$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\widehat{\nabla}^{2}\bar{\alpha}^{F}\bar{\alpha}^{F}\rangle|\leq\epsilon^{\frac{1}{2}}\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\widehat{\nabla}\alpha||_{L^{4}(S)}\sup_{\underline{u}}||\widehat{\nabla}^{2}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}$
(7.81)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\widehat{\nabla}\bar{\alpha}^{F}\widehat{\nabla}\bar{\alpha}^{F}\rangle|\leq\epsilon\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||^{2}_{L^{4}(S)}$
(7.82) $\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\widehat{\nabla}\bar{\alpha}^{F}\varphi\rangle|\leq\epsilon\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||_{L^{4}(S)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}$
(7.83) $\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\widehat{\nabla}_{3}\bar{\alpha}^{F}\varphi\rangle|\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.84)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\alpha},\alpha^{F}(\rho^{F},\sigma^{F})\bar{\alpha}\rangle|\leq\epsilon^{\frac{1}{2}}||\nabla\bar{\alpha}||_{L^{2}(\underline{H})}\sup_{u,\underline{u}}||\alpha^{F}||_{L^{4}(S)}||\widehat{\nabla}(\rho^{F},\sigma^{F})||_{L^{4}(S)}||\bar{\alpha}||_{L^{4}(S)}\leq$
(7.85)
$\displaystyle\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$

Therefore the coupled Yang-Mills-Einstein terms are under control. Now we
control the remaining terms using the estimates on the connection coefficients
(lemma 6.1-6.7) as follows

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\bar{\beta}\nabla\varphi\rangle|\leq\epsilon\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\bar{\beta}||_{L^{4}(S)}||\nabla\varphi||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.86)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\nabla\bar{\beta}\varphi\rangle|\leq\epsilon\sup_{u}||\nabla\bar{\beta}||^{2}_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.87)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},\nabla\bar{\alpha}\varphi\rangle|\leq\epsilon^{\frac{1}{2}}\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{\underline{u}}||\nabla\bar{\alpha}||_{L^{2}(\underline{H})}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
(7.88)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\beta},K\bar{\alpha}\rangle|\leq\epsilon\sup_{u}||\nabla\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||K||_{L^{4}(S)}||\bar{\alpha}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}).$ (7.89)

Now we estimate the most dangerous terms

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\alpha},\nabla\bar{\alpha}\mbox{tr}\chi\rangle|\leq\sup_{u,\underline{u}}||\mbox{tr}\chi||_{L^{\infty}(S)}\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}},$
(7.90)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\alpha},\nabla\bar{\alpha}\omega\rangle|\leq\sup_{u,\underline{u}}||\omega||_{L^{\infty}(S)}\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}},$
(7.91)

where we notice that the connection coefficients that appear multiplied with
the top derivative of $\bar{\alpha}$ satisfy $\nabla_{3}$ equation and
therefore solely determined by the initial data $\mathcal{O}_{0}$. This allows
us to use the Grönwall’s inequality to complete the energy estimate. The
remaining terms are harmless

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\alpha},\nabla\Psi\varphi\rangle|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),~{}|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\bar{\alpha},\nabla\varphi\Psi\rangle|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
(7.92)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle(\eta+\underline{\eta})\nabla\bar{\beta},\nabla{\bar{\alpha}}\rangle|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
(7.93)

where $\Psi\in(\beta,\bar{\beta},\rho,\sigma,\alpha)$ and $\varphi$ is any
connection coefficients belonging to the set
$(\widehat{\chi},\underline{\widehat{\chi}},\mbox{tr}\chi,\mbox{tr}\underline{\chi},\underline{\eta},\omega,\eta,\underline{\omega})$.
Collecting all the terms, we may now estimate $ER14$ as follows

$\displaystyle|ER14|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}.$
(7.94)

Therefore collecting all the terms we obtain

$\displaystyle
2\int_{H_{u}}|\nabla\bar{\beta}|^{2}+\int_{\underline{H}_{\underline{u}}}|\nabla\bar{\alpha}|^{2}\leq
2\int_{H_{0}}|\nabla\bar{\beta}|^{2}+\int_{\underline{H}_{0}}|\nabla\bar{\alpha}|^{2}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$
$\displaystyle\leq
C(\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

This completes the proof of the lemma. $\square$

###  7.2 Energy estimates for the Yang-Mills curvature

Now that we have completed the energy estimates for the Weyl curvature, we
move on to estimating the energy associated with the Yang-Mills curvature. For
this purpose, we could use the canonical stress-energy tensor associated with
the Yang-Mills theory. However, we will follow the direct integration by parts
procedure using the null Yang-Mills equations. In the Yang-Mills case, we will
not do the energy estimates separately for each order but only for the topmost
derivatives. If the estimates at the top order close, so do the lower orders
since the hyperbolic structure of the null Yang-Mills equations is preserved
after commuting with derivatives.

###### Lemma 7.3

The horizontal derivatives of the null Yang-Mills curvature satisfy the
following $L^{2}$ energy estimates

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}\leq
C(\mathcal{F}_{0})+\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
(7.95)
$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}$
(7.96) $\displaystyle\leq
C(\mathcal{W}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

for $0\leq I\leq 2$, $I\in\mathbb{Z}$.

_Proof._ In order to prove this estimate, we use the integration identities
(7.1-7.2) once again. First, we identify the pairs
$(\alpha^{F},\rho^{F},\sigma^{F})$ and apply the identities (7.1-7.2) to
obtain

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}=\int_{H_{0}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}$
(7.97)
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\rho^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)}_{ER15}$
$\displaystyle-\underbrace{2\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}\alpha^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\alpha^{F}\rangle+\langle\widehat{\nabla}^{I}\rho^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}\sigma^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\sigma^{F}\rangle\right)}_{ER16}.$

We can directly estimate $ER15$ by using the estimates of the connection
coefficients (lemma 6.1) and the definition of $\mathcal{F}$ since all the
Yang-Mills curvature coefficients can be estimated on $H$ by

$\displaystyle|ER15|\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}).$
(7.98)

In order to estimate $ER16$, we use the null Yang-Mills equations

$\displaystyle
ER16=-2\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}\alpha^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\alpha^{F}\rangle+\langle\widehat{\nabla}^{I}\rho^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}\sigma^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\sigma^{F}\rangle\right)$
(7.99)
$\displaystyle=\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla^{I}\alpha^{F},-\frac{1}{2}\widehat{\nabla}^{I}(\mbox{tr}\underline{\chi}\alpha^{F})-\widehat{\nabla}\widehat{\nabla}^{I}\rho^{F}+~{}^{*}\widehat{\nabla}\widehat{\nabla}^{I}\sigma^{F}-2\widehat{\nabla}^{I}(~{}^{*}\underline{\eta}\sigma^{F})+2\widehat{\nabla}^{I}(\underline{\eta}\rho^{F})\right.$
$\displaystyle\left.+2\widehat{\nabla}^{I}(\underline{\omega}\alpha^{F})-\widehat{\nabla}^{I}(\widehat{\chi}\cdot\bar{\alpha}^{F})+[\widehat{\nabla}_{3},\widehat{\nabla}^{I}]\alpha^{F}+[\widehat{\nabla},\widehat{\nabla}^{I}]\rho^{F}-[^{*}\widehat{\nabla},\widehat{\nabla}^{I}]\sigma^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}\rho^{F},-\hat{div}\widehat{\nabla}^{I}\alpha^{F}-\widehat{\nabla}^{I}(\mbox{tr}\chi\rho^{F})-\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}]\rho^{F}+[\hat{div},\widehat{\nabla}^{I}]\alpha^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}\sigma^{F},-\hat{curl}\widehat{\nabla}^{I}\alpha^{F}-\widehat{\nabla}^{I}(\mbox{tr}\chi\sigma^{F})+\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot~{}^{*}\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}]\sigma^{F}-[\hat{curl},\widehat{\nabla}^{I}]\alpha^{F}\rangle\right)$
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla^{I}\alpha^{F},-\frac{1}{2}\widehat{\nabla}^{I}(\mbox{tr}\underline{\chi}\alpha^{F})-2\widehat{\nabla}^{I}(~{}^{*}\underline{\eta}\sigma^{F})+2\widehat{\nabla}^{I}(\underline{\eta}\rho^{F})\right.$
$\displaystyle\left.+2\widehat{\nabla}^{I}(\underline{\omega}\alpha^{F})-\widehat{\nabla}^{I}(\widehat{\chi}\cdot\bar{\alpha}^{F})+[\widehat{\nabla}_{3},\widehat{\nabla}^{I}]\alpha^{F}+[\widehat{\nabla},\widehat{\nabla}^{I}]\rho^{F}-[^{*}\widehat{\nabla},\widehat{\nabla}^{I}]\sigma^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}\rho^{F},-\widehat{\nabla}^{I}(\mbox{tr}\chi\rho^{F})-\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}]\rho^{F}+[\hat{div},\widehat{\nabla}^{I}]\alpha^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}\sigma^{F},-\widehat{\nabla}^{I}(\mbox{tr}\chi\sigma^{F})+\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot~{}^{*}\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}]\sigma^{F}-[\hat{curl},\widehat{\nabla}^{I}]\alpha^{F}\rangle\right)$
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},((\eta+\underline{\eta})(\widehat{\nabla}^{I}\rho^{F}+\widehat{\nabla}^{I}\sigma^{F})\rangle.$

We will sketch how we handle each term. There will be terms of the following
type that are estimated easily

$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\varphi\widehat{\nabla}^{I}\Phi^{F}\rangle\leq\int_{\mathcal{D}_{u,\underline{u}}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\varphi||_{L^{\infty}(S)}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.100)
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\widehat{\nabla}^{I}\varphi\Phi^{F}\rangle\leq\int_{0}^{u}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(H)}||\widehat{\nabla}^{I}\varphi||_{L^{2}(H)}||\Phi^{F}||_{L^{\infty}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.101)
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\widehat{\nabla}\varphi\widehat{\nabla}\Psi\rangle\leq\int_{\mathcal{D}_{u,\underline{u}}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\widehat{\nabla}\varphi||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$

where $\Phi^{F}\in(\alpha^{F},\rho^{F},\sigma^{F})$ and therefore are
controllable on $H$. For the connection coefficients, we use the lemma
(6.1-6.7). Now we need to control the commutator terms

$\displaystyle[\widehat{\nabla}_{3},\widehat{\nabla}^{2}]\alpha^{F}=[\widehat{\nabla}_{3},\widehat{\nabla}]\widehat{\nabla}\alpha^{F}+\widehat{\nabla}[\widehat{\nabla}_{3},\widehat{\nabla}]\alpha^{F}$
(7.102)

