# High-frequency solutions to the constraint equations

Arthur Touati Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette,
France<EMAIL_ADDRESS>
###### Abstract

We construct high-frequency initial data for the Einstein vacuum equations in
dimension 3+1 by solving the constraint equations on $\mathbb{R}^{3}$. Our
family of solutions $(\bar{g}_{\lambda},K_{\lambda})_{\lambda\in(0,1]}$ is
defined through a high-frequency expansion similar to the geometric optics
approach and converges in a particular sense to the data of a null dust. In
order to solve the constraint equations, we use their conformal formulation
and the main challenge of our proof is to adapt this method to the high-
frequency context. In particular, the parameters of the conformal formulation
are oscillating. The main application of this article is our companion article
[Tou22] where we construct high-frequency gravitational waves in generalised
wave gauge.

###### Contents

  1. 1 Introduction
    1. 1.1 Backreaction for the constraint equations
    2. 1.2 The conformal method
    3. 1.3 Preliminaries
      1. 1.3.1 Geometric notations
      2. 1.3.2 Function spaces and asymptotically Euclidean manifold
      3. 1.3.3 Elliptic estimates
      4. 1.3.4 High-frequency notations
    4. 1.4 Acknowledgment
  2. 2 Statement of the results
    1. 2.1 The background
    2. 2.2 Solving the constraint equations
    3. 2.3 Application to high-frequency vacuum spacetimes
  3. 3 Strategy of proof
    1. 3.1 High-frequency expansion of the parameters
    2. 3.2 High-frequency expansion of the solutions
    3. 3.3 The TT-tensor
    4. 3.4 Outline of the proof
  4. 4 High-frequency conformal class
    1. 4.1 Definitions and first computations
    2. 4.2 Expansion of differential operators
      1. 4.2.1 The Laplace-Beltrami operator
      2. 4.2.2 The divergence operator
      3. 4.2.3 The conformal Killing operator
      4. 4.2.4 The conformal Laplacian
  5. 5 Approximate solution to the constraint equations
    1. 5.1 The approximate Hamiltonian constraint
      1. 5.1.1 The $\lambda^{0}$ Hamiltonian level
      2. 5.1.2 The $\lambda^{1}$ Hamiltonian level
    2. 5.2 The approximate momentum constraint
    3. 5.3 An almost TT-tensor
    4. 5.4 An exact TT-tensor
      1. 5.4.1 Definition of $\sigma^{(2)}$ and $\mathfrak{f}$
      2. 5.4.2 Solving for $Y$
  6. 6 Exact solution to the constraint equations
    1. 6.1 System for the remainders
      1. 6.1.1 Definition of $W^{(3)}(\widetilde{W})$
      2. 6.1.2 The system for $\widetilde{W}$ and $\widetilde{\varphi}$
    2. 6.2 Fixed point argument
  7. 7 Proof of the main theorem
    1. 7.1 The metric $\bar{g}_{\lambda}$ and proof of (2.12)
    2. 7.2 The tensor $K_{\lambda}$ and proof of (2.13)

##  1 Introduction

In this article, we construct high-frequency initial data for the Einstein
vacuum equations

$\displaystyle R_{\mu\nu}(g)=0,$ (1.1)

where $R_{\mu\nu}(g)$ is the Ricci tensor of $g$, a Lorentzian metric on the
manifold $[0,1]\times\mathbb{R}^{3}$. Initial data for (1.1) on
$\Sigma_{0}=\\{t=0\\}$ are given by a Riemannian metric $\bar{g}$ and a
symmetric 2-tensor $K$. In the spacetime $([0,1]\times\mathbb{R}^{3},g)$ that
$(\Sigma_{0},\bar{g},K)$ gives birth to, $\bar{g}$ will be the restriction of
$g$ to $\Sigma_{0}$ and $K$ will be the second fundamental form of
$\Sigma_{0}$, that is $K=-\frac{1}{2}\mathcal{L}_{T}g$ where $T$ is the unit
normal to $\Sigma_{0}$ for $g$. A necessary condition for $(\bar{g},K)$ to be
the set of initial data to a solution of (1.1) is that $(\bar{g},K)$ solves
the following vacuum constraint equations:

