyle\partial_{\rho}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X}),$
(9.77) $\displaystyle\partial_{z}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{z}-(\alpha_{K})_{z}-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X}),$

where

$\displaystyle(\alpha_{K})_{\rho}$ $\displaystyle=\frac{1}{4}\rho
X_{K}^{-2}((\partial_{\rho}X_{K})^{2}-(\partial_{z}X_{K})^{2}+(\partial_{\rho}Y_{K})^{2}-(\partial_{z}Y_{K})^{2}),$
$\displaystyle(\alpha_{K})_{z}$ $\displaystyle=\frac{1}{4}\rho
X_{K}^{-2}((\partial_{\rho}X_{K})(\partial_{z}X_{K})+(\partial_{\rho}Y_{K})(\partial_{z}Y_{K})),$

where $\alpha_{\rho}$ and $\alpha_{z}$ satisfy

$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{\rho}$
$\displaystyle=\frac{1}{4}(\partial_{\rho}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}+(\partial_{\rho}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{z}\sigma)((\partial^{2}_{\rho,z}\sigma))$
(9.78)
$\displaystyle+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))),$

$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{z}$
$\displaystyle=-\frac{1}{4}(\partial_{z}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}-(\partial_{z}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{\rho}\sigma)((\partial^{2}_{\rho,z}\sigma))$
(9.79)
$\displaystyle+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))$

Moreover $\lambda=\overset{\circ}{\lambda}+\lambda_{K}$ satisfies the second
order equation

$\displaystyle 2\partial^{2}_{\rho}\lambda+2\partial^{2}_{z}\lambda$
$\displaystyle=-\partial^{2}_{\rho}\log{X}-\partial^{2}_{z}\log{X}+\sigma^{-1}((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})+F_{4}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma},\overset{\circ}{\lambda})(\rho,z)$
(9.80)
$\displaystyle-\frac{1}{2}X^{-2}((\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}+(\theta_{\rho})^{2}+(\theta_{z})^{2}),$

Note that if we try to solve $\overset{\circ}{\lambda}$ by integrating
directly equations (9.77), then we do not know a posteriori whether $\alpha$
satisfies the compatibility condition (9.75) and thus $d\alpha=0$.

  1. 1.

In this case, we proceed as in [17] and we solve the following system of
equations

$\displaystyle\partial_{\rho}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X}),$
(9.81) $\displaystyle\partial_{z}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{z}-(\alpha_{K})_{z}-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X})$
$\displaystyle-\int_{\rho}^{\infty}\left(\beta_{2}\wedge\left(d\overset{\circ}{\lambda}-(\alpha-\alpha_{K})-\frac{1}{2}d\log(1+\overset{\circ}{X})\right)\right)_{\rho,z}(\tau,z)\,d\tau.$

  2. 2.

Next, we show that the solution map verifies the original system of equations:
(9.77).

  3. 3.

Finally, we apply Theorem 7 in order to obtain that $\overset{\circ}{\lambda}$
satisfies the second order equation (9.80).

The remaining of this section is devoted to the proof of the above steps. This
will allow us to obtain Proposition 33. We also note that we follow the same
steps in Section 12 of [17] and we write down details in order to be self-
contained.

####  9.8.1 Solving the modified equations (9.81)

We begin by introducing some notations: let $\overline{\delta}_{0}>0$ and let
$\alpha_{0}\in(0,1)$. Let $0<\delta_{0}\leq\overline{\delta}_{0}$ be obtained
by Proposition 32. Let
$\displaystyle\mathscr{L}:B_{\delta_{0}}(\mathcal{L}_{\lambda})\times[0,\delta_{0}[\to\mathcal{L}_{\lambda}$
be the mapping defined by

$\mathscr{L}(\overset{\circ}{\lambda},\delta)(\rho,z):=-\int_{\rho}^{\infty}\left(\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X})\right)(\tau,z)\,d\tau.$
(9.82)

Here,
$(\sigma,\overset{\circ}{X},\overset{\circ}{\Theta})=(\sigma,\overset{\circ}{X},\overset{\circ}{\Theta})(\overset{\circ}{\lambda},\delta)$.
We omit the dependence on the $(\overset{\circ}{\lambda};\delta)$ in order to
lighten the expressions.
Now, we state the following lemma

######  Lemma 87.

Let $(\sigma,\overset{\circ}{X},\overset{\circ}{\Theta})$ be the solution map
obtained by Proposition 32. Then, $\forall(\overset{\circ}{\lambda},\delta)\in
B_{\delta_{0}}(\mathcal{L}_{\lambda})\times[0,\delta_{0}[$, the one-form
$\alpha$ defined by (9.78) and (9.79) satisfies

$d\alpha=\beta_{2}\wedge\left(d\lambda-\alpha-\frac{1}{2}d\log X\right).$

where $\beta_{2}$ is defined by (9.76).

###### Proof.

We check the assumptions of Theorem 7

  1. 1.

The regularity assumptions are satisfied since the quantities
$\displaystyle\left(\overset{\circ}{\sigma},B,\overset{\circ}{X},\overset{\circ}{Y},\overset{\circ}{\Theta}\right)(\overset{\circ}{\lambda},\delta)$
as well as $\overset{\circ}{\lambda}$ lie in the right spaces.

  2. 2.

$\displaystyle\left(\overset{\circ}{\sigma},B,\overset{\circ}{X},\overset{\circ}{Y},\overset{\circ}{\Theta}\right)(\overset{\circ}{\lambda},\delta)$
solve their respective equations on $\overline{\mathscr{B}}$. Therefore, the
original metric data $(X,W,\theta,\sigma)(\overset{\circ}{\lambda},\delta)$
solve their respective equations on $\mathscr{B}$.

  3. 3.

Since $\sigma_{K}=\rho,$ $d\sigma_{K}=d\rho$ on $\mathscr{B}$. Therefore
$|d\sigma_{K}|\neq 0$. Now, we choose $\delta_{0}>0$ sufficiently small so
that $|d\sigma|\neq 0$. Indeed,

$\displaystyle\left|d\sigma\right|^{2}$
$\displaystyle=\left|d\left(\rho(1+\overset{\circ}{\sigma})\right)\right|^{2}$
$\displaystyle=(1+\overset{\circ}{\sigma}+\rho\partial_{\rho}\overset{\circ}{\sigma})^{2}+(\rho\partial_{z}\overset{\circ}{\sigma})^{2}.$

On $\mathscr{B}$, we have

$\displaystyle\left|d\sigma\right|^{2}$
$\displaystyle\geq(1+\overset{\circ}{\sigma})^{2}+2\rho(\partial_{\rho}\overset{\circ}{\sigma})(1+\overset{\circ}{\sigma})$
$\displaystyle\geq 1-4\delta_{0}.$

The latter is obtained by assuming that $\delta_{0}<1$ and by the control of
$L^{\infty}$ norm of $\rho\partial_{\rho}\overset{\circ}{\sigma}$. Therefore,
we choose $\delta_{0}$ so that the latter is positive.