Now

$\displaystyle[\widehat{\nabla}_{3},\widehat{\nabla}]\Phi^{F}\sim\bar{\beta}\Phi^{F}+\bar{\alpha}^{F}\Phi^{F}+\bar{\alpha}^{F}(\rho^{F}+\sigma^{F})\Phi^{F}+(\eta+\underline{\eta})\nabla_{3}\Phi^{F}-\bar{\chi}\nabla\Phi^{F}+\bar{\chi}\eta\Phi^{F},$

and therefore

$\displaystyle[\widehat{\nabla}_{3},\widehat{\nabla}]\widehat{\nabla}\alpha^{F}\sim\bar{\beta}\widehat{\nabla}\alpha^{F}+\bar{\alpha}^{F}\widehat{\nabla}\alpha^{F}+\bar{\alpha}^{F}(\rho^{F}-\sigma^{F})\widehat{\nabla}\alpha^{F}+(\eta+\underline{\eta})\widehat{\nabla}_{3}\widehat{\nabla}\alpha^{F}$
(7.103)
$\displaystyle-\bar{\chi}\nabla\widehat{\nabla}\alpha^{F}+\bar{\chi}\eta\widehat{\nabla}\alpha^{F}$
$\displaystyle\sim\bar{\beta}\widehat{\nabla}\alpha^{F}+\bar{\alpha}^{F}\widehat{\nabla}\alpha^{F}+\bar{\alpha}^{F}(\rho^{F}-\sigma^{F})\widehat{\nabla}\alpha^{F}+(\eta+\underline{\eta})\widehat{\nabla}\widehat{\nabla}_{3}\alpha^{F}$
$\displaystyle-\bar{\chi}\nabla\widehat{\nabla}\alpha^{F}+\bar{\chi}\eta\widehat{\nabla}\alpha^{F}+(\eta+\underline{\eta})[\widehat{\nabla}_{3},\widehat{\nabla}]\alpha^{F},$

where $\widehat{\nabla}\widehat{\nabla}_{3}\alpha^{F}$ is evaluated by means
of the null Yang-Mills equations. Now

$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},[\widehat{\nabla}_{3},\widehat{\nabla}^{I}]\Phi^{F}\rangle$
(7.104)

would contain the following terms

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\bar{\alpha}^{F}\widehat{\nabla}\Phi^{F}\rangle|\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\bar{\alpha}^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\bar{\alpha}^{F}(\rho^{F}+\sigma^{F})\widehat{\nabla}\alpha^{F}\rangle|\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\bar{\alpha}^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}||\rho^{F},\sigma^{F})||_{L^{\infty}(S)}$
$\displaystyle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\bar{\alpha}^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}||\widehat{\nabla}(\rho^{F},\sigma^{F})||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\varphi\varphi\widehat{\nabla}\bar{\alpha}^{F}\rangle|\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\varphi||_{L^{\infty}(S)}||\varphi||_{L^{4}(S)}||\widehat{\nabla}\bar{\alpha}^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\varphi\widehat{\nabla}^{I}\Phi^{F}\rangle|\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||^{2}_{L^{2}(S)}||\varphi||_{L^{\infty}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.105)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\widehat{\nabla}^{I}\Phi^{F}\bar{\beta}\widehat{\nabla}\Phi^{F}|\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\bar{\beta}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$

Now note the following commutation relation

$\displaystyle[\widehat{\nabla}_{4},\widehat{\nabla}^{2}](\rho^{F},\sigma^{F})=[\widehat{\nabla}_{4},\widehat{\nabla}]\widehat{\nabla}(\rho^{F},\sigma^{F})+\widehat{\nabla}[\widehat{\nabla}_{4},\widehat{\nabla}](\rho^{F},\sigma^{F})$
(7.106)

Now

$\displaystyle[\widehat{\nabla}_{4},\widehat{\nabla}](\rho^{F},\sigma^{F})\sim\beta(\rho^{F},\sigma^{F})+\alpha^{F}(\rho^{F},\sigma^{F})+\alpha^{F}(\rho^{F}+\sigma^{F})(\rho^{F},\sigma^{F})+(\eta+\underline{\eta})\nabla_{4}(\rho^{F},\sigma^{F})$
$\displaystyle-\chi\nabla(\rho^{F},\sigma^{F})+\chi\underline{\eta}(\rho^{F},\sigma^{F}),$

and therefore

$\displaystyle[\widehat{\nabla}_{4},\widehat{\nabla}]\widehat{\nabla}(\rho^{F},\sigma^{F})\sim\beta\widehat{\nabla}(\rho^{F},\sigma^{F})+\alpha^{F}\widehat{\nabla}(\rho^{F},\sigma^{F})+\alpha^{F}(\rho^{F}+\sigma^{F})\widehat{\nabla}(\rho^{F},\sigma^{F})$
$\displaystyle+(\eta+\underline{\eta})\widehat{\nabla}_{4}\widehat{\nabla}(\rho^{F},\sigma^{F})-\chi\nabla\widehat{\nabla}(\rho^{F},\sigma^{F})+\bar{\chi}\eta\widehat{\nabla}(\rho^{F},\sigma^{F})$
$\displaystyle\sim\beta\widehat{\nabla}(\rho^{F},\sigma^{F})+\alpha^{F}\widehat{\nabla}(\rho^{F},\sigma^{F})+{\alpha}^{F}(\rho^{F}-\sigma^{F})\widehat{\nabla}(\rho^{F},\sigma^{F})+(\eta+\underline{\eta})\widehat{\nabla}\widehat{\nabla}_{4}(\rho^{F},\sigma^{F})-\bar{\chi}\nabla\widehat{\nabla}(\rho^{F},\sigma^{F})$
$\displaystyle+\bar{\chi}\eta\widehat{\nabla}(\rho^{F},\sigma^{F})+(\eta+\underline{\eta})\left(\beta+\alpha^{F}(\rho^{F}+\sigma^{F}))(\rho^{F},\sigma^{F})+(\eta+\underline{\eta})\widehat{\nabla}_{4}(\rho^{F},\sigma^{F})\right.$
$\displaystyle\left.-\chi\widehat{\nabla}(\rho^{F},\sigma^{F})+\chi\underline{\eta}(\rho^{F},\sigma^{F})\right),$

where $\widehat{\nabla}\widehat{\nabla}_{4}(\rho^{F},\sigma^{F})$ can be
further simplified by means of the null Yang-Mills equations. The resulting
terms are easy to estimate

$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\Phi^{F}\widehat{\nabla}\Phi^{F}\rangle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\Phi^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\Phi^{F}(\rho^{F}+\sigma^{F})\widehat{\nabla}\Phi^{F}\rangle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\Phi^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}||(\rho^{F},\sigma^{F})||_{L^{\infty}(S)}$
$\displaystyle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\Phi^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}||\widehat{\nabla}(\rho^{F},\sigma^{F})||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\varphi\varphi\widehat{\nabla}\Phi^{F}\rangle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}||\varphi||_{L^{\infty}(S)}||\varphi||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\varphi\widehat{\nabla}^{I}\Phi^{F}\rangle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$

where once again $\Phi^{F}\in(\alpha^{F},\rho^{F},\sigma^{F})$ and $\varphi$
is any connection coefficients belonging to the set
$(\widehat{\chi},\underline{\widehat{\chi}},\mbox{tr}\chi,\mbox{tr}\underline{\chi},\underline{\eta},\omega,\eta,\underline{\omega})$.
Now note the following

$\displaystyle[\widehat{\nabla},\widehat{\nabla}^{2}]\Phi^{F}=\widehat{\nabla}[\widehat{\nabla},\widehat{\nabla}]\Phi^{F}+[\widehat{\nabla},\widehat{\nabla}]\widehat{\nabla}\Phi^{F}\sim\widehat{\nabla}((K+\sigma^{F})\Phi^{F})+(K+\sigma^{F})\widehat{\nabla}\Psi$
(7.107)

and therefore

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},[\widehat{\nabla},\widehat{\nabla}^{2}]\Phi^{F}\rangle|\sim|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\Phi^{F},\widehat{\nabla}((K+\sigma^{F})\Phi^{F})+(K+\sigma^{F})\widehat{\nabla}\Phi^{F}\rangle|$
(7.108)
$\displaystyle\leq\int_{u,\underline{u}}||\widehat{\nabla}^{I}\Phi^{F}||_{L^{2}(S)}\left(||\nabla
K||_{L^{2}(S)}||\Phi^{F}||_{L^{\infty}(S)}+||K||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}+||\widehat{\nabla}\sigma^{F}||_{L^{4}(S)}||\Phi^{F}||_{L^{4}(S)}\right.$
$\displaystyle\left.+||\sigma^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}\right)\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$

where $||\nabla K||_{L^{2}(S)},~{}||K||_{L^{4}(S)}$ are controlled by
$\mathcal{W}$ and $\mathcal{F}$ by the virtue of the null Hamiltonian
constraint (3.33, corollary 1). Collecting all the terms together, we obtain

$\displaystyle|ER16|\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$
(7.109)

and therefore

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}=\int_{H_{0}}|\widehat{\nabla}^{I}\alpha^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}$
(7.110) $\displaystyle+\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\leq
C(\mathcal{W}_{0})+\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}).$

Now in order to prove the second estimate of the lemma, we collect the triple
$(\bar{\alpha},\rho^{F},\sigma^{F})$ and apply the integration identities
(7.1-7.2)

$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}=\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}$
(7.111)
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\rho^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})}_{ER17}$
$\displaystyle-2\underbrace{\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\bar{\alpha}^{F}\rangle+\langle\widehat{\nabla}^{I}\rho^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}\sigma^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\sigma^{F}\rangle\right)}_{ER18}.$

Once again notice that the term involving
$|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}$ contains connection coefficients
$(\omega,\mbox{tr}\chi)$ which satisfy $\nabla_{3}$ evolution equations and
therefore are estimated solely by means of the initial data allowing us to use
Grönwall’s inequality. $ER17$ is estimated a follows

$\displaystyle|ER17|\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}\bar{\alpha}^{F}||^{2}_{L^{2}(\bar{H)}}d\underline{u}^{{}^{\prime}}.$
(7.112)