$\displaystyle R(\bar{g})+(\mathrm{tr}_{\bar{g}}K)^{2}-|K|^{2}_{\bar{g}}$
$\displaystyle=0,$ (1.2)
$\displaystyle-\mathrm{div}_{\bar{g}}K+\mathrm{d}\mathrm{tr}_{\bar{g}}K$
$\displaystyle=0,$ (1.3)

where $R(\bar{g})$ denotes the scalar curvature of $\bar{g}$. Equation (1.2)
is the Hamiltonian constraint, and equation (1.3) is the momentum constraint.
Together, they form a coupled system of non-linear elliptic partial
differential equations and we refer to Chapter 7 of [CB09] for a complete
presentation of their mathematical study.

###  1.1 Backreaction for the constraint equations

We construct a family $(\bar{g}_{\lambda},K_{\lambda})_{\lambda\in(0,1]}$ of
solutions to (1.2)-(1.3) which oscillate at frequency $\lambda^{-1}$. The mean
feature of this family is its high-frequency limit, i.e its behaviour when
$\lambda$ tends to 0. It converges in the following weak sense

$\displaystyle\bar{g}_{\lambda}$
$\displaystyle\to\bar{g}_{0},\quad\text{uniformly in $L^{\infty}$,}$
$\displaystyle\nabla\bar{g}_{\lambda}$
$\displaystyle\rightharpoonup\nabla\bar{g}_{0},\quad\text{weakly in
$L^{2}_{loc}$,}$ (1.4) $\displaystyle K_{\lambda}$
$\displaystyle\rightharpoonup K_{0},\quad\text{weakly in $L^{2}_{loc}$,}$

to a fixed solution $(\bar{g}_{0},K_{0})$ of the null dust maximal constraint
equations:

$\displaystyle R(\bar{g}_{0})-|K_{0}|^{2}_{\bar{g}_{0}}$
$\displaystyle=2|\nabla u_{0}|^{2}_{\bar{g}_{0}}F_{0}^{2},$ (1.5)
$\displaystyle-\mathrm{div}_{\bar{g}_{0}}K_{0}$ $\displaystyle=|\nabla
u_{0}|_{\bar{g}_{0}}F_{0}^{2}\mathrm{d}u_{0},$ (1.6)
$\displaystyle\mathrm{tr}_{\bar{g}_{0}}K_{0}$ $\displaystyle=0,$ (1.7)

where $u_{0}$ and $F_{0}$ are scalar functions defined on $\mathbb{R}^{3}$.
Therefore, this article provides the first example of backreaction for the
constraint equations: a sequence of solutions to the vacuum equations
describes, in the high-frequency limit, a matter model.

This phenomenon is at the heart of Burnett’s conjecture in general relativity.
In [Bur89], he conjectured that if a sequence $(g_{\lambda})_{\lambda}$ of
solutions to (1.1) converges weakly in $H^{1}_{loc}$ to a metric $g_{0}$, then
$g_{0}$ solves the massless Einstein-Vlasov system

$\displaystyle R_{\alpha\beta}(g_{0})$
$\displaystyle=\int_{g_{0}^{-1}(p,p)=0}f(x,p)p_{\alpha}p_{\beta}\mathrm{d}\mu_{g_{0}},$
(1.8) $\displaystyle
p^{\alpha}\partial_{\alpha}f-p^{\alpha}p^{\beta}\Gamma(g_{0})^{\rho}_{\alpha\beta}\partial_{p^{\rho}}f$
$\displaystyle=0,$ (1.9)

for a density $f$ defined on the tangent bundle and where
$\mathrm{d}\mu_{g_{0}}$ is the measure on the tangent bundle. The second
equation is the Vlasov equation for the density of massless particles $f$. In
his article, Burnett also asked the reverse question : given $g_{0}$ a
solution to (1.8)-(1.9), can one construct a sequence
$(g_{\lambda})_{\lambda}$ of solutions to (1.1) converging weakly in $H^{1}$
to $g_{0}$ ? The present article provides a positive answer to this question
at the level of the spacelike data for (1.1) and for a discrete version of the
massless Einstein-Vlasov system, namely the Einstein-null dust system