  4. 4.

$\tensor{{T}}{{}_{\alpha\beta}}$ is divergence free and satisfies (5.16) (See
Section 5.3 ) with respect to $g$ given by (5.1).

Therefore, we apply Theorem 7 to obtain that

  * •

$\alpha$ satisfies (9.75),

  * •

if $\lambda$ satisfies (9.77), then $\lambda$ also satisfies (9.80).

∎

The remaining of this section is to prove the following result

######  Lemma 88.

Let $\overline{\delta}_{0}>0$ and let $\alpha_{0}\in(0,1)$. There exists
$0<\delta_{0}\leq\overline{\delta}_{0}$ such that
$\displaystyle\forall\overset{\circ}{\lambda}\in
B_{\delta_{0}}(\mathcal{L}_{\lambda})\,;\,\forall\delta\in[0,\delta_{0}[$,

  * •

$\displaystyle\mathscr{L}(\overset{\circ}{\lambda},\delta)$ is well-defined
and lies in $\displaystyle B_{\delta_{0}}(\mathcal{L}_{\lambda})$,

  * •

$\displaystyle\mathscr{L}(\overset{\circ}{\lambda},\delta)$ solves (9.81) and
verifies

$\displaystyle\left|\left|\mathscr{L}(\overset{\circ}{\lambda},\delta)\right|\right|_{\mathcal{L}_{\lambda}}\leq
C\left(\left|\left|\overset{\circ}{\lambda}\right|\right|_{\mathcal{L}_{\lambda}}^{2}+\delta\right),$
$\displaystyle\left|\left|\mathscr{L}(\overset{\circ}{\lambda}_{1},\delta_{1})-\mathscr{L}(\overset{\circ}{\lambda}_{2},\delta_{2})\right|\right|_{\mathcal{L}_{\lambda}}\leq
C\left(\left(\left|\left|\overset{\circ}{\lambda}_{1}\right|\right|_{\mathcal{L}_{\lambda}}+\left|\left|\overset{\circ}{\lambda}_{2}\right|\right|_{\mathcal{L}_{\lambda}}\right)\left|\left|\overset{\circ}{\lambda}_{1}-\overset{\circ}{\lambda}_{2}\right|\right|_{\mathcal{L}_{\lambda}}+|\delta_{1}-\delta_{2}|\right),$

###### Proof.

The proof is similar to what has been done before: the estimates are proven in
each region
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}},\overline{\mathscr{B}_{N}},\overline{\mathscr{B}_{S}}$.

  1. 1.

First of all, we claim that there exists $C(\alpha_{0})>0$ such that

$\left|\left|r^{2}(\alpha-\alpha_{K})\right|\right|_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})}+\left|\left|r^{2}d\log(1+\overset{\circ}{X})\right|\right|_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})}\leq
C\delta+C\left|\left|\left(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\cdots\mathcal{L}_{\Theta}}$

     * •

We start with the term $\alpha-\alpha_{K}$. We write

$\displaystyle(\alpha-\alpha_{K})_{\rho}$
$\displaystyle=\frac{1}{4}\left(|\partial\sigma|^{2}(\partial_{\rho}\sigma)\sigma
X^{-2}(\partial_{\rho}X)^{2}-\rho X_{K}^{-2}(\partial_{\rho}X_{K})^{2}\right)$
$\displaystyle+\frac{1}{4}(\partial_{\rho}\sigma)|\partial\sigma|^{-2}\sigma\frac{-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}+|\partial\sigma|^{-2}(\partial_{\rho}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+|\partial\sigma|^{-2}(\partial_{z}\sigma)((\partial^{2}_{\rho,z}\sigma))$
$\displaystyle+\frac{1}{2}|\partial\sigma|^{-2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z})))-\frac{1}{4}\rho
X_{K}^{-2}(-(\partial_{z}X_{K})^{2}+(\partial_{\rho}Y_{K})^{2}-(\partial_{z}Y_{K})^{2})$

Recall from Lemma 55 and Lemma 56 that $\displaystyle\partial_{\rho}\log
X_{K}$ and thus $\displaystyle\partial_{\rho}\log X$ behave like
$\displaystyle\frac{1}{\rho}$ near the axis.
Furthermore, since the renormalised unknowns lie in the right space, the
$\hat{C}^{1,\alpha_{0}}$ estimates is straightforward for all the terms except
the following:

$\displaystyle\frac{1}{4}\left(|\partial\sigma|^{-2}(\partial_{\rho}\sigma)\sigma
X^{-2}(\partial_{\rho}X)^{2}-\rho X_{K}^{-2}(\partial_{\rho}X_{K})^{2}\right)$

In order to estimate the latter, we write

$\displaystyle\frac{1}{4}\left(|\partial\sigma|^{-2}(\partial_{\rho}\sigma)\sigma
X^{-2}(\partial_{\rho}X)^{2}-\rho X_{K}^{-2}(\partial_{\rho}X_{K})^{2}\right)$
$\displaystyle=\frac{1}{4}\left(|\partial\sigma|^{-2}(\partial_{\rho}\sigma)\sigma(\partial_{\rho}\log
X)^{2}-\rho(\partial_{\rho}\log X_{K})^{2}\right)$
$\displaystyle=\frac{1}{4}\left(1-\rho|\partial\sigma|^{-2}\left(\partial_{\rho}\overset{\circ}{\sigma}(1+\overset{\circ}{\sigma})+\rho|\partial\overset{\circ}{\sigma}|^{2}\right)\right)\rho\left(\partial_{\rho}\log(1+\overset{\circ}{X})\right)^{2}+\frac{1}{2}\rho\left(\partial_{\rho}\log
X_{K}\right)\left(\partial_{\rho}\log(1+\overset{\circ}{X})\right)$
$\displaystyle-\frac{1}{4}|\partial\sigma|^{-2}\left((1+\overset{\circ}{\sigma})\partial_{\rho}\overset{\circ}{\sigma}+\rho|\partial\overset{\circ}{\sigma}|^{2}\right)\rho^{2}\left(\partial_{\rho}\log
X_{K}\right)^{2}.$

Since $\rho\partial_{\rho}\log X_{K}$ is bounded and smooth near the axis and
all the remaining terms are $C^{1,\alpha}$ controlled. We obtain the desired
estimate.

     * •

The term $\displaystyle r^{2}d\log(1+\overset{\circ}{X})$ is easily controlled
by $\left|\left|\overset{\circ}{X}\right|\right|_{\mathcal{L}_{X}}$ in
$\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})$.