Once again $ER18$ is controlled by means of commuted null Yang-Mills equations

$\displaystyle\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}\bar{\alpha}^{F}\rangle+\langle\widehat{\nabla}^{I}\rho^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}\sigma^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}\sigma^{F}\rangle\right)$
(7.113)
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},-\frac{1}{2}\widehat{\nabla}^{I}(\mbox{tr}\chi\bar{\alpha}^{F})-2\widehat{\nabla}^{I}(~{}^{*}\underline{\eta}\cdot\sigma^{F})-2\widehat{\nabla}^{I}(\underline{\eta}\cdot\rho^{F})+2\widehat{\nabla}^{I}(\omega\bar{\alpha}^{F})-\widehat{\nabla}^{I}(\underline{\widehat{\chi}}\cdot\alpha^{F})\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}\rho^{F},\widehat{\nabla}^{I}(\mbox{tr}\underline{\chi}\rho^{F})+\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot\bar{\alpha}^{F})\rangle+\langle\widehat{\nabla}^{I}\sigma^{F},-\widehat{\nabla}^{I}(\mbox{tr}\underline{\chi}\sigma^{F})+\widehat{\nabla}^{I}((\eta-\underline{\eta})\cdot~{}^{*}\bar{\alpha}^{F})\rangle\right)$
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},[\widehat{\nabla}_{4},\widehat{\nabla}^{I}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},[\widehat{\nabla},\widehat{\nabla}^{I}](\rho^{F},\sigma^{F})\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\rho^{F},[\widehat{\nabla}_{3},\widehat{\nabla}^{I}]\rho^{F}\rangle$
(7.114)
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\rho^{F},[\widehat{\nabla},\widehat{\nabla}^{I}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\sigma^{F},[\widehat{\nabla},\widehat{\nabla}^{I}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},((\eta+\underline{\eta})(\widehat{\nabla}^{I}\rho^{F}+\widehat{\nabla}^{I}\sigma^{F})\rangle$

The terms that appear with the top order derivative of $\bar{\alpha}^{F}$ are
$\mbox{tr}\chi,\omega,$ and $\underline{\widehat{\chi}}$ which are determined
solely by $\mathcal{O}_{0}$. We use the following commutation relation

$\displaystyle[\widehat{\nabla}_{4},\widehat{\nabla}]\bar{\alpha}^{F}\sim\beta\bar{\alpha}^{F}+\alpha^{F}\bar{\alpha}^{F}+\alpha^{F}(\rho^{F}-\sigma^{F})\bar{\alpha}^{F}+(\eta+\underline{\eta})\nabla_{4}\bar{\alpha}^{F}-\bar{\chi}\nabla\bar{\alpha}^{F}+\chi\underline{\eta}\bar{\alpha}^{F}$

to evaluate $[\widehat{\nabla}_{4},\widehat{\nabla}^{2}]\bar{\alpha}^{F}$

$\displaystyle[\widehat{\nabla}_{4},\widehat{\nabla}^{2}]\bar{\alpha}^{F}=[\widehat{\nabla}_{4},\widehat{\nabla}]\widehat{\nabla}\bar{\alpha}^{F}+\widehat{\nabla}[\widehat{\nabla}_{4},\widehat{\nabla}]\bar{\alpha}^{F}.$
(7.115)

These will contain terms of the type

$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},\widehat{\nabla}^{I}\varphi\bar{\alpha}^{F}\rangle\leq||\widehat{\nabla}^{I}\bar{\alpha}^{F}||_{L^{2}(\mathcal{D}_{u,\underline{u}})}\sup_{u,\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||_{L^{4}(S)}||\widehat{\nabla}\varphi||_{L^{2}(\mathcal{D}_{u,\underline{u}})}\leq\epsilon^{1/2}C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},\widehat{\nabla}\varphi\widehat{\nabla}\bar{\alpha}^{F}\rangle\leq\int_{\mathcal{D}_{u,\underline{u}}}||\widehat{\nabla}^{I}\bar{\alpha}^{F}||_{L^{2}(S)}||\widehat{\nabla}\varphi||_{L^{4}(S)}||\widehat{\nabla}\bar{\alpha}^{F}||_{L^{4}(S)}\leq\epsilon^{1/2}C(\mathcal{O}_{0},\mathcal{F})$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},\nabla
K\bar{\alpha}^{F}\rangle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),~{}\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},K\widehat{\nabla}\bar{\alpha}^{F}\rangle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F})$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},K\widehat{\nabla}\bar{\alpha}^{F}\rangle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),~{}\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},\nabla
K\bar{\alpha}^{F}\rangle\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{F}),$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\alpha^{F},\varphi\widehat{\nabla}\bar{\alpha}^{F}\rangle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{F}),~{}\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},\nabla\varphi\bar{\alpha}^{F}\rangle\leq\epsilon^{1/2}C(\mathcal{O}_{0},\mathcal{F})$
$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}\bar{\alpha}^{F},(\omega,\mbox{tr}\chi)\widehat{\nabla}^{I}\bar{\alpha}^{F}\rangle\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}\bar{\alpha}^{F}||_{L^{2}(\underline{H})},$

Notice that the last term is controllable by Grönwall’s inequality and
connection coefficients in the last integral do not contain $\eta$ and
$\underline{\omega}$. Therefore, collecting all the terms, we obtain

$\displaystyle|ER18|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}.$
(7.116)

Substituting this estimate in the main energy identity (7.111) we obtain

$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}=\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}\bar{\alpha}^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}\rho^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}\sigma^{F}|^{2}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$
$\displaystyle\leq
C(\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

This completes the proof of the lemma. $\square$

###  7.3 Energy estimates for $\nabla_{4}\alpha,\nabla_{3}\underline{\alpha}$

We need to estimate the remaining terms of the curvature energy.

###### Lemma 7.4

The null derivatives of the null components of the Weyl curvature satisfy the
following $L^{2}$ energy estimates

$\displaystyle\int_{H_{u}}|\nabla_{4}\alpha|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon
C(\mathcal{O}_{0}\mathcal{W},\mathcal{F}),$ (7.117)
$\displaystyle\int_{\underline{H}_{\underline{u}}}|\nabla_{3}\bar{\alpha}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon
C(\mathcal{O}_{0}\mathcal{W},\mathcal{F})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$
(7.118)
$\displaystyle+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

_Proof._ We proceed exactly the same way as the previous case only we commute
the null derivatives with the evolution equations of the Weyl curvature
components. First we identify the set of pairs $(\alpha,\beta)$ and
$(\bar{\alpha},\bar{\beta})$ and apply the integration identities (7.1-7.2)

$\displaystyle\int_{H_{u}}|\nabla_{4}\alpha|^{2}+2\int_{\underline{H}_{\underline{u}}}|\nabla_{4}\beta|^{2}=\int_{H_{0}}|\nabla_{4}\alpha|^{2}+2\int_{\underline{H}_{0}}|\nabla_{4}\beta|^{2}$
(7.119)
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}|\nabla_{4}\alpha|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}2|\nabla_{4}\beta|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)}_{ER19}$
$\displaystyle-2\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla_{4}\alpha,\nabla_{3}\nabla_{4}\alpha\rangle+2\langle\nabla_{4}\beta,\nabla_{4}\nabla_{4}\beta\rangle\right)}_{ER20}$

Now notice that $\nabla_{4}\beta$ contains the term $\nabla\alpha$ which can
not be controlled on $\underline{H}$ and therefore we needed to include
$||\nabla_{4}\beta||_{L^{2}(\underline{H}_{0}})$ term in the initial data for
the Weyl curvature. Fortunately, $\nabla_{4}\beta$ appears in the error term
$ER19$ within the bulk integral multiplied by $\omega$ and $\mbox{tr}\chi$ and
therefore harmless. We first estimate the error terms

$\displaystyle|ER19|=|\int_{\mathcal{D}_{u,\underline{u}}}|\nabla_{4}\alpha|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}2|\nabla_{4}\beta|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)|$
(7.120) $\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{4}\beta||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

where notice the connection coefficients multiplying $|\nabla_{4}\beta|^{2}$
in the bulk integral is solely determined by the initial data
$\mathcal{O}_{0}$. This allows us to utilize Grönwall’s estimate. Now for the
term $ER20$, we need to use the commuted evolution equations

$\displaystyle
ER20=-2\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla_{4}\alpha,\nabla_{3}\nabla_{4}\alpha\rangle+2\langle\nabla_{4}\beta,\nabla_{4}\nabla_{4}\beta\rangle\right)$
(7.121)
$\displaystyle=-2\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla_{4}\alpha,\nabla\hat{\otimes}\nabla_{4}\beta-\frac{1}{2}\nabla_{4}(\mbox{tr}\underline{\chi}\alpha)+4\nabla_{4}(\underline{\omega}\alpha)-3\nabla_{4}(\widehat{\chi}\rho+~{}^{*}\chi\sigma)+\nabla_{4}((\zeta+4\eta)\hat{\otimes}\beta)\right.$
$\displaystyle\left.+\frac{1}{2}\nabla_{4}(D_{3}R_{44}-D_{4}R_{33})\gamma+[\nabla_{3},\nabla_{4}]\alpha+[\nabla_{4},\nabla]\hat{\otimes}\beta\rangle+2\langle\nabla_{4}\beta,div(\nabla_{4}\alpha)-2\nabla_{4}(\mbox{tr}\chi\beta)-2\nabla_{4}(\omega\beta)\right.$
$\displaystyle\left.+\nabla_{4}(\eta\alpha)-\frac{1}{2}\nabla_{4}(D_{b}R_{44}-D_{4}R_{4b})+[\nabla_{4},div]\alpha\rangle\right)$
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla_{4}\alpha,-\frac{1}{2}\nabla_{4}(\mbox{tr}\underline{\chi}\alpha)+4\nabla_{4}(\underline{\omega}\alpha)-3\nabla_{4}(\widehat{\chi}\rho+~{}^{*}\chi\sigma)+\nabla_{4}((\eta+\underline{\eta})\hat{\otimes}\beta)\right.$
$\displaystyle\left.+\frac{1}{2}\nabla_{4}(D_{3}R_{44}-D_{4}R_{33})\gamma+[\nabla_{3},\nabla_{4}]\alpha+[\nabla_{4},\nabla]\hat{\otimes}\beta\rangle+\langle\nabla_{4}\beta,-2\nabla_{4}(\mbox{tr}\chi\beta)-2\nabla_{4}(\omega\beta)\right.$
$\displaystyle\left.+\nabla_{4}(\eta\alpha)-\frac{1}{2}\nabla_{4}(D_{b}R_{44}-D_{4}R_{4b})+[\nabla_{4},div]\alpha\rangle\right)+\int_{\mathcal{D}_{u,\underline{u}}}\langle(\eta+\underline{\eta})\nabla_{4}\beta,\nabla_{4}\alpha\rangle.$