$\displaystyle R_{\alpha\beta}(g_{0})$
$\displaystyle=F_{0}^{2}\partial_{\alpha}u_{0}\partial_{\beta}u_{0},$ (1.10)
$\displaystyle g_{0}^{-1}(\mathrm{d}u_{0},\mathrm{d}u_{0})$ $\displaystyle=0,$
(1.11) $\displaystyle
2g_{0}^{\rho\sigma}\partial_{\rho}u_{0}\partial_{\sigma}F_{0}+(\Box_{g_{0}}u_{0})F_{0}$
$\displaystyle=0.$ (1.12)

The equations (1.5)-(1.7) are the constraint equations that spacelike data for
the system (1.10)-(1.12) need to solve, in the particular case where the
initial hypersurface is maximal. This explains why we refered to the system
(1.5)-(1.7) as the null dust maximal constraint equations. Note that we
slightly abuse notations: in this article the scalar functions $F_{0}$ and
$u_{0}$ will be defined on $\Sigma_{0}$ only, while in (1.10)-(1.12) they are
defined on a whole spacetime.

The two aspects of Burnett’s conjecture, the direct one and the indirect one,
have been studied in several works and different settings. Assuming
$\mathbb{U}(1)$ symmetry, Huneau and Luk address the direct conjecture by
means of microlocal defect measure in [HL19] (see also [GdC21]). Under the
same symmetry, they construct high-frequency vacuum spacetimes converging to
$N$ null dusts in [HL18]. The first result without symmetry assumptions was
obtained by Luk and Rodnianski in [LR20], where they address both sides of
Burnett’s conjecture in double null gauge. This choice of gauge restricts the
class of kinetic spacetimes they consider to 2 null dusts.

In a companion paper [Tou22], the author constructs high-frequency vacuum
spacetimes in generalised wave gauge, paving the way to a proof of Burnett’s
conjecture in this gauge. This motivates the need for the high-frequency
initial data solving (1.2)-(1.3) provided by this article. The link between
this article and [Tou22] and its mathematical implications will be further
discussed in Section 2.3, after we state the main result in Section 2.2.

Finally, note that Burnett’s conjecture and the backreaction it describes for
the Einstein vacuum equations falls into the widest category of non-linear
effects induced by homogenization, that is through the interaction of multiple
scales. This type of effects can be encountered in the study of virtually all
non-linear equations coming from physics. Examples are porous media,
hydrodynamics, quantum mechanics etc. We refer to [Tar09] for a rich
presentation of the field of homogenization.

###  1.2 The conformal method

The constraint equations (1.2)-(1.3) are underdetermined, and in order to
solve them we use the conformal method. Introduced by Lichnerowicz in [Lic44],
this method is based on a conformal formulation of the data $(\bar{g},K)$ and
it identifies free parameters. In particular, this method transforms
(1.2)-(1.3) into a determined system of equations composed of a vectorial
equation and a scalar equation. The idea giving its name to the method is to
prescribe the conformal class of $\bar{g}$, i.e to fix a Riemannian metric
$\gamma$ on $\mathbb{R}^{3}$ and to solve for the scalar function $\varphi$
such that

$\displaystyle\bar{g}$ $\displaystyle=\varphi^{4}\gamma.$ (1.13)

We say that a tensor is a TT-tensor if it is traceless divergence free and
symmetric and one can show that the space of TT-tensor depends only on the
conformal class of the metric. Therefore, the next step in the conformal
method is to decompose the symmetric 2-tensor $K$ in connection with its trace
and divergence features. More precisely we fix a scalar function $\tau$ and a
TT-tensor $\sigma$ for the metric $\gamma$ and solve for the vector field $W$
such that