     * •

We apply the dominated convergence theorem and obtain that
$\mathscr{L}(\overset{\circ}{\lambda},\delta)$ is well defined and lies in
$\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})$.
Moreover, we have the bounds

$\left|\left|\mathscr{L}(\overset{\circ}{\lambda},\delta)\right|\right|_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})}\leq
C\left|\left|\left(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\cdots\mathcal{L}_{\Theta}}.$

  2. 2.

Now, we show that $\mathscr{L}(\overset{\circ}{\lambda},\delta)$ solves
(9.81). We compute,

$\displaystyle\partial_{\rho}\mathscr{L}(\overset{\circ}{\lambda},\delta)$
$\displaystyle=\left(\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X})\right)(\rho,z)$
$\displaystyle\partial_{z}\mathscr{L}(\overset{\circ}{\lambda},\delta)$
$\displaystyle=-\int_{\rho}^{\infty}\partial_{z}\left(\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X})\right)(\tau,z)\,d\tau$

By Lemma 87, $\alpha$ verifies

$\displaystyle d(\alpha-\alpha_{K})$ $\displaystyle=d\alpha$
$\displaystyle=\beta_{2}\wedge\left(d\lambda-\alpha-\frac{1}{2}d\log X\right)$
$\displaystyle=\beta_{2}\wedge\left(d\overset{\circ}{\lambda}-d\lambda_{K}-\alpha-\frac{1}{2}d\log
X_{K}-\frac{1}{2}d\log(1+\overset{\circ}{X})\right).$

We recall that

$d\lambda_{K}=\alpha_{K}-\frac{1}{2}d\log X_{K}.$

Hence,

$\displaystyle d(\alpha-\alpha_{K})$
$\displaystyle=\beta_{2}\wedge\left(d\overset{\circ}{\lambda}-(\alpha-\alpha_{K})-\frac{1}{2}d\log(1+\overset{\circ}{X})\right)$

Hence,

$\displaystyle\partial_{z}\mathscr{L}(\overset{\circ}{\lambda},\delta)$
$\displaystyle=\alpha_{z}(\rho,z)-(\alpha_{z})_{K}(\rho,z)-\frac{1}{2}\partial_{z}(1+\overset{\circ}{X})-\int_{\rho}^{\infty}(d\alpha)_{\rho
z}(\tau,z)\,d\tau$
$\displaystyle=\alpha_{z}(\rho,z)-(\alpha_{z})_{K}(\rho,z)-\frac{1}{2}\partial_{z}(1+\overset{\circ}{X})$
$\displaystyle-\int_{\rho}^{\infty}\left(\beta_{2}\wedge\left(d\overset{\circ}{\lambda}-(\alpha-\alpha_{K})-\frac{1}{2}d\log(1+\overset{\circ}{X})\right)_{\rho
z}(\tau,z)\right)\,d\tau.$

  3. 3.

The estimates in the region $\overline{\mathscr{B}_{N}}$ follow in the same
manner up to computations for the change of coordinates from $(\rho,z)$ to
$(s,\chi)$. We refer to the proof of Lemma 12.2.2 in [17] for details.

$\left|\left|\mathscr{(}\overset{\circ}{\lambda},\delta)\right|\right|_{\hat{C}^{1,\alpha_{0}}(\overline{\mathscr{B}_{N}})}\leq
C\left|\left|\left(\overset{\circ}{\sigma},B,(\overset{\circ}{X},\overset{\circ}{Y}),\overset{\circ}{\Theta}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\cdots\mathcal{L}_{\Theta}}.$

  4. 4.

The remaining estimates follow in the same manner.

  5. 5.

Finally, we choose $\delta_{0}>0$ sufficiently small that
$\displaystyle\mathscr{L}(\overset{\circ}{\lambda},\delta)\in
B_{\delta_{0}}(\mathcal{L}_{\lambda})$.

∎

####  9.8.2 Solving the original equations (5.53)

Now, let $\delta_{0}>0$ and
$(\overset{\circ}{\lambda}(\delta))_{\delta\in[0,\delta_{0}[}$ be the one-
parameter family of solutions to (9.81). In this section, we will show that
the latter actually satisfies (9.77), that is

$d\overset{\circ}{\lambda}=\alpha-\alpha_{K}-\frac{1}{2}d\log(1+\overset{\circ}{X}).$

To this end, let $z\in\mathbb{R}$ be fixed and set

$f_{z}(\rho):=\left(\partial_{z}\overset{\circ}{\lambda}-(\alpha_{z}-(\alpha_{K})_{z})-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X})\right)(\rho,z)$

We state the following lemma

######  Lemma 89.

$\forall z\in\mathbb{R}$ we have

$\forall\rho\geq 0\;,\;f_{z}(\rho)=0.$

###### Proof.

  * •

First of all, since $\overset{\circ}{\lambda}(\delta)$ solves (9.81), we have

$\displaystyle f_{z}(\rho)$
$\displaystyle=\left(\partial_{z}\overset{\circ}{\lambda}-(\alpha_{z}-(\alpha_{K})_{z})-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X})\right)(\rho,z)$
$\displaystyle=\int_{\rho}^{\infty}\,\left(\beta_{2}\wedge\left(d\overset{\circ}{\lambda}-(\alpha-\alpha_{K})-\frac{1}{2}d\log(1+\overset{\circ}{X})\right)\right)_{\rho,z}(\tau,z)\,d\tau$
$\displaystyle=\int_{\rho}^{\infty}\,\left(((\beta_{2})_{\rho}d\rho+(\beta_{2})_{z}dz)\wedge\left(\partial_{z}\overset{\circ}{\lambda}-(\alpha-\alpha_{K})_{z}-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X})\right)\right)_{\rho,z}(\tau,z)\,d\tau$
$\displaystyle=\int_{\rho}^{\infty}\,(\beta_{2})_{\rho}(\tau,z)f_{z}(\tau)\,d\tau.$

  * •

Now, since $\overset{\circ}{\sigma}\in\mathcal{L}_{\sigma}$, we have

$\left|(\beta_{2})_{\rho}\right|\leq Cr^{-3}.$

Therefore $\forall\rho_{0}>0$, there exists $c_{0}=c_{0}(\rho_{0})>0$ such
that

$\forall(\rho,z)\in[\rho_{0},\infty[\times\mathbb{R}\;,\;\left|(\beta_{2})_{\rho}\right|\leq
C\rho^{-3}.$

Moreover, there exists $c_{1}>0$ such that

$\forall(\rho,z)\in\overline{\mathscr{B}}\;\;,\;\;\left|f_{z}(\rho)\right|\leq
c_{1}.$

Hence,

$\displaystyle\left|f_{z}(\rho)\right|$ $\displaystyle\leq
c_{0}\int_{\rho}^{\infty}\,\left|f_{z}(\tau)\right|\tau^{-3}\,d\tau$
$\displaystyle\leq\frac{c_{0}c_{1}}{2}\rho^{-2}.$

  * •

Therefore

$\forall
n\in\mathbb{N}\;,\;\forall(\rho,z)\in[\rho_{0},\infty[\times\mathbb{R}\;\left|f_{z}(\rho)\right|\leq
c_{1}\left(\frac{c_{0}}{2\rho^{2}}\right)^{n}\frac{1}{n!}$

  * •

Hence, when $n\to\infty$, the right hand side goes to $0$ and thus,
$\forall\rho_{0}>0\;,\;\forall(\rho,z)\in[\rho_{0},\infty[\times\mathbb{R}\;:\;f_{z}(\rho)=0.$

Finally, by continuity of $f_{z}$, $f_{z}$ vanishes on $[0,\infty[$. ∎

Therefore, by the previous Lemma, we have

$\partial_{z}\overset{\circ}{\lambda}=\alpha_{z}-(\alpha_{K})_{z}-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X}).$

####  9.8.3 Conclusion

Finally, we combine the previous results in order to prove Proposition 33:

###### Proof.

  1. 1.