Now write down the commutators explicitly

$\displaystyle[\nabla_{3},\nabla_{4}]\alpha\sim\sigma\alpha+(\rho^{F}\rho^{F}+\sigma^{F}\sigma^{F})\alpha+\omega\nabla_{3}\alpha+\underline{\omega}\nabla_{4}\alpha+(\eta-\underline{\eta})\nabla\alpha,$
(7.122)
$\displaystyle[\nabla_{4},\nabla]\beta\sim(\beta+\alpha^{F}(\rho^{F}+\sigma^{F}))\beta+(\eta+\underline{\eta})\nabla_{4}\beta-\chi\nabla\beta+\chi\underline{\eta}\beta,$
$\displaystyle[\nabla_{4},\nabla]\alpha\sim(\beta+\alpha^{F}(\rho^{F}+\sigma^{F}))\alpha+(\eta+\underline{\eta})\nabla_{4}\alpha-\chi\nabla\beta+\chi\underline{\eta}\alpha$

and use the following null evolution equation for $\beta$

$\displaystyle\nabla_{4}\beta_{a}+2\mbox{tr}\chi\beta_{a}=(div\alpha)_{a}-2\omega\beta_{a}+(\eta\cdot\alpha)_{a}-\frac{1}{2}(\mathcal{D}_{a}R_{44}-\mathcal{D}_{4}R_{4a}),$
(7.123)

where the Yang-Mills source term may be evaluated as follows

$\displaystyle
D_{b}R_{44}-D_{4}R_{4b}\sim\langle\alpha^{F},\widehat{\nabla}_{b}\alpha^{F}\rangle-\chi_{bc}\mathfrak{T}_{c4}+\eta_{b}\mathfrak{T}_{44}-2\omega\mathfrak{T}_{4b}-(\widehat{\nabla}_{4}\alpha^{F}_{b}\cdot(\rho^{F}+\sigma^{F})$
$\displaystyle-\alpha^{F}_{b}\cdot(\widehat{\nabla}_{4}\rho^{F}+\widehat{\nabla}_{4}\sigma^{F})).$

In addition, we will use the null evolution equations for the connection
coefficients whenever they are available. In the error term $ER20$, we note
that $\nabla_{4}\underline{\eta}$ appears. We can use the estimate from lemma
6.7 to control this term (this was the whole point of proving lemma 6.4-6.7).
Note
$\langle\nabla_{4}\alpha,\nabla_{4}(D_{3}R_{44}-D_{4}R_{33})\gamma\rangle=0$
(since $\nabla_{4}$ commutes with $\gamma$). Now we may estimate each term
separately

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha,\nabla_{4}\varphi\alpha\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\nabla_{4}\varphi||_{L^{4}(S)}||\alpha||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.124)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha,\varphi\nabla_{4}\alpha\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||^{2}_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}\leq\epsilon
C(\mathcal{O},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha\varphi\nabla_{4}\Psi\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}||\nabla_{4}\Psi||_{L^{2}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha\nabla_{4}\varphi\Psi\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\nabla_{4}\varphi||_{L^{4}(S)}||\Psi||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha\varphi\nabla\Psi\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}||\nabla\Psi||_{L^{2}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\alpha,\varphi\Phi^{F}\widehat{\nabla}_{4}\alpha^{F}\rangle|\leq\epsilon\sup_{u}||\nabla\alpha||_{L^{2}(H)}\sup_{u}||\widehat{\nabla}_{4}\alpha^{F}||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}$
$\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\alpha,\varphi\alpha^{F}\widehat{\nabla}_{4}\Phi^{F}\rangle|\leq\epsilon\sup_{u}||\nabla\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty(S)}}||\nabla\alpha^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{2}(S)}$
$\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla\alpha,\varphi\nabla_{4}\alpha\rangle|\leq\epsilon\sup_{u}||\nabla\alpha||_{L^{2}(H)}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha,\varphi\Phi^{F}\widehat{\nabla}_{4}\alpha^{F}\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u}||\widehat{\nabla}_{4}\alpha^{F}||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{4}(S)}$
$\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{4}\alpha,\varphi\alpha^{F}\widehat{\nabla}_{4}\Phi^{F}\rangle|\leq\epsilon\sup_{u}||\nabla_{4}\alpha||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty(S)}}||\nabla\alpha^{F}||_{L^{4}(S)}||\widehat{\nabla}\Phi^{F}||_{L^{2}(S)}$
$\displaystyle\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$

where $\Psi:(\alpha,\beta,\rho,\sigma)$ and
$\Phi^{F}:=(\alpha^{F},\rho^{F},\sigma^{F})$. We have also used the
$||\nabla_{4}\varphi||_{L^{4}(S)}$ estimate from corollary (2). Collecting all
the terms, we obtain

$\displaystyle|ER20|\leq\epsilon C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$
(7.125)

and therefore

$\displaystyle\int_{H_{u}}|\nabla_{4}\alpha|^{2}+2\int_{\underline{H}_{\underline{u}}}|\nabla_{4}\beta|^{2}\leq\int_{H_{0}}|\nabla_{4}\alpha|^{2}+2\int_{\underline{H}_{0}}|\nabla_{4}\beta|^{2}$
(7.126) $\displaystyle+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{4}\beta||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$

by means of Grönwall’s inequality. Now we need to prove the second part of the
lemma. Once again we identify the pair $(\bar{\alpha},\bar{\beta})$ and apply
the integration identities (7.1-7.2) to yield

$\displaystyle
2\int_{H_{u}}|\nabla_{3}\bar{\beta}|^{2}+\int_{\underline{H}_{\underline{u}}}|\nabla_{3}\bar{\alpha}|^{2}=2\int_{H_{0}}|\nabla_{3}\bar{\beta}|^{2}+\int_{\underline{H}_{0}}|\nabla_{3}\bar{\alpha}|^{2}$
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}2|\nabla_{3}\bar{\beta}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}|\nabla_{3}\bar{\alpha}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)}_{ER21}$
$\displaystyle-2\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}\left(2\langle\nabla_{3}\bar{\beta},\nabla_{3}\nabla_{3}\bar{\beta}\rangle+\langle\nabla_{3}\bar{\alpha},\nabla_{4}\nabla_{3}\bar{\alpha}\rangle\right)}_{ER22}.$

We first control $ER21$. Notice that $\underline{\omega}$ does not satisfy a
$\nabla_{3}$ equation and as a result
$||\underline{\omega}||_{L^{\infty}(S)}\leq
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$. Luckily
$|\nabla_{3}\bar{\beta}|^{2}$ is controlled on $H$ and therefore the first
term of $ER21$ can be estimated as
$\int_{0}^{u}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}du^{{}^{\prime}}$.
For the second term, once again $|\nabla_{3}\bar{\alpha}|^{2}$ is multiplied
by $\omega$ and $\mbox{tr}\chi$ which are solely determined by the norm of the
initial data $\mathcal{O}_{0}$. Therefore, we have

$\displaystyle|ER21|\leq
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\int_{0}^{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}du^{{}^{\prime}}+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$
(7.127) $\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

Utilizing the null Bianchi equations, $ER22$ may be written as follows

$\displaystyle
ER22=-2\int_{\mathcal{D}_{u,\underline{u}}}\left(2\langle\nabla_{3}\bar{\beta},\nabla_{3}\nabla_{3}\bar{\beta}\rangle+\langle\nabla_{3}\bar{\alpha},\nabla_{4}\nabla_{3}\bar{\alpha}\rangle\right)$
$\displaystyle=-2\int_{\mathcal{D}_{u,\underline{u}}}\left(2\langle\nabla_{3}\bar{\beta},-(div\nabla_{3}\bar{\alpha})-2\nabla_{3}(\mbox{tr}\underline{\chi}\bar{\beta})-2\nabla_{3}(\underline{\omega}\bar{\beta})+\nabla_{3}(\underline{\eta}\cdot\bar{\alpha})\right.$
$\displaystyle\left.+\frac{1}{2}\nabla_{3}(\mathcal{D}_{a}R_{33}-\mathcal{D}_{3}R_{3a})+[div,\nabla_{3}]\bar{\alpha}\rangle\right.$
$\displaystyle\left.+\langle\nabla_{3}\bar{\alpha},-(\nabla\hat{\otimes}\nabla_{3}\bar{\beta})-\frac{1}{2}\nabla_{3}(\mbox{tr}\chi\bar{\alpha})+4\nabla_{3}(\omega\bar{\alpha})-3\nabla_{3}(\underline{\widehat{\chi}}\rho-~{}^{*}\underline{\widehat{\chi}}\sigma)+\nabla_{3}((\zeta-4\underline{\eta})\hat{\otimes}\bar{\beta})\right.$
$\displaystyle\left.+\frac{1}{2}\nabla_{3}(\mathcal{D}_{4}R_{33}-\mathcal{D}_{3}R_{34})\gamma+[\nabla_{4},\nabla_{3}]\bar{\alpha}+[\nabla,\nabla_{3}]\hat{\otimes}\bar{\beta}\rangle\right)$
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla_{3}\bar{\beta},-2\nabla_{3}(\mbox{tr}\underline{\chi}\bar{\beta})-2\nabla_{3}(\underline{\omega}\bar{\beta})+\nabla_{3}(\underline{\eta}\cdot\bar{\alpha})+\frac{1}{2}\nabla_{3}(\mathcal{D}_{a}R_{33}-\mathcal{D}_{3}R_{3a})+[div,\nabla_{3}]\bar{\alpha}\rangle\right.$
$\displaystyle\left.+\langle\nabla_{3}\bar{\alpha},-\frac{1}{2}\nabla_{3}(\mbox{tr}\chi\bar{\alpha})+4\nabla_{3}(\omega\bar{\alpha})-3\nabla_{3}(\underline{\widehat{\chi}}\rho-~{}^{*}\underline{\widehat{\chi}}\sigma)+\nabla_{3}((\zeta-4\underline{\eta})\hat{\otimes}\bar{\beta})\right.$
$\displaystyle\left.+[\nabla_{4},\nabla_{3}]\bar{\alpha}+[\nabla,\nabla_{3}]\hat{\otimes}\bar{\beta}\rangle+\langle(\eta+\underline{\eta})\nabla_{3}\bar{\beta},\nabla_{3}\bar{\alpha}\rangle\right),$

where
$\langle\nabla_{3}\bar{\alpha},\frac{1}{2}\nabla_{3}(D_{4}R_{33}-D_{3}R_{44})\gamma\rangle=0$
due to $\gamma-$trace-less property of $\bar{\alpha}$. Now we want to estimate
each term separately. First, note the commutators property