$\displaystyle K$
$\displaystyle=\varphi^{-2}(\sigma+L_{\gamma}W)+\frac{1}{3}\varphi^{4}\gamma\tau$
(1.14)

where $L_{\gamma}W$ is defined in (4.30). The exponents appearing in (1.13)
and (1.14) are linked to the dimension of the manifold, here 3, see Chapter 6
of [CB09] for their general expression. Once the parameters $\gamma$, $\tau$
and $\sigma$ are chosen, the constraint equations (1.2)-(1.3) rewrite as the
following coupled system of non-linear elliptic equations for $(\varphi,W)$:

$\displaystyle 8\Delta_{\gamma}\varphi$
$\displaystyle=R(\gamma)\varphi+\frac{2}{3}\tau^{2}\varphi^{5}-\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\varphi^{-7},$
(1.15) $\displaystyle\mathrm{div}_{\gamma}L_{\gamma}W$
$\displaystyle=\frac{2}{3}\varphi^{6}\mathrm{d}\tau.$ (1.16)

A wealth of literature has been produced over the years on the construction of
solutions to (1.15)-(1.16) on various Riemannian manifolds. Note that if the
mean curvature $\tau$ is constant (CMC setting), then (1.15) and (1.16)
decouple and the construction of solutions is simplified. The results on
(1.15)-(1.16) can thus be categorized into CMC, near CMC (where
$\frac{\mathrm{d}\tau}{\tau}$ is small) or far from CMC results, where no
assumption is made on $\tau$. On compact manifolds, CMC results are obtained
in [Ise95]. Near CMC results on asymptotically Euclidean manifolds are
obtained in [CBIY00], where the authors also treat the constraint equations
with matter. For far from CMC results, we refer the reader to [Max09] and
[DIMM14], where the case of compact manifolds and asymptotically Euclidean
manifolds are respectively treated.

Our main challenge is to adapt the conformal method to the high-frequency
context, i.e choose the parameters so that the resulting solution
$(\bar{g}_{\lambda},K_{\lambda})$ displays the behaviour (1.4). Not only will
the solutions $(\varphi,W)$ be defined by high-frequency ansatz, but also the
parameters $\gamma$, $\tau$ and $\sigma$. In particular, we will construct an
oscillating TT-tensor, i.e solve $\mathrm{div}_{\gamma}\sigma=0$ and
$\mathrm{tr}_{\gamma}\sigma=0$, with $\gamma$ itself oscillating.

###  1.3 Preliminaries

We fix here our notations and present the analytic setting of this article.

####  1.3.1 Geometric notations

Throughout this article, the notation $\Sigma_{0}$ refers to the manifold
$\mathbb{R}^{3}$. On $\Sigma_{0}$ we consider the usual Euclidean coordinates
$x=(x^{1},x^{2},x^{3})$, and we denote by $e$ the standard Euclidean metric,
i.e

$\displaystyle
e=\left(\mathrm{d}x^{1}\right)^{2}+\left(\mathrm{d}x^{2}\right)^{2}+\left(\mathrm{d}x^{3}\right)^{2}.$

Latin indices are used for the Euclidean coordinates and therefore runs from 1
to 3. In this article, repeated indices are always summed over. A second frame
adapted to the background structure will be defined in Section 2.1.

If $f$ is a scalar function, we define its gradient by $\nabla
f=(\partial_{1}f,\partial_{2}f,\partial_{3}f)$. If $h$ is a Riemannian metric
on $\Sigma_{0}$ we define

$\displaystyle|\nabla f|_{h}^{2}=h^{ij}\partial_{i}f\partial_{j}f.$

In the particular case of the Euclidean metric we simply write $|\nabla
f|^{2}$. Moreover if $T$ and $S$ are two symmetric 2-tensors on $\Sigma_{0}$,
we define $|T\cdot S|_{h}=h^{ij}h^{k\ell}T_{ik}S_{j\ell}$ and
$|T|^{2}_{h}=|T\cdot T|_{h}$. The trace of $T$ with respect to $h$ is defined
by $\mathrm{tr}_{h}T=h^{ij}T_{ij}$.