First of all, we apply Proposition 32 to find a one-parameter family of
solutions
$\displaystyle\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\right.$
$\left.\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\overset{\circ}{\lambda},\delta\right)$
to their respective equations which depends continuously on
$(\overset{\circ}{\lambda},\delta)\in
B_{\delta_{0}}(\mathcal{L}_{\lambda})\times[0,\delta_{0}[$.

  2. 2.

We apply Lemma 88 in order to show that $\mathscr{L}$ satisfies the
assumptions of Theorem 5.

  3. 3.

We apply Theorem 5 with:

$\mathcal{T}=\mathscr{L}\;,\;\mathcal{L}=\mathcal{L}_{\lambda}\;,\;\mathcal{P}=[0,\overline{\delta}_{0}[$

Therefore, after choosing $0<\delta_{0}\leq\overline{\delta}_{0}$, we obtain a
one-parameter family of solutions
$(\overset{\circ}{\lambda}(\delta))_{\delta\in[0,\delta_{0}[}$ satisfying

$\displaystyle\left|\left|\overset{\circ}{\lambda}(\delta)\right|\right|_{\mathcal{L}_{\lambda}}\leq
C\delta,$
$\displaystyle\left|\left|\overset{\circ}{\lambda}(\delta_{1})-\overset{\circ}{\lambda}(\delta_{2})\right|\right|_{\mathcal{L}_{\lambda}}\leq
C|\delta_{1}-\delta_{2}|,$

  4. 4.

Henceforth, the solution map
$\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)$
can be seen as a one-parameter family depending on $\delta$ in the following
way

$\displaystyle\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\delta\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}}\leq
C(\alpha_{0})\left(||\overset{\circ}{\lambda}(\delta)||^{2}_{\mathcal{L}_{\lambda}}+\delta\right)$
$\displaystyle\leq C(\delta^{2}+\delta)\leq C\delta.$

and $\displaystyle\forall\delta_{i}\in[0,\delta_{0}[$,

$\displaystyle\left|\left|\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\delta_{1}\right)-\left(\overset{\circ}{\sigma},\left(0,0,B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta}\right)\left(\delta_{2}\right)\right|\right|_{\mathcal{L}_{\sigma}\times\mathcal{L}_{B}\times\mathcal{L}_{X}\times\mathcal{L}_{Y}\times\mathcal{L}_{\Theta}}$
$\displaystyle\leq
C(\alpha_{0})\left(\left(\left|\left|\overset{\circ}{\lambda}(\delta_{1})\right|\right|_{\mathcal{L}_{\lambda}}+\left|\left|\overset{\circ}{\lambda}(\delta_{2})\right|\right|_{\mathcal{L}_{\lambda}}\right)\left|\left|\overset{\circ}{\lambda}(\delta_{1})-\overset{\circ}{\lambda}(\delta_{2})\right|\right|_{\mathcal{L}_{\lambda}}+\left|\delta_{1}-\delta_{2}\right|\right)$
$\displaystyle\leq C(\alpha_{0})\delta.$

∎

##  10 Proof of the Main Result

By the previous section, we obtain the following result

######  Proposition 34.

Let $\delta_{0}>0$ and let
$\displaystyle\delta\ni[0,\delta_{0}[\to\left(\overset{\circ}{\sigma},B,\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)(\delta)$
be the one parameter family family of solutions to the reduced EV-system
obtained by Proposition 33. Then,

  * •

the metric $g_{\delta}$ given by

$g_{\delta}:=-V_{\delta}dt^{2}+2W_{\delta}dtd\phi+X_{\delta}d\phi^{2}+e^{2\lambda_{\delta}}\left(d\rho^{2}+dz^{2}\right)$

where

$\displaystyle\lambda_{\delta}$
$\displaystyle:=\overset{\circ}{\lambda}(\delta)+\lambda_{K}$ (10.1)
$\displaystyle X_{\delta}:$
$\displaystyle=X_{K}(1+\overset{\circ}{X}(\delta))$ $\displaystyle
W_{\delta}:$
$\displaystyle=X_{\delta}(\overset{\circ}{\Theta}(\delta)+X_{K}^{-1}W_{K})$
$\displaystyle V_{\delta}$
$\displaystyle:=\frac{\sigma^{2}_{\delta}-W^{2}_{\delta}}{X_{\delta}}\quad\text{where}\quad\sigma_{\delta}:=\rho(1+\overset{\circ}{\sigma}(\delta)),$

  * •

and the distribution function $f^{\delta}$ given by

$f^{\delta}(t,\phi,\rho,z,p^{\rho},p^{\phi},p^{z}):=\Phi(\varepsilon_{\delta},\ell_{z})\Psi_{\eta}(\rho-\rho_{1}((\varepsilon_{\delta},(\ell_{z})_{\delta}),(X_{\delta},W_{\delta},\sigma_{\delta})))$

where

$\displaystyle\varepsilon_{\delta}$
$\displaystyle:=\frac{\sigma_{\delta}}{\sqrt{X_{\delta}}}(1+|p|^{2})^{\frac{1}{2}}-\frac{W_{\delta}}{X_{\delta}}(\ell_{z})_{\delta},$
$\displaystyle(\ell_{z})_{\delta}$ $\displaystyle:=\sqrt{X_{\delta}}p^{\phi},$

and $\rho_{1}$ is the second largest solution of the equation

$E_{(\ell_{z})_{\delta}}(X_{\delta},W_{\delta},\sigma_{\delta},\rho,0)=\varepsilon_{\delta}.$

solve the Einstein-Vlasov system on
$\mathcal{M}=\mathbb{R}\times\mathbb{S}^{1}\times\mathscr{B}$.

It remains to prove the following

  1. 1.

The spacetime $(\mathcal{M},g_{\delta})$ is $C^{2,\alpha}-$extendable to a
black hole spacetime in the sense of Definition 4,

  2. 2.

The extended solution $(\tilde{\mathcal{M}},\tilde{g}_{\delta})$ is
asymptotically flat.