$\displaystyle[\nabla_{3},\nabla_{4}]\bar{\alpha}\sim\sigma\bar{\alpha}+(\rho^{F}\rho^{F}+\sigma^{F}\sigma^{F})\bar{\alpha}+\omega\nabla_{3}\bar{\alpha}+\underline{\omega}\nabla_{4}\bar{\alpha}+(\eta-\underline{\eta})\nabla\bar{\alpha},$
(7.128)
$\displaystyle[\nabla_{3},\nabla]\bar{\beta}\sim(\bar{\beta}+\bar{\alpha}^{F}(\rho^{F}+\sigma^{F}))\bar{\beta}+(\eta+\underline{\eta})\nabla_{3}\bar{\beta}-\bar{\chi}\nabla\bar{\beta}+\bar{\chi}\eta\bar{\beta},$
$\displaystyle[\nabla_{3},\nabla]\bar{\alpha}\sim(\bar{\beta}+\bar{\alpha}^{F}(\rho^{F}+\sigma^{F}))\bar{\alpha}+(\eta+\underline{\eta})\nabla_{4}\bar{\alpha}-\bar{\chi}\nabla\bar{\alpha}+\bar{\chi}\eta\bar{\alpha}.$

Also, recall the null Bianchi equations that we shall make use of

$\displaystyle\nabla_{3}\bar{\beta}_{a}+2\mbox{tr}\underline{\chi}\bar{\beta}_{a}=-(div\bar{\alpha})_{a}-2\underline{\omega}\bar{\beta}_{a}+(\underline{\eta}\cdot\bar{\alpha})_{a}+\frac{1}{2}(\mathcal{D}_{a}R_{33}-\mathcal{D}_{3}R_{3a})$
(7.129)
$\displaystyle\nabla_{4}\bar{\alpha}_{ab}+\frac{1}{2}\mbox{tr}\chi\bar{\alpha}=-(\nabla\hat{\otimes}\bar{\beta})_{ab}+4\omega\bar{\alpha}_{ab}-3(\underline{\widehat{\chi}}_{ab}\rho-~{}^{*}\underline{\widehat{\chi}}_{ab}\sigma)+((\zeta-4\underline{\eta})\hat{\otimes}\bar{\beta})_{ab}$
(7.130)
$\displaystyle+\frac{1}{2}(\mathcal{D}_{4}R_{33}-\mathcal{D}_{3}R_{34})\gamma_{ab}+\frac{1}{2}\left(D_{a}R_{3b}+D_{b}R_{3a}\right),$

and

$\displaystyle
D_{b}R_{33}-D_{3}R_{3b}\sim\langle\bar{\alpha}^{F},\widehat{\nabla}\bar{\alpha}^{F}\rangle-\bar{\chi}\bar{\alpha}^{F}\cdot(\rho^{F}+\sigma^{F})+\underline{\eta}|\bar{\alpha}^{F}|^{2}-2\underline{\omega}\bar{\alpha}^{F}\cdot(\rho^{F}+\sigma^{F})$
(7.131)
$\displaystyle-\widehat{\nabla}_{3}(\bar{\alpha}^{F}\cdot\rho^{F}+\bar{\alpha}^{F}\cdot\sigma^{F}).$

Note an extremely important point that
$||\eta-\underline{\eta}||_{L^{\infty}}$ is completely determined by
$C(\mathcal{O}_{0})$ since $\eta-\underline{\eta}$ satisfies a $\nabla_{3}$
equation

$\displaystyle\nabla_{3}(\eta-\underline{\eta})\sim-\nabla\underline{\omega}-\bar{\chi}\cdot(\eta+\zeta)+\underline{\omega}(\zeta-\eta)-\frac{1}{2}\bar{\beta}+\frac{1}{2}\sigma^{F}\bar{\alpha}^{F},$
(7.132)

where $\zeta=\frac{1}{2}(\eta-\underline{\eta})$ After commuting with
$\nabla$, a direct application of the transport inequality (proposition 5.2),
and lemma 6.1-6.4 yields

$\displaystyle||\nabla(\eta-\underline{\eta})||_{L^{2}(S)}\leq
C(\mathcal{O}_{0}).$ (7.133)

Application of the trace inequality yields

$\displaystyle||\nabla(\eta-\underline{\eta})||_{L^{4}(S)}\leq
C\left(||\nabla(\eta-\underline{\eta})||_{L^{4}(S_{0,\underline{u}})}+||\nabla(\eta-\underline{\eta})||^{1/2}_{L^{2}(\underline{H})}||\widehat{\nabla}_{3}\nabla(\eta-\underline{\eta})||^{1/4}_{L^{2}(\underline{H})}(\nabla||\eta-\underline{\eta}||_{L^{2}(\underline{H})}\right.$
$\displaystyle\left.+||\nabla^{2}(\eta-\underline{\eta})||_{L^{2}(\underline{H})})^{1/4}\right)$
$\displaystyle\leq
C(\mathcal{O}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F},\mathcal{W}(S),\mathcal{F}(S))$
$\displaystyle\leq C(\mathcal{O}_{0})$

after choosing sufficiently small $\epsilon$. Now use the Sobolev inequality
(5.6) to yield111111So effectively in this setting, we have an improvement
$||\eta||_{L^{\infty}(S)}\leq C(\mathcal{O}_{0})$ too.

$\displaystyle||\eta-\underline{\eta}||_{L^{\infty}(S)}\leq
C(\mathcal{O}_{0}).$ (7.134)

This is extremely important to handle the term
$\int_{D_{u,\underline{u}}}\langle\nabla_{3}\bar{\alpha},(\eta-\underline{\eta})\nabla\bar{\alpha}\rangle$
that arises as a result of commutation. The remaining terms can be estimated
similarly to those of the previous cases. We need only take care of the terms
where $\nabla_{3}\bar{\alpha}$ appears quadratic and keep track of the
associated coefficients. We estimate each term using lemma 6.1-6.7 as follows

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},(\nabla_{3}\varphi)\bar{\beta}|\leq\epsilon\sup_{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}du^{{}^{\prime}}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.135)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\varphi\nabla_{3}\bar{\beta}\rangle|\leq\int_{0}^{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}$
(7.136) $\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\nabla_{3}\varphi\bar{\alpha}\rangle|\leq\epsilon\sup_{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.137)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\varphi\nabla_{3}\bar{\alpha}\rangle|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
(7.138)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta}\widehat{\nabla}^{2}_{3}\bar{\alpha}^{F}\Phi^{F}\rangle|\leq||\nabla_{3}\bar{\beta}||_{L^{2}(\mathcal{D}_{u,\underline{u}})}||\widehat{\nabla}^{2}_{3}\bar{\alpha}^{F}||_{L^{2}(\mathcal{D}_{u,\underline{u}})}\sup_{u,\underline{u}}||\Phi^{F}||_{L^{\infty}(S)}$
(7.139)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{F}_{0},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta}\widehat{\nabla}_{3}\widehat{\nabla}\bar{\alpha}^{F}\Phi^{F}\rangle|\leq||\nabla_{3}\bar{\beta}||_{L^{2}(\mathcal{D}_{u,\underline{u}})}||\widehat{\nabla}_{3}\widehat{\nabla}\bar{\alpha}^{F}||_{L^{2}(\mathcal{D}_{u,\underline{u}})}\sup_{u,\underline{u}}||\Phi^{F}||_{L^{\infty}(S)}$
(7.140)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{F}_{0},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\varphi\widehat{\nabla}_{3}\bar{\alpha}^{F}\Phi^{F}\rangle|\leq\epsilon\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}||\widehat{\nabla}_{3}\bar{\alpha}^{F}||_{L^{4}(S)}||\Phi^{F}||_{L^{4}(S)}$
$\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\widehat{\nabla}_{3}\bar{\alpha}^{F}\widehat{\nabla}_{3}\Phi^{F}\rangle|\leq\epsilon\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}\sup_{u,\underline{u}}||\widehat{\nabla}_{3}\bar{\alpha}^{F}||_{L^{4}(S)}||\widehat{\nabla}_{3}\Phi^{F}||_{L^{4}(S)}$
$\displaystyle\leq\epsilon
C(\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\alpha},\mbox{tr}\chi\nabla_{3}\bar{\alpha}+\omega\nabla_{3}\bar{\alpha})\rangle|\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{\underline{H}}d\underline{u}^{{}^{\prime}},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\alpha}\varphi\nabla_{3}\Psi\rangle\leq\epsilon^{\frac{1}{2}}\sup_{\underline{u}}||\nabla_{3}\bar{\alpha}||_{L^{2}(\underline{H})}\sup_{u}||\nabla\Psi||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty(S)}}$
(7.141)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\alpha},\omega\nabla_{3}\bar{\alpha}\rangle|\leq
C(\mathcal{O}_{0}\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}},$
(7.142)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\alpha},(\eta-\underline{\eta})\nabla\bar{\alpha}\rangle|\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}},$
(7.143)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\varphi\nabla_{4}\bar{\alpha}\rangle|\leq\epsilon\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}||\nabla_{4}\bar{\alpha}||_{L^{2}(H)}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}$
(7.144) $\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\nabla_{3}\bar{\beta},\varphi\nabla\bar{\alpha}\rangle|\leq\epsilon^{\frac{1}{2}}\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}\sup_{\underline{u}}||\nabla\bar{\alpha}||_{L^{2}(\underline{H})}\sup_{u,\underline{u}}||\varphi||_{L^{\infty}(S)}$
(7.145)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle(\eta+\underline{\eta})\nabla_{3}\bar{\beta},\nabla_{3}\bar{\alpha}\rangle|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)},$
(7.146)

where we have utilized the available Bianchi equations and null Yang-Mills
equations (e.g., we have $\nabla_{4}\bar{\alpha}$ which does not contain
derivatives of $\bar{\alpha}$ thereby allowing us to control it on $H$ since
we have $L^{4}(S)$ of $\bar{\alpha}$ under control). Once again, we observe
that the connection coefficients $\omega$ and $\mbox{tr}\chi$ multiplying the
most dangerous term
$\langle\nabla_{3}\bar{\alpha},\nabla_{3}\bar{\alpha}\rangle$ are determined
completely by their initial value. Collecting all the terms, we obtain

$\displaystyle
2\int_{H_{u}}|\nabla_{3}\bar{\beta}|^{2}+\int_{\underline{H}_{\underline{u}}}|\nabla_{3}\bar{\alpha}|^{2}\leq
2\int_{H_{0}}|\nabla_{3}\bar{\beta}|^{2}+\int_{\underline{H}_{0}}|\nabla_{3}\bar{\alpha}|^{2}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||^{2}_{L^{2}(H)}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\nabla_{3}\bar{\beta}||_{L^{2}(H)}+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}+C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})\int_{0}^{\underline{u}}||\nabla_{3}\bar{\alpha}||_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}$

which upon utilizing Grönwall’s inequality and smallness of $\epsilon$ yields

$\displaystyle
2\int_{H_{u}}|\nabla_{3}\bar{\beta}|^{2}+\int_{\underline{H}_{\underline{u}}}|\nabla_{3}\bar{\alpha}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0}).$ (7.147)

This concludes the proof of the lemma. $\square$

###  7.4 Energy estimates for
$\widehat{\nabla}_{4}\alpha^{F},\widehat{\nabla}^{2}_{4}\alpha^{F},\widehat{\nabla}_{4}\widehat{\nabla}\alpha^{F},\widehat{\nabla}_{3}\underline{\alpha}^{F},\widehat{\nabla}^{2}_{3}\underline{\alpha}^{F},\widehat{\nabla}_{3}\widehat{\nabla}\underline{\alpha}^{F}$

Now we are left with estimating the similar estimates for the Yang-Mills
curvature components. Since the proof goes exactly a similar way as the Weyl
curvature case, we will only sketch the proof.