####  1.3.2 Function spaces and asymptotically Euclidean manifold

If $x\in\mathbb{R}^{3}$ we set $\langle
x\rangle\vcentcolon=(1+|x|^{2})^{\frac{1}{2}}$ with
$|x|=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$. We define the following
weighted Sobolev spaces on $\mathbb{R}^{3}$.

######  Definition 1.1 (Weighted Sobolev spaces).

For $1\leq p<+\infty$, $\delta\in\mathbb{R}$ and $k\in\mathbb{N}$ we define
the space $W^{k,p}_{\delta}$ as the completion of $C^{\infty}_{c}$ for the
norm

$\displaystyle\left\|u\right\|_{W^{k,p}_{\delta}}$
$\displaystyle=\sum_{0\leq|\alpha|\leq k}\left\|\langle
x\rangle^{\delta+|\alpha|}\nabla^{\alpha}u\right\|_{L^{p}},$

where the $L^{p}$ norm is defined with the Euclidean volume element. We extend
this definition to tensors of any type by summing over all components in the
Euclidean coordinates. Some special cases are
$H^{k}_{\delta}\vcentcolon=W^{k,2}_{\delta}$ and
$L^{p}_{\delta}\vcentcolon=W^{0,p}_{\delta}$.

We also define the following $L^{\infty}$-based spaces:

######  Definition 1.2.

For $k\in\mathbb{N}$ and $\delta\in\mathbb{R}$ we define $C^{k}_{\delta}$ as
the completion of $C^{\infty}_{c}$ for the norm

$\displaystyle\left\|u\right\|_{C^{k}_{\delta}}$
$\displaystyle=\sum_{0\leq|\alpha|\leq k}\left\|\langle
x\rangle^{\delta+|\alpha|}\nabla^{\alpha}u\right\|_{L^{\infty}},$

In the following proposition we recall some useful facts about these spaces
(see [CB09] for the proofs).

######  Proposition 1.1.

Let $s,s^{\prime},s_{1},s_{2},m\in\mathbb{N}$,
$\delta,\delta^{\prime},\delta_{1},\delta_{2},\beta\in\mathbb{R}$ and $1\leq
p<+\infty$.

  1. 1.

If $s\leq\min(s_{1},s_{2})$, $s<s_{1}+s_{2}-\frac{3}{p}$ and
$\delta<\delta_{1}+\delta_{2}+\frac{3}{p}$ we have the continuous embedding

$\displaystyle W^{s_{1},p}_{\delta_{1}}\times W^{s_{2},p}_{\delta_{2}}\subset
W^{s,p}_{\delta}.$

  2. 2.

If $m<s-\frac{3}{p}$ and $\beta\leq\delta+\frac{3}{p}$ we have the continuous
embedding

$\displaystyle W^{s,p}_{\delta}\subset C^{m}_{\beta}.$

The background metric and the solution of the constraint equations we will
produce are asymptotically Euclidean, meaning that they converge in some sense
to the Euclidean metric at infinity. We give the definition of [CBIY00].

######  Definition 1.3 (Asymptotically Euclidean initial data).

Let $\bar{g}$ be a metric on $\mathbb{R}^{3}$ and $K$ a symmetric 2-tensor on
$\mathbb{R}^{3}$. If $k>\frac{5}{2}$ and $\delta>-\frac{3}{2}$, we say that
the pair $(\bar{g},K)$ is $H^{k}_{\delta}$ asymptotically flat if

$\displaystyle\bar{g}-e\in H^{k}_{\delta}\quad\text{and}\quad K\in
H^{k-1}_{\delta+1}.$

Note that the restrictions on $k$ and $\delta$ in the previous definition
ensure that $\bar{g}$ is $C^{1}$ and $\bar{g}-e$ tends to 0 at infinity, since
Proposition 1.1 implies
$H^{k}_{\delta}\xhookrightarrow{}C^{1}_{\delta+\frac{3}{2}}$.