First of all, we claim that

######  Lemma 90.

there exists $\Omega_{\delta}\in\mathbb{R}$ such that

$\left.\frac{W_{\delta}}{X_{\delta}}\right|_{\mathscr{H}}=-\Omega.$

###### Proof.

We recall that $\forall\delta\geq 0$, we have

$\partial_{z}\left(\frac{W_{\delta}}{X_{\delta}}\right)=\frac{\sigma_{\delta}}{X_{\delta}}\left(\partial_{\rho}Y_{\delta}+(B_{\rho})_{\delta}\right).$

On the horizon, the right hand side vanishes, since $\sigma_{\delta}$
vanishes. Therefore, there exists $\Omega_{\delta}\in\mathbb{R}$ such that

$\forall
z\in]-\beta,\beta[\quad,\quad\frac{W_{\delta}}{X_{\delta}}(0,z)=-\Omega_{\delta}$

∎

We recall the following results from [17, Section 13]

######  Proposition 35.

Let $\delta\in[0,\delta_{0}[$. Then,

  1. 1.

$\displaystyle\left.\left(e^{2\lambda}-\rho^{-2}X\right)\right|_{\mathscr{A}}=0$.

  2. 2.

there exists $\kappa(\delta)>0$ such that
$\displaystyle\left.\left(e^{2\lambda}-\kappa_{\delta}^{-2}\rho^{-2}\left(V-2\Omega_{\delta}W-\Omega_{\delta}^{2}X\right)\right)\right|_{\mathscr{H}}=0.$

  3. 3.

On the set $\overline{\mathscr{B}_{N}}$, we have

$\left.\left((\chi^{2}+s^{2})e^{2\lambda}-s^{-2}X\right)\right|_{\left\\{s=0\right\\}\cup\left\\{\chi=0\right\\}}=0.$

  4. 4.

On the set $\overline{\mathscr{B}_{N}}$, we have

$\left.\left((\chi^{2}+s^{2})e^{2\lambda}-\kappa_{\delta}^{-2}\chi^{-2}\left(V-2\Omega_{\delta}W-\Omega_{\delta}^{2}X\right)\right)\right|_{\left\\{s=0\right\\}\cup\left\\{\chi=0\right\\}}=0.$

  5. 5.

On the set $\overline{\mathscr{B}_{S}}$, we have

$\left.\left(((\chi^{\prime})^{2}+(s^{\prime})^{2})e^{2\lambda}-(s^{\prime})^{-2}X\right)\right|_{\left\\{s^{\prime}=0\right\\}\cup\left\\{\chi^{\prime}=0\right\\}}=0.$

  6. 6.

On the set $\overline{\mathscr{B}_{S}}$, we have

$\left.\left(((\chi^{\prime})^{2}+(s^{\prime})^{2})e^{2\lambda}-\kappa_{\delta}^{-2}(\chi^{\prime})^{-2}\left(V-2\Omega_{\delta}W-\Omega_{\delta}^{2}X\right)\right)\right|_{\left\\{s^{\prime}=0\right\\}\cup\left\\{\chi^{\prime}=0\right\\}}=0.$

######  Lemma 91.

Assume that $f(\rho,z)$ is a smooth function on $\overline{\mathscr{B}_{A}}$,
resp. $\overline{\mathscr{B}_{H}}$, that is smooth when considered as a
function on $\mathbb{R}^{n}$ with the metric
$d\rho^{2}+\rho^{2}d\mathbb{S}^{n-2}+dz^{2}$ for $n>2$. Then,
$f(\rho,z)=g(\rho^{2},z)$ for some smooth function on
$\overline{\mathscr{B}_{A}}$, resp $\overline{\mathscr{B}_{H}}$.
If $f(s,\chi)$ is a smooth function on $\overline{\mathscr{B}_{N}}$, that is
smooth when considered as a function on $\mathbb{R}^{n}$ with the metric
$ds^{2}+s^{2}d\phi_{1}^{2}+d\chi^{2}+\chi^{2}d\phi_{2}^{2}$ for $n>2$. Then,
$f(s,\chi)=g(s^{2},\chi)$ for some smooth function on
$\overline{\mathscr{B}_{N}}$.

The above proposition and lemma, together with Proposition 4 yield the
extendability of $(\mathcal{M},g_{\delta})$, for all $\delta\geq 0$. More
precisely, we obtain the following result

######  Proposition 36.

The spacetime $(\mathcal{M},g_{\delta})$ is $C^{2,\alpha}-$extendable to a
Lorentzian manifold with corners $(\tilde{\mathcal{M}},\tilde{g}_{\delta})$
which is stationary and axisymmetric, and whose boundary corresponds to a
bifurcate Killing event horizon.

###### Proof.

  1. 1.

We will only write the details for the extendability near the axis. The other
extensions are obtained in the same way.

  2. 2.

By construction, all the metric coefficients are $C^{2,\alpha}$ on
$\mathscr{B}$.

  3. 3.

By Proposition 4, the Kerr metric is $C^{2,\alpha}$ (it is in fact
$C^{\infty}$) extendable in the sense of Definition 4.

  4. 4.

By Lemma 91, in a neighbourhood of the axis of symmetry, we have

$\displaystyle\overset{\circ}{X}(\delta)(\rho,z)$
$\displaystyle=\overset{\circ}{X}_{\delta,\mathscr{A}}(\rho^{2},z),$
$\displaystyle\overset{\circ}{\Theta}(\delta)(\rho,z)$
$\displaystyle=\overset{\circ}{\Theta}_{\delta,\mathscr{A}}(\rho^{2},z),$
$\displaystyle\overset{\circ}{\sigma}(\delta)(\rho,z)$
$\displaystyle=\overset{\circ}{\sigma}_{\delta,\mathscr{A}}(\rho^{2},z),$

for $C^{2,\alpha}$ functions,
$\overset{\circ}{X}_{\delta,\mathscr{A}},\overset{\circ}{\Theta}_{\delta,\mathscr{A}},\overset{\circ}{\sigma}_{\delta,\mathscr{A}}$
defined on $\tilde{\mathscr{A}}$.

  5. 5.

In $\tilde{\mathscr{A}}$, we have

$\left.X_{\delta}\right|_{\tilde{\mathscr{A}}}=\rho^{2}X_{\mathscr{A}}(\rho^{2},z)(1+\overset{\circ}{X}(\delta)(\rho,z)),$

where $X_{\mathscr{A}}(0,z)>0$. Since $\delta$ is small so that
$||\overset{\circ}{X}(\delta)||_{\mathcal{L}_{X}}<1$, then there exists a
$C^{2,\alpha}$ function
$X_{\delta,\mathscr{A}}:\tilde{\mathscr{A}}\to\mathbb{R}$ such that
$X_{\delta,\mathscr{A}}(0,z)>0$ and

$\left.X_{\delta}\right|_{\tilde{\mathscr{A}}}(\rho,z)=\rho^{2}X_{\delta,\mathscr{A}}(\rho^{2},z).$

Hence, $X_{\delta}$ verifies the third point of Definition 4.