###### Lemma 7.5

The null derivatives of the null components of the Weyl curvature satisfy the
following $L^{2}$ energy estimates

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.148)
$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$ (7.149)
$\displaystyle+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}d\underline{u},$
$\displaystyle\int_{H_{u}}|\widehat{\nabla}_{4}\widehat{\nabla}\alpha^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$ (7.150)
$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}_{3}\widehat{\nabla}\bar{\alpha}^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$ (7.151)
$\displaystyle+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}_{3}\widehat{\nabla}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}d\underline{u}$

for $1\leq I\leq 2,I\in\mathbb{Z}$.

_Proof._ We only sketch the proof for $I=2$ since that is the most non-trivial
case for the Yang-Mills fields. Once again we identify the triple
$(\alpha^{F},\rho^{F},\sigma^{F})$ and apply the integration identities
(7.1-7.2)

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}=\int_{H_{0}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}$
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)}_{ER23}$
$\displaystyle-\underbrace{2\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}_{4}\alpha^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}_{4}\alpha^{F}\rangle+\langle\widehat{\nabla}^{I}_{4}\rho^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}_{4}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}_{4}\sigma^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}_{4}\sigma^{F}\rangle\right)}_{ER24}.$

Using the estimates on the connection coefficients, the first error term
$ER_{23}$ is estimated easily

$\displaystyle|ER23|=|\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)|$
$\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}||\widehat{\nabla}^{I}_{4}\alpha^{F}||^{2}_{L^{2}(H)}+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\rho^{F}||^{2}_{L^{2}(\underline{H})}+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\sigma^{F}||^{2}_{L^{2}(\underline{H})}$
$\displaystyle\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\rho^{F}||^{2}_{L^{2}(\underline{H})}+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\sigma^{F}||^{2}_{L^{2}(\underline{H})}$

In order to estimate $ER24$, we utilize the null Yang-Mills equations and
proceed in an exactly similar way as the previous case

$\displaystyle\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla^{I}_{4}\alpha^{F},-\frac{1}{2}\widehat{\nabla}^{I}_{4}(\mbox{tr}\underline{\chi}\alpha^{F})-\widehat{\nabla}\widehat{\nabla}^{I}_{4}\rho^{F}+~{}^{*}\widehat{\nabla}\widehat{\nabla}^{I}_{4}\sigma^{F}-2\widehat{\nabla}^{I}_{4}(~{}^{*}\underline{\eta}\sigma^{F})+2\widehat{\nabla}^{I}_{4}(\underline{\eta}\rho^{F})\right.$
$\displaystyle\left.+2\widehat{\nabla}^{I}_{4}(\underline{\omega}\alpha^{F})-\widehat{\nabla}^{I}_{4}(\widehat{\chi}\cdot\bar{\alpha}^{F})+[\widehat{\nabla}_{3},\widehat{\nabla}^{I}_{4}]\alpha^{F}+[\widehat{\nabla},\widehat{\nabla}^{I}_{4}]\rho^{F}-[^{*}\widehat{\nabla},\widehat{\nabla}^{I}_{4}]\sigma^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}_{4}\rho^{F},-\hat{div}\widehat{\nabla}^{I}_{4}\alpha^{F}-\widehat{\nabla}^{I}_{4}(\mbox{tr}\chi\rho^{F})-\widehat{\nabla}^{I}_{4}((\eta-\underline{\eta})\cdot\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}_{4}]\rho^{F}+[\hat{div},\widehat{\nabla}^{I}_{4}]\alpha^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}_{4}\sigma^{F},-\hat{curl}\widehat{\nabla}^{I}_{4}\alpha^{F}-\widehat{\nabla}^{I}_{4}(\mbox{tr}\chi\sigma^{F})+\widehat{\nabla}^{I}_{4}((\eta-\underline{\eta})\cdot~{}^{*}\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}_{4}]\sigma^{F}-[\hat{curl},\widehat{\nabla}^{I}_{4}]\alpha^{F}\rangle\right)$
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\nabla^{I}_{4}\alpha^{F},-\frac{1}{2}\widehat{\nabla}^{I}_{4}(\mbox{tr}\underline{\chi}\alpha^{F})-2\widehat{\nabla}^{I}_{4}(~{}^{*}\underline{\eta}\sigma^{F})+2\widehat{\nabla}^{I}_{4}(\underline{\eta}\rho^{F})\right.$
$\displaystyle\left.+2\widehat{\nabla}^{I}_{4}(\underline{\omega}\alpha^{F})-\widehat{\nabla}^{I}_{4}(\widehat{\chi}\cdot\bar{\alpha}^{F})+[\widehat{\nabla}_{3},\widehat{\nabla}^{I}_{4}]\alpha^{F}+[\widehat{\nabla},\widehat{\nabla}^{I}_{4}]\rho^{F}-[^{*}\widehat{\nabla},\widehat{\nabla}^{I}_{4}]\sigma^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}_{4}\rho^{F},-\widehat{\nabla}^{I}_{4}(\mbox{tr}\chi\rho^{F})-\widehat{\nabla}^{I}_{4}((\eta-\underline{\eta})\cdot\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}_{4}]\rho^{F}+[\hat{div},\widehat{\nabla}^{I}_{4}]\alpha^{F}\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}_{4}\sigma^{F},-\widehat{\nabla}^{I}_{4}(\mbox{tr}\chi\sigma^{F})+\widehat{\nabla}^{I}_{4}((\eta-\underline{\eta})\cdot~{}^{*}\alpha^{F})+[\widehat{\nabla}_{4},\widehat{\nabla}^{I}_{4}]\sigma^{F}-[\hat{curl},\widehat{\nabla}^{I}_{4}]\alpha^{F}\rangle\right)$
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{4}\alpha^{F},((\eta+\underline{\eta})(\widehat{\nabla}^{I}_{4}\rho^{F}+\widehat{\nabla}^{I}_{4}\sigma^{F})\rangle.$

Now Consider the equation for $\alpha^{F}$ after commuting
$\widehat{\nabla}_{4}$ once

$\displaystyle\widehat{\nabla}_{3}\widehat{\nabla}_{4}\alpha^{F}=-\frac{1}{2}(\nabla_{4}(\mbox{tr}\underline{\chi})\alpha^{F}+\mbox{tr}\underline{\chi}\widehat{\nabla}_{4}\alpha^{F})-\widehat{\nabla}\widehat{\nabla}_{4}\rho^{F}+~{}^{*}\widehat{\nabla}\widehat{\nabla}_{4}\sigma^{F}-2~{}^{*}\nabla_{4}\eta\sigma^{F}$
$\displaystyle-2\eta\widehat{\nabla}_{4}\sigma^{F}+2\nabla_{4}\eta\rho^{F}+2\eta\widehat{\nabla}_{4}\rho^{F}+2\widehat{\nabla}_{4}\underline{\omega}\alpha^{F}+2\underline{\omega}\widehat{\nabla}_{4}\alpha^{F}-\widehat{\nabla}_{4}\widehat{\chi}\cdot\bar{\alpha}^{F}-\widehat{\chi}\cdot\widehat{\nabla}_{4}\bar{\alpha}^{F}$
$\displaystyle=-\frac{1}{2}(-\frac{1}{2}\mbox{tr}\chi\mbox{tr}\underline{\chi}+2\omega\mbox{tr}\underline{\chi}+\underbrace{2div\underline{\eta}}_{II}+2|\underline{\eta}|^{2}_{\gamma}+2\rho-\widehat{\chi}\cdot\underline{\widehat{\chi}})\alpha^{F}-\frac{1}{2}\mbox{tr}\underline{\chi}\widehat{\nabla}_{4}\alpha^{F}$
$\displaystyle-\underbrace{\widehat{\nabla}\widehat{\nabla}_{4}\rho^{F}+~{}^{*}\widehat{\nabla}\widehat{\nabla}_{4}\sigma^{F}}_{I}-2~{}^{*}(-\chi\cdot(\eta-\underline{\eta})-\beta-\frac{1}{2}\mathfrak{T}(\cdot,e_{4}))\sigma^{F}$
$\displaystyle-2\eta(-\hat{curl}\alpha^{F}-\mbox{tr}\chi\sigma^{F}+(\eta-\underline{\eta})\cdot~{}^{*}\alpha^{F})+2(-\chi\cdot(\eta-\underline{\eta})-\beta^{W}-\frac{1}{2}\mathfrak{T}(\cdot,e_{4}))\rho^{F}$
$\displaystyle+2\eta(-\hat{div}\alpha^{F}-\mbox{tr}\chi\rho^{F}-(\eta-\underline{\eta})\cdot\alpha^{F})$
$\displaystyle+2(2\omega\underline{\omega}+\frac{3}{4}|\eta-\underline{\eta}|^{2}-\frac{1}{4}(\eta-\underline{\eta})\cdot(\eta+\underline{\eta})-\frac{1}{8}|\eta+\underline{\eta}|^{2}+\frac{1}{2}\rho+\frac{1}{4}\mathfrak{T}_{43})\alpha^{F}$
$\displaystyle+2\underline{\omega}\widehat{\nabla}_{4}\alpha^{F}-(-\mbox{tr}\chi\widehat{\chi}-2\omega\widehat{\chi}-\alpha)$
$\displaystyle-\widehat{\chi}\cdot(-\frac{1}{2}\mbox{tr}\chi\bar{\alpha}^{F}-\widehat{\nabla}\rho^{F}-~{}^{*}\widehat{\nabla}\sigma^{F}-2~{}^{*}\underline{\eta}\cdot\sigma^{F}-2\underline{\eta}\cdot\rho^{F}+2\omega\bar{\alpha}^{F}-\underline{\widehat{\chi}}\cdot\alpha^{F})$
$\displaystyle+[\widehat{\nabla}_{3},\widehat{\nabla}_{4}]\alpha^{F}-[\widehat{\nabla}_{4},\widehat{\nabla}]\rho^{F}+[\widehat{\nabla}_{4},~{}^{*}\widehat{\nabla}]\sigma^{F}.$