  6. 6.

By (10.1), we have

$W_{\delta}=X_{\delta}(\overset{\circ}{\Theta}(\delta)+X_{K}^{-1}W_{K})$

and

$V_{\delta}=\frac{\sigma^{2}_{\delta}-W^{2}_{\delta}}{X_{\delta}}\quad\text{where}\quad\sigma_{\delta}:=\rho(1+\overset{\circ}{\sigma}(\delta)),$

Again, we use the extendability properties of $X_{K}$ and $W_{K}$ in order to
obtain

$\displaystyle\left.W_{\delta}\right|_{\tilde{\mathscr{A}}}$
$\displaystyle=\left.X_{\delta}\right|_{\tilde{\mathscr{A}}}(\overset{\circ}{\Theta}(\delta)(\rho,z)+X_{\mathscr{A}}(\rho^{2},z)^{-1}W_{\mathscr{A}}(\rho^{2},z))$
$\displaystyle=\rho^{2}X_{\delta,\mathscr{A}}(\rho^{2},z)(\overset{\circ}{\Theta}_{\delta,\mathscr{A}}(\rho^{2},z)+X_{\mathscr{A}}(\rho^{2},z)^{-1}W_{\mathscr{A}}(\rho^{2},z))$
$\displaystyle=\rho^{2}W_{\delta,\mathscr{A}}(\rho^{2},z).$

Hence, $W_{\delta}$ verifies the second point of Definition 4.

  7. 7.

The first point follows from the definition of $V_{\delta}$.

  8. 8.

As for the fourth point, we apply the first point of Proposition 35:

$\displaystyle\left.\left(e^{2\lambda}-\rho^{-2}X\right)\right|_{\mathscr{A}}=0.$

Finally, we apply Proposition 3 to conclude. ∎

######  Remark 30.

The regularity of the metric coefficients depend on the regularity of the
distribution function, which depends on the regularity of the profile $\Phi$.
Moreover, the way one extends the resulting spacetimes to a larger spacetimes
with event horizon depends on the regularity of the profile $\Phi$. Hence, if
$\Phi$ is $C^{k}$, then one can $C^{k+2,\alpha}$-extend.

Finally, we state the following result

######  Proposition 37.

$\forall\delta\in[0,\delta_{0}[$, the spacetime obtained by Proposition 36,
$(\tilde{\mathcal{M}},\tilde{g}_{\delta})$, is asymptotically flat in the
sense of Definition 9.

##  Appendix A Coordinate Systems for asymptotic flatness

In this appendix, we recall from [16, Appendix A] the different change of
coordinates used in order to define "spatial infinity". This allows us to
define a notion of asymptotic flatness adapted to our work.
Assume that $(\mathcal{M},g)$ is extendable to a regular black hole spacetime
with corners $(\tilde{\mathcal{M}},\tilde{g})$. $\tilde{\mathcal{M}}$ is given
by

$\tilde{\mathcal{M}}=\mathbb{R}_{t}\times(0,2\pi)_{\phi}\times\overline{\mathscr{B}}_{\rho,z},$

where $\overline{\mathscr{B}}$ is covered by
$\overline{\mathscr{B}_{H}},\overline{\mathscr{B}_{A}},\overline{\mathscr{B}_{N}}$
and $\overline{\mathscr{B}_{S}}$. Let
$K\subset\subset\overline{\mathscr{B}_{A}}$ be a compact subset of
$\overline{\mathscr{B}_{A}}$ such that

$\overline{\mathscr{B}_{H}}\cup\overline{\mathscr{B}_{N}}\cup\overline{\mathscr{B}_{S}}\nsubseteq\overline{\mathscr{B}}\backslash
K.$

We introduce the coordinates $(t,x,y,z)\in\mathbb{R}\times\mathbb{R}^{3}$ in
the region $\mathbb{R}\times(0,2\pi)\times\overline{\mathscr{B}}\backslash K$
defined by

$\displaystyle x:=\rho\cos\phi,$ (A.1) $\displaystyle y:=\rho\sin\phi.$

Therefore, a stationary axisymmetric spacetime $(\mathcal{M},g)$ is
asymptotically flat if the $(t,x,y,z)$ coordinates defined above, the metric
$\tilde{g}$ has the following expansion in the region
$\mathbb{R}\times(0,2\pi)\times\overline{\mathscr{B}}\backslash K$

$\tilde{g}=\left(1+O\left(r^{-1}\right)\right)\left(-dt^{2}+dx^{2}+dy^{2}+dz^{2}\right)+O\left(r^{-2}\right)\left(dtdx+dtdy+dxdy\right).$

##  Appendix B Classical inequalities and estimates

######  Theorem 8 ([9]).

If $\delta<0$, then there exists $C>0$ such that $\forall u\in
W^{1,p}_{\delta}(\mathbb{R}^{3})$

$||u||_{p,\delta}\leq C||u_{r}||_{p,\delta-1},$

where $u_{r}$ is defined by

$u_{r}:=\left<r\right>^{-1}(x,y,z)^{t}\cdot\partial u.$

######  Lemma 92.

Let $f\;:\;\mathbb{R}^{2}\to\mathbb{R}$ be a $C^{2}$ spherically symmetric
function and $(\rho,\phi)$denote polar coordinates on $\mathbb{R}^{2}$. Then,
for any point $(x_{0},y_{0})\in\mathbb{R}^{2}$ we have

$\left(x_{0}^{2}+y_{0}^{2}\right)^{-1/2}\left.\partial_{\rho}f\right|_{(x,y)=(x_{0},y_{0})}=\left.\partial_{xx}f\right|_{(x,y)=\left(0,\left(x_{0}^{2}+y_{0}^{2}\right)^{1/2}\right)}$

We recall Theorem $6.6$ of [27] concerning Schauder estimates:

######  Theorem 9.

Let $\Omega$ be a $C^{2,\alpha}$ domain in $\mathbb{R}^{n}$ and let $u\in
C^{2,\alpha}(\overline{\Omega})$ be a solution of $\displaystyle
Lu=f\;\text{in}\;\Omega$, where $f\in C^{0,\alpha}(\overline{\Omega})$, $L$ is
an elliptic operator and the coefficients of $L$ satisfy, for positive
constants $\lambda$, $\Lambda$,

$a^{ij}\xi_{i}\xi_{j}\geq\lambda|\xi|^{2}\quad\forall
x\in\Omega,\xi\in\mathbb{R}^{n},$

and

$|a^{ij}|_{0,\alpha;\Omega},|b^{i}|_{0,\alpha;\Omega},|c|_{0,\alpha;\Omega}\leq\Lambda.$

Let $\phi(x)\in C^{2,\alpha}(\overline{\Omega})$ and suppose $u=\phi$ on
$\partial\Omega$. Then

$|u|_{2;\alpha;\Omega}\leq
C(n,\alpha,\lambda,\Lambda,\Omega)\left(|u|_{0;\alpha;\Omega}+|\phi|_{0;\alpha;\Omega}+|f|_{0;\alpha;\Omega}\right).$

We recall Theorem $10.3$ of [34] concerning Schauder estimates for the
Laplacian in $\mathbb{R}^{n}$

######  Theorem 10 (Newtonian estimates).

Let $f\in C_{c}^{k,\alpha}(\mathbb{R}^{n})$ with
$k\in\mathbb{N},\;\alpha\in(0,1)$ and let $K_{f}$ be given by

$K_{f}(x):=\int_{\mathbb{R}^{n}}\,G(x,y)f(y)\,dy\quad\text{where }\quad
G(x,y):=-\frac{1}{|x-y|^{n-2}}$

Then,

$K_{f}\in C^{k+2,\alpha}(\mathbb{R}^{n})$

and there exists $C=C(k,\alpha,n)>0$

$||K_{f}||_{C^{k+2,\alpha}(\mathbb{R}^{n})}\leq
C||f||_{C^{k,\alpha}(\mathbb{R}^{n})}.$

######  Theorem 11 (Calderon-Zygmund estimates).

Let $f\in L^{p}(\mathbb{R}^{n})$ and let $K_{f}$ be as defined in Theorem 10.
Then, $K_{f}$ has weak second derivative in $L^{p}(\mathbb{R}^{n})$ and we
have

$||K_{f}||_{\dot{W}^{2,p}(\mathbb{R}^{n})}\leq
C||f||_{L^{p}(\mathbb{R}^{n})}.$

##  Appendix C Classical Carter-Robinson theory

In the absence of a matter field, the metric quantities
$X,W,\sigma,\theta,\lambda$ satisfy the following equations on $\mathscr{B}$:

  * •

$\sigma=\rho$. Hence,

$\Delta_{\mathbb{R}^{3}}\sigma=0.$

  * •

$\theta$ is given by

$\theta=\frac{X^{2}}{\sigma}(\partial_{z}(X^{-1}W)d\rho-\partial_{\rho}(X^{-1}W)dz).$

  * •

$W$ satisfies the equation

$\partial_{\rho}(X^{-1}W)d\rho+\partial_{z}(X^{-1}W)dz=\frac{\rho}{X^{2}}((\partial_{\rho}Y)dz-(\partial_{z}Y)d\rho).$

  * •

$(X,Y)$ satisfies the following system on $\mathbb{R}^{3}$

$\left\\{\begin{aligned}
\rho^{-1}\partial_{\rho}(\rho\partial_{\rho}X)+\rho^{-1}\partial_{z}(\rho\partial_{z}X)&=\frac{(\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}-(\partial_{\rho}Y)^{2}-(\partial_{z}Y)^{2}}{X},\\\
\rho^{-1}\partial_{\rho}(\rho\partial_{\rho}Y)+\rho^{-1}\partial_{z}(\rho\partial_{z}Y)&=\frac{2(\partial_{\rho}Y)(\partial_{\rho}X)+2(\partial_{z}Y)(\partial_{z}X)}{X}.\end{aligned}\right.$
(C.1)

  * •

$\lambda$ satisfies the equation

$\left\\{\begin{aligned}
\partial_{\rho}\lambda&=\frac{1}{4}\rho{X}^{-2}((\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\partial_{\rho}Y)^{2}-(\partial_{z}Y)^{2})-\frac{1}{2}\partial_{\rho}\log{X},\\\
\partial_{\rho}\lambda&=\frac{1}{4}\rho{X}^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\partial_{\rho}Y)(\partial_{z}Y))-\frac{1}{2}\partial_{z}\log{X}.\end{aligned}\right.$
(C.2)

In particular $(X_{K},Y_{K},\sigma_{K},\theta_{K},\lambda_{K})$ satisfies the
above equations.

[Symbols]