Now notice the structure of the previous equation. the principal terms denoted
by $I$ are preserved after we take another $\widehat{\nabla}_{4}$ derivative
and therefore are cancelled after integrating by parts. Since
$\nabla^{I}_{4}\alpha^{F}$ is controlled over $H$, we can always gain a factor
of $\epsilon$. The most problematic terms in the above expression after taking
another $\widehat{\nabla}_{4}$ derivative are

$\displaystyle\nabla_{4}\underline{\eta},\nabla_{4}\omega,\nabla_{4}\nabla\underline{\eta}.$
(7.152)

We control $\nabla_{4}\underline{\eta}$ and $\nabla_{4}\omega$ by their
$L^{4}(S)$ norms. $\nabla_{4}\nabla\underline{\eta}$ can be controlled by its
$||\nabla_{4}\nabla\underline{\eta}||_{L^{2}(\underline{H})}$ norm. Since
$\nabla_{4}\nabla\underline{\eta}$ is already of top order it appears with
innocuous terms that may be estimated as follows

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\nabla^{I}_{4}\alpha^{F}\nabla_{4}\nabla\underline{\eta}\alpha^{F}|\leq\epsilon^{\frac{1}{2}}\sup_{u}||\nabla^{I}_{4}\alpha^{F}||_{L^{2}(H)}\sup_{\underline{u}}||\nabla_{4}\nabla\underline{\eta}||_{L^{2}(\underline{H})}\sup_{u,\underline{u}}||\alpha^{F}||_{L^{4}(S)}\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
(7.153)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\nabla^{I}_{4}\alpha^{F}\nabla_{4}\varphi\widehat{\nabla}_{4}\alpha^{F}|\leq\epsilon\sup_{u}||\nabla^{I}_{4}\alpha^{F}||_{L^{2}(H)}\sup_{u,\underline{u}}||\widehat{\nabla}_{4}\alpha^{F}||_{L^{4}(S)}||\nabla_{4}\varphi||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$

where we have utilized the lemma 6.7 to control the $\nabla_{4}\varphi$ in
$L^{4}(S)$. Similar terms arise from the equations for
$\widehat{\nabla}^{2}_{4}(\rho^{F},\sigma^{F})$ where most problematic terms
are of the above type. Since these estimates involve $\alpha^{F}$, we will
always gain a factor of $\epsilon$. By collecting all the terms, we may obtain

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}+\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}\leq\int_{H_{0}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{4}\rho^{F}|^{2}+\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{4}\sigma^{F}|^{2}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\rho^{F}||^{2}_{L^{2}(\underline{H})}+C(\mathcal{O}_{0})||\widehat{\nabla}^{I}_{4}\sigma^{F}||^{2}_{L^{2}(\underline{H})}$

which yields

$\displaystyle\int_{H_{u}}|\widehat{\nabla}^{I}_{4}\alpha^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$ (7.154)

through Grönwall estimate and smallness of $\epsilon$. Exact same calculations
but commuting the e.o.m with $\widehat{\nabla}$ in the second time yield the
estimate for
$\int_{H_{u}}|\widehat{\nabla}_{4}\widehat{\nabla}\alpha^{F}|^{2}$. This
completes the first part of the lemma. The second part is proved in a similar
way. Now we have to pay attention to the terms that are associated with the
top derivatives of $\bar{\alpha}^{F}$. Application of the integration
identities for the triple $(\bar{\alpha}^{F},\rho^{F},\sigma^{F})$ yields

$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}_{3}\rho^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}_{3}\sigma^{F}|^{2}=\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}_{3}\rho^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}_{3}\sigma^{F}|^{2}$
$\displaystyle+\underbrace{\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}(2\omega-\frac{1}{2}\mbox{tr}\chi)+\int_{\mathcal{D}_{u,\underline{u}}}|\widehat{\nabla}^{I}_{3}\rho^{F}|^{2}(2\underline{\omega}-\frac{1}{2}\mbox{tr}\underline{\chi})}_{ER25}$
$\displaystyle-2\underbrace{\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}\rangle+\langle\widehat{\nabla}^{I}_{3}\rho^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}_{3}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}_{3}\sigma^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}_{3}\sigma^{F}\rangle\right)}_{ER26}.$

As usual notice that the connection coefficients multiplying
$|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}$ satisfy $\nabla_{3}$
equations and there are completely determined by their initial data. Therefore
we obtain

$\displaystyle|ER25|\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})\sup_{u}(||\widehat{\nabla}^{I}_{3}\rho^{F}||_{L^{2}(H)}+||\widehat{\nabla}^{I}_{3}\sigma^{F}||_{L^{2}(H)}).$
(7.155)

For $ER26$ we utilize the evolution equations

$\displaystyle\int_{D_{u},\underline{u}}\left(\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},\widehat{\nabla}_{4}\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}\rangle+\langle\widehat{\nabla}^{I}_{3}\rho^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}_{3}\rho^{F}\rangle+\langle\widehat{\nabla}^{I}_{3}\sigma^{F},\widehat{\nabla}_{3}\widehat{\nabla}^{I}_{3}\sigma^{F}\rangle\right)$
$\displaystyle\sim\int_{\mathcal{D}_{u,\underline{u}}}\left(\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},-\frac{1}{2}\widehat{\nabla}^{I}_{3}(\mbox{tr}\chi\bar{\alpha}^{F})-2\widehat{\nabla}^{I}_{3}(~{}^{*}\underline{\eta}\cdot\sigma^{F})-2\widehat{\nabla}^{I}_{3}(\underline{\eta}\cdot\rho^{F})+2\widehat{\nabla}^{I}_{3}(\omega\bar{\alpha}^{F})-\widehat{\nabla}^{I}_{3}(\underline{\widehat{\chi}}\cdot\alpha^{F})\rangle\right.$
$\displaystyle\left.+\langle\widehat{\nabla}^{I}_{3}\rho^{F},\widehat{\nabla}^{I}_{3}(\mbox{tr}\underline{\chi}\rho^{F})+\widehat{\nabla}^{I}_{3}((\eta-\underline{\eta})\cdot\bar{\alpha}^{F})\rangle+\langle\widehat{\nabla}^{I}_{3}\sigma^{F},-\widehat{\nabla}^{I}(\mbox{tr}\underline{\chi}\sigma^{F})+\widehat{\nabla}^{I}_{3}((\eta-\underline{\eta})\cdot~{}^{*}\bar{\alpha}^{F})\rangle\right)$
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},[\widehat{\nabla}_{4},\widehat{\nabla}^{I}_{3}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},[\widehat{\nabla},\widehat{\nabla}^{I}_{3}](\rho^{F},\sigma^{F})\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\rho^{F},[\widehat{\nabla}_{3},\widehat{\nabla}^{I}_{3}]\rho^{F}\rangle$
$\displaystyle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\rho^{F},[\widehat{\nabla},\widehat{\nabla}^{I}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\sigma^{F},[\widehat{\nabla},\widehat{\nabla}^{I}_{3}]\bar{\alpha}^{F}\rangle+\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\alpha^{F},((\eta+\underline{\eta})(\widehat{\nabla}^{I}_{3}\rho^{F}+\widehat{\nabla}^{I}_{3}\sigma^{F})\rangle.$

Now we notice some of the key features of the commuted equations. Consider the
$\widehat{\nabla}_{3}$ commuted equation for $\bar{\alpha}^{F}$