## References

  * [1] Ellery Ames, Håkan Andréasson, and Anders Logg.  On axisymmetric and stationary solutions of the self-gravitating Vlasov system.  Classical and Quantum Gravity, 33(15):155008, 2016.
  * [2] Håkan Andréasson, David Fajman, and Maximilian Thaller.  Static Solutions to the Einstein–Vlasov System with a Nonvanishing Cosmological Constant.  SIAM Journal on Mathematical Analysis, 47(4):2657–2688, 2015.
  * [3] Håkan Andréasson, Markus Kunze, and Gerhard Rein.  Existence of axially symmetric static solutions of the Einstein-Vlasov system.  Communications in mathematical physics, 308(1):23, 2011.
  * [4] Håkan Andréasson, Markus Kunze, and Gerhard Rein.  Rotating, stationary, axially symmetric spacetimes with collisionless matter.  Communications in Mathematical Physics, 329(2):787–808, 2014.
  * [5] Håkan Andréasson, Alan D. Rendall, and Marsha Weaver.  Existence of CMC and constant areal time foliations in $T^{2}$ symmetric spacetimes with Vlasov matter.  Comm. Partial Differential Equations, 29(1-2):237–262, 2004.
  * [6] J. M. Baarden.  Rapidly rotating stars, disks and black holes.  In Les Houches Summer School of Theoretical Physics: Black Holes, pages 241–290, 1973.
  * [7] J. M. Bardeen.  Timelike and null geodesics in the Kerr metric.  In Les Houches Summer School of Theoretical Physics: Black Holes, pages 215–240, 1973.
  * [8] James M Bardeen, William H Press, and Saul A Teukolsky.  Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation.  The Astrophysical Journal, 178:347–370, 1972.
  * [9] Robert Bartnik.  The mass of an asymptotically flat manifold.  Communications on pure and applied mathematics, 39(5):661–693, 1986\.
  * [10] Jürgen Batt, W Faltenbacher, and E Horst.  Stationary spherically symmetric models in stellar dynamics.  Archive for Rational Mechanics and Analysis, 93(2):159–183, 1986\.
  * [11] Léo Bigorgne, David Fajman, Jérémie Joudioux, Jacques Smulevici, and Maximilian Thaller.  Asymptotic stability of Minkowski space-time with non-compactly supported massless Vlasov matter.  arXiv preprint arXiv:2003.03346, 2020.
  * [12] Brandon Carter.  Republication of: Black hole equilibrium states.  General Relativity and Gravitation, 41(12):2873, 2009.
  * [13] Brandon Carter.  Republication of: Black hole equilibrium states part ii. general theory of stationary black hole states.  General Relativity and Gravitation, 42(3):653–744, 2010.
  * [14] Subrahmanyan Chandrasekhar.  The mathematical theory of black holes.  In Oxford, UK: Clarendon (1992) 646 p., OXFORD, UK: CLARENDON (1985) 646 P., 1985.
  * [15] Subrahmanyan Chandrasekhar.  The mathematical theory of black holes, volume 69.  Oxford University Press, 1998.
  * [16] Otis Chodosh and Yakov Shlapentokh-Rothman.  Stationary axisymmetric black holes with matter.  arXiv preprint arXiv:1510.08024, 2015.
  * [17] Otis Chodosh and Yakov Shlapentokh-Rothman.  Time-periodic Einstein-Klein-Gordon bifurcations of Kerr.  Communications in Mathematical Physics, 356(3):1155–1250, 2017\.
  * [18] Yvonne Choquet-Bruhat.  Problème de Cauchy pour le système intégro-différentiel d’Einstein-Liouville.  In Annales de l’institut Fourier, volume 21, pages 181–201, 1971\.
  * [19] Piotr T Chruściel, João Lopes Costa, and Markus Heusler.  Stationary black holes: uniqueness and beyond.  Living Reviews in Relativity, 15(1):7, 2012.
  * [20] Mihalis Dafermos.  A note on the collapse of small data self-gravitating massless collisionless matter.  Journal of Hyperbolic Differential Equations, 3(04):589–598, 2006\.
  * [21] Mihalis Dafermos and Alan D Rendall.  Strong cosmic censorship for $T^{2}$-symmetric cosmological spacetimes with collisionless matter.  arXiv preprint gr-qc/0610075, 2006.
  * [22] Mihalis Dafermos and Alan D Rendall.  Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter.  Communications on Pure and Applied Mathematics, 69(5):815–908, 2016\.
  * [23] L.C. Evans.  Partial Differential Equations.  Graduate studies in mathematics. American Mathematical Society, 1998.
  * [24] David Fajman.  The nonvacuum Einstein flow on surfaces of negative curvature and nonlinear stability.  Communications in Mathematical Physics, 353(2):905–961, 2017.
  * [25] David Fajman, Jérémie Joudioux, and Jacques Smulevici.  The Stability of the Minkowski space for the Einstein-Vlasov system.  arXiv preprint arXiv:1707.06141, 2017.
  * [26] F Fayos and Ch Teijón.  Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time.  General Relativity and Gravitation, 40(11):2433–2460, 2008.
  * [27] David Gilbarg and Neil S Trudinger.  Elliptic partial differential equations of second order.  springer, 2015.
  * [28] Markus Heusler.  Black hole uniqueness theorems, volume 6.  Cambridge University Press, 1996.
  * [29] Werner Israel.  Relativistic kinetic theory of a simple gas.  Journal of Mathematical Physics, 4(9):1163–1181, 1963.
  * [30] Fatima Ezzahra Jabiri.  Static spherically symmetric Einstein-Vlasov bifurcations of the Schwarzschild spacetime.  Ann. Henri Poincaré, 22(7):2355–2406, 2021.
  * [31] J. H. Jeans.  On the theory of star-streaming and the structure of the universe.  Monthly Notices of the Royal Astronomical Society, 76:70–84, 1915\.
  * [32] James Hopwood Jeans.  Problems of cosmogony and stellar dynamics.  CUP Archive, 2017.
  * [33] Leon Lichtenstein.  Gleichgewichtsfiguren rotierender Flüssigkeiten.  Springer-Verlag, 2013.
  * [34] Elliott H Lieb, Michael Loss, et al.  Graduate studies in mathematics.  Analysis, 14, 2001.
  * [35] Hans Lindblad and Martin Taylor.  Global stability of Minkowski space for the Einstein–Vlasov system in the harmonic gauge.  arXiv preprint arXiv:1707.06079, 2017.
  * [36] Pierre-Louis Lions and Benoît Perthame.  Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system.  Inventiones mathematicae, 105(1):415–430, 1991.
  * [37] Yasushi Mino.  Perturbative approach to an orbital evolution around a supermassive black hole.  Physical Review D, 67(8):084027, 2003.
  * [38] Charles W Misner, Kip S Thorne, and John Archibald Wheeler.  Gravitation.  Princeton University Press, 2017.
  * [39] Barrett O’Neill.  The geometry of Kerr black holes.  Courier Corporation, 2014.
  * [40] Klaus Pfaffelmoser.  Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data.  Journal of Differential Equations, 95(2):281–303, 1992.
  * [41] Gerhard Rein.  Static solutions of the spherically symmetric Vlasov–Einstein system.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 115, pages 559–570. Cambridge University Press, 1994.
  * [42] Gerhard Rein and Alan D Rendall.  Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data.  Communications in mathematical physics, 150(3):561–583, 1992.
  * [43] Gerhard Rein and Alan D Rendall.  Smooth static solutions of the spherically symmetric Vlasov-Einstein system.  In Annales de l’IHP Physique théorique, volume 59, pages 383–397, 1993.
  * [44] Hans Ringström.  On the Topology and Future Stability of the Universe.  OUP Oxford, 2013.
  * [45] Olivier Sarbach and Thomas Zannias.  The geometry of the tangent bundle and the relativistic kinetic theory of gases.  Classical and Quantum Gravity, 31(8):085013, 2014.
  * [46] Jack Schaeffer.  A Class of Counterexamples to Jeans’ Theorem for the Vlasov–Einstein System.  Communications in mathematical physics, 204(2):313–327, 1999.
  * [47] Jacques Smulevici.  Strong cosmic censorship for $T^{2}$-symmetric spacetimes with cosmological constant and matter.  In Annales Henri Poincaré, volume 9, pages 1425–1453. Springer, 2008.
  * [48] Jacques Smulevici.  On the area of the symmetry orbits of cosmological spacetimes with toroidal or hyperbolic symmetry.  Analysis & PDE, 4(2):191–245, 2011.
  * [49] John Lighton Synge.  The energy tensor of a continuous medium.  Trans. Roy. Soc. Canada, page 127, 1934.
  * [50] Gerald E Tauber and JW Weinberg.  Internal state of a gravitating gas.  Physical Review, 122(4):1342, 1961.
  * [51] Martin Taylor.  The global nonlinear stability of Minkowski space for the massless Einstein–Vlasov system.  Annals of PDE, 3(1):9, 2017.
  * [52] Robert M Wald.  General relativity.  University of Chicago press, 2010.
  * [53] Marsha Weaver.  On the area of the symmetry orbits in $T^{2}$-2 symmetric spacetimes with Vlasov matter.  Classical and Quantum Gravity, 21(4):1079, 2004.
  * [54] Gilbert Weinstein.  On rotating black holes in equilibrium in general relativity.  Communications on pure and applied mathematics, 43(7):903–948, 1990\.
  * [55] Gilbert Weinstein.  The stationary axisymmetric two-body problem in general relativity.  Communications on pure and applied mathematics, 45(9):1183–1203, 1992.