$\displaystyle\widehat{\nabla}_{4}\widehat{\nabla}_{3}\bar{\alpha}^{F}=-\frac{1}{2}(-\frac{1}{2}\mbox{tr}\underline{\chi}\mbox{tr}\chi+2\underline{\omega}\mbox{tr}\chi+2div\eta+2|\eta|^{2}+2\rho-\widehat{\chi}\cdot\underline{\widehat{\chi}})\bar{\alpha}^{F}$
$\displaystyle-\frac{1}{2}\mbox{tr}\chi\nabla_{3}\bar{\alpha}^{F}-\widehat{\nabla}\widehat{\nabla}_{3}\rho^{F}-~{}^{*}\widehat{\nabla}\widehat{\nabla}_{3}\sigma^{F}-2~{}^{*}\widehat{\nabla}_{3}\underline{\eta}\cdot\sigma^{F}-2~{}^{*}\underline{\eta}\cdot\widehat{\nabla}_{3}\sigma^{F}-2\widehat{\nabla}_{3}\underline{\eta}\cdot\rho^{F}$
$\displaystyle-2\underline{\eta}\cdot\widehat{\nabla}_{3}\rho^{F}+2\widehat{\nabla}_{3}\omega\bar{\alpha}^{F}+2\omega\widehat{\nabla}_{3}\bar{\alpha}^{F}-\widehat{\nabla}_{3}\underline{\widehat{\chi}}\cdot\alpha^{F}-\underline{\widehat{\chi}}\cdot\widehat{\nabla}_{3}\alpha^{F}$
$\displaystyle=-\frac{1}{2}(-\frac{1}{2}\mbox{tr}\underline{\chi}\mbox{tr}\chi+2\underline{\omega}\mbox{tr}\chi+2div\eta+2|\eta|^{2}+2\rho-\widehat{\chi}\cdot\underline{\widehat{\chi}})\bar{\alpha}^{F}$
$\displaystyle-\frac{1}{2}\mbox{tr}\chi\nabla_{3}\bar{\alpha}^{F}-\widehat{\nabla}\widehat{\nabla}_{3}\rho^{F}-~{}^{*}\widehat{\nabla}\widehat{\nabla}_{3}\sigma^{F}-2~{}^{*}(-\bar{\chi}\cdot(\underline{\eta}-\eta)+\bar{\beta}^{W}+\frac{1}{2}\mathfrak{T}(\cdot,e_{3}))\cdot\sigma^{F}$
$\displaystyle-2~{}^{*}\underline{\eta}\cdot(-\hat{curl}\bar{\alpha}^{F}-\mbox{tr}\underline{\chi}\sigma^{F}+(\eta-\underline{\eta})\cdot~{}^{*}\bar{\alpha}^{F})-2(-\bar{\chi}\cdot(\underline{\eta}-\eta)+\bar{\beta}^{W}+\frac{1}{2}\mathfrak{T}(\cdot,e_{3}))\cdot\rho^{F}$
$\displaystyle-2\underline{\eta}\cdot(-\hat{div}\bar{\alpha}^{F}+\mbox{tr}\underline{\chi}\rho^{F}+(\eta-\underline{\eta})\cdot\bar{\alpha}^{F})$
$\displaystyle+2(2\omega\underline{\omega}+\frac{3}{4}|\eta-\underline{\eta}|^{2}+\frac{1}{4}(\eta-\underline{\eta})\cdot(\eta+\underline{\eta})-\frac{1}{8}|\eta+\underline{\eta}|^{2}+\frac{1}{2}\rho^{W}+\frac{1}{4}\mathfrak{T}_{43})\bar{\alpha}^{F}$
$\displaystyle+2\omega\widehat{\nabla}_{3}\bar{\alpha}^{F}-(-\mbox{tr}\underline{\chi}\underline{\widehat{\chi}}-2\underline{\omega}\underline{\widehat{\chi}}-\bar{\alpha}^{W})\cdot\alpha^{F}$
$\displaystyle-\underline{\widehat{\chi}}\cdot(-\frac{1}{2}\mbox{tr}\underline{\chi}\alpha^{F}-\widehat{\nabla}\rho^{F}+~{}^{*}\widehat{D}\sigma^{F}-2~{}^{*}\eta\sigma^{F}+2\eta\rho^{F}+2\underline{\omega}\alpha^{F}-\widehat{\chi}\cdot\bar{\alpha}^{F})$
$\displaystyle+[\widehat{\nabla}_{4},\widehat{\nabla}_{3}]\bar{\alpha}^{F}+[\widehat{\nabla},\widehat{\nabla}_{3}]\rho^{F}+[\widehat{\nabla},\widehat{\nabla}_{3}]\sigma^{F}.$

Similar to the previous case, after another application of $\nabla_{3}$
produces $\nabla\nabla_{3}\eta$. But since this is at the level of top order
derivative it contains $\widehat{\nabla}^{I}\bar{\alpha}^{F}$ in addition to
an algebraic term $\bar{\alpha}^{F}$. Now we may control $\bar{\alpha}^{F}$ in
$L^{4}(S)$ and $\nabla\nabla_{3}\eta$ in $L^{2}(H)$. This way we gain a factor
of $\epsilon$. Similarly, we control $\nabla_{3}\eta$ and
$\nabla_{3}\underline{\omega}$ in $L^{4}(S)$ using lemma 6.7. The most
dangerous terms are estimated as follows

$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\langle\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F},(\omega,\mbox{tr}\chi)\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}\rangle|\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})},$
(7.156)
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\nabla^{I}_{3}\bar{\alpha}^{F}\nabla\nabla_{3}\eta\bar{\alpha}^{F}|\leq\epsilon^{\frac{1}{2}}\sup_{u}||\nabla^{I}_{3}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}\sup_{\underline{u}}||\nabla\nabla_{3}\eta||_{L^{2}(H)}\sup_{u,\underline{u}}||\bar{\alpha}^{F}||_{L^{4}(S)}$
(7.157)
$\displaystyle\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$
$\displaystyle|\int_{\mathcal{D}_{u,\underline{u}}}\nabla^{I}_{3}\bar{\alpha}^{F}\nabla_{3}\varphi\widehat{\nabla}_{3}\bar{\alpha}^{F}|$
(7.158)
$\displaystyle\leq\epsilon\sup_{u}||\nabla^{I}_{3}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}\sup_{u,\underline{u}}||\widehat{\nabla}_{3}\bar{\alpha}^{F}||_{L^{4}(S)}||\nabla_{3}\varphi||_{L^{4}(S)}\leq\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F}),$

where $\varphi$ denotes the connection coefficients. Collecting all the terms,
we obtain

$\displaystyle|ER26|\leq\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}$
(7.159)
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\rho^{F}||_{L^{2}(H)}+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\sigma^{F}||_{L^{2}(H)}$

and therefore

$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}_{3}\rho^{F}|^{2}+\int_{H_{u}}|\widehat{\nabla}^{I}_{3}\sigma^{F}|^{2}\leq\int_{\underline{H}_{0}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}_{3}\rho^{F}|^{2}+\int_{H_{0}}|\widehat{\nabla}^{I}_{3}\sigma^{F}|^{2}$
$\displaystyle\leq
C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||^{2}_{L^{2}(\underline{H})}d\underline{u}^{{}^{\prime}}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\rho^{F}||_{L^{2}(H)}+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\sigma^{F}||_{L^{2}(H)}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\rho^{F}||_{L^{2}(H)}$
$\displaystyle+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})||\widehat{\nabla}^{I}_{3}\sigma^{F}||_{L^{2}(H)}.$

Smallness of $\epsilon$ yields

$\displaystyle\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}|^{2}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})+\epsilon^{\frac{1}{2}}C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})+\epsilon
C(\mathcal{O}_{0},\mathcal{W},\mathcal{F})$
$\displaystyle+C(\mathcal{O}_{0})\int_{0}^{\underline{u}}||\widehat{\nabla}^{I}_{3}\bar{\alpha}^{F}||_{L^{2}(\underline{H})}d\underline{u}$

Exact same procedure but commuting the e.o.m with $\widehat{\nabla}$ in the
second time yields the estimate for
$\int_{\underline{H}_{\underline{u}}}|\widehat{\nabla}_{3}\widehat{\nabla}\bar{\alpha}^{F}|^{2}$.
This concludes the proof of the lemma and the energy estimates associated with
the Weyl and Yang-Mills curvature. $\square$

###### Corollary 7.1

$\mathcal{W}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0}),~{}\mathcal{F}\leq
C(\mathcal{O}_{0},\mathcal{W}_{0},\mathcal{F}_{0})$.

_Proof._ A direct consequence of lemma 7.1\- 7.5, Grönwall’s inequality and
the fact that $u\in[0,\epsilon]$ and $\underline{u}\in[0,J]$. $\square$

This concludes the proof of the estimates for the Weyl and Yang-Mills
curvature throughout $\mathcal{D}_{u,\underline{u}}$ in terms of the initial
data. Once these estimates are derived, we can choose the bootstrap constant
$\Delta$ (5.1) large enough (but finite) to close the argument since the final
estimates do not depend on the bootstrap constant $\Delta$.

##  8 Concluding Remarks

Here we have obtained ‘semi-global’ estimates for the coupled Einstein-Yang-
Mills equations in a gauge-invariant way (in the sense of Yang-Mills gauge
theory). The null structure of the Einstein-Yang-Mills equations plays a
crucial role in achieving this result. This lays the platform to study several
problems associated with the coupled Einstein-Yang-Mills dynamics in the
immediate future. As we have mentioned, the double null framework is
appropriate for the radiation problem associated with gravity or/and Yang-
Mills (they both have the same characteristics) equations. Since the Yang-
Mills equations are themselves non-linear, they can counterbalance the gravity
(e.g., in the case of regular soliton-like solutions). As a consequence, all
three possibilities (black hole solution, regular solutions, and naked
singular solutions) are open in the context of an evolution problem. Therefore
it would be interesting to obtain sharp criteria demarcating all three
regimes.

Another interesting perspective would be to study the Wang-Yau quasi-local
energy [35, 36] contained in the space-like domain bounded by the membrane
$S_{u,\underline{u}}$. Ultimately, the formation of singularities is
associated with the focusing of energy, and since the dynamics of the
topological $2-$ spheres $S_{u,\underline{u}}$ is one of the central parts of
the analysis of singularity formation, it is only natural to understand the
evolution of the energy contained within it. Therefore, we need a definition
of energy in a fully relativistic setting. Luckily as we have mentioned
before, [35, 36] constructed a notion of quasi-local energy associated with a
topological $2-$ sphere $S_{u,\underline{u}}$. It was proven in [37] that such
quasi-local energy while evolving along the incoming null direction reproduces
the Bel-Robinson energy and the matter stress-energy tensor (at different
orders of course) at the limit of approaching vertex. Motivated by this
result, it is only natural to study the evolution of this quasi-local energy
in the outgoing null direction and observe the behavior in the focusing regime
(note an expression of the quasi-local energy was obtained in [38] in the
presence of a gauge field). We expect an alternate notion of trapped surface
formation through the study of this quasi-local energy.

Lastly, we want to mention the fact that our estimates associated with the
Yang-Mills curvature components are completely gauge-invariant. In particular,
we obtain estimates for the fully gauge covariant angular derivatives. This
essentially hints at an apparent similarity with the Maxwell and Yang-Mills
theory despite the fact that the latter is a fully non-linear theory because
the gauge covariant derivative hides the information of the connection and the
non-linear coupling shows up only as the commutator of the fully gauge
covariant derivatives. This does not cause a problem in the context of
obtaining estimates since all the associated inequalities are formulated in
terms of the gauge covariant derivatives (and the estimates required for a
local existence theory for coupled Yang-Mills equations can be obtained in a
gauge-invariant way; note that at the end one ought to choose a gauge and work
with the equation for connection. However, our proffered choice of gauge is
temporal gauge where the Yang-Mills equations take the form of a symmetric
hyperbolic system and the spatial connection can be determined in terms of the
gauge invariant norms of the Yang-Mills curvature). There is of course a
physical motivation behind this. Since the double null framework in some sense
encodes the information about the physical nature of the Yang-Mills fields,
the choice of the gauge should not matter and one should expect to obtain
gauge-invariant estimates as is done in the current context. In this framework
of the gauge-invariant estimates, therefore, the exterior stability of the
Minkowski space under coupled gravity-Yang-Mills perturbations is expected to
hold. In other words, we conjecture “Exterior stability of the Minkowski space
holds under the coupled gravity-Yang-Mills perturbations. Since we have
developed the framework in this article, we want to pursue such stability
conjecture in the near future.

##  9 Acknowledgement

P.M. is supported by the Center of Mathematical Sciences and Applications,
Department of Mathematics at Harvard University.

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