######  Remark 1.1.

While the assumptions in Definition 1.3 are adapted to the inversion of
elliptic operators, the decay assumption $\delta>-\frac{3}{2}$ is too weak to
be able to define the ADM mass of $\bar{g}$. Introduced in [ADM08], the ADM
mass of an asymptotically Euclidean manifold is defined by

$\displaystyle M(\bar{g})=\frac{1}{16\pi}\lim_{R\to+\infty}\int_{\partial
B_{R}}\sum_{i,j}\left(\partial_{i}\bar{g}_{ij}-\partial_{j}\bar{g}_{ii}\right)\nu_{j}\mathrm{d}\mu,$

where $\nu$ is the outgoing normal to the spheres $\partial B_{R}$ and
$\mathrm{d}\mu$ the measure on these spheres. Since the volume of the spheres
$\partial B_{R}$ grows like $R^{2}$, we need to ask for
$\delta\geq-\frac{1}{2}$ in Definition 1.3 in order to be able to define
$M(\bar{g})$ in general. See also Remark 2.1.

####  1.3.3 Elliptic estimates

Solving the constraint equations with the conformal method requires to invert
the elliptic operators $\Delta_{\gamma}$ and
$\mathrm{div}_{\gamma}L_{\gamma}$, for $\gamma$ a Riemannian metric. The first
one is the Laplace-Beltrami operator acting on scalar functions and the second
one is the conformal Laplacian acting on vector field, see Section 4.2 for
their exact definition. The metric $\gamma$ will be defined in Section 4.1 but
we can already say that it will be close to a background metric $\bar{g}_{0}$,
itself close to the Euclidean metric $e$ on $\mathbb{R}^{3}$. We will benefit
from this fact and invert $\Delta=\Delta_{e}$ and $\mathrm{div}_{e}L_{e}$
rather than $\Delta_{\gamma}$ and $\mathrm{div}_{\gamma}L_{\gamma}$, which
have oscillating coefficients. The following proposition gives the desired
inversion properties. The proof of its first part can be found in [McO79]
while the second part is proved in [CBIY00].

######  Proposition 1.2.

If $-\frac{3}{2}<\delta<-\frac{1}{2}$ then
$\Delta:H^{2}_{\delta}\longrightarrow L^{2}_{\delta+2}$ and
$\mathrm{div}_{e}L_{e}:H^{2}_{\delta}\longrightarrow L^{2}_{\delta+2}$ are
isomorphisms.

####  1.3.4 High-frequency notations

In this article we consider high-frequency quantities, i.e tensors of all
types including scalar functions, metrics, 1-forms etc. A quantity is said to
be high-frequency if it admits an expansion in powers of the small parameter
$\lambda$ with coefficients of the form

$\displaystyle\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)f$ (1.17)

where $f$ depends only on $x\in\Sigma_{0}$ and $\mathrm{T}$ is an oscillating
function, i.e an element of

$\displaystyle\left\\{\theta\in\mathbb{R}\longmapsto\sin(k\theta)~{}\middle|~{}k\in\mathbb{N}\right\\}\cup\left\\{\theta\in\mathbb{R}\longmapsto\cos(k\theta)~{}\middle|~{}k\in\mathbb{N}\right\\}.$
(1.18)

When considering a high-frequency quantity such as a tensor $S$, we denote by
$S^{(i)}$ the coefficients of $\lambda^{i}$ in the expansion defining $S$,
which thus expands formally as

$\displaystyle S=\sum_{i\in\mathbb{Z}}\lambda^{i}S^{(i)}.$

Note that $S^{(i)}$ is a tensor of the same type as $S$. Moreover, if
$j\in\mathbb{Z}$ we define $S^{(\geq j)}$ by $S^{(\geq j)}=\sum_{k\geq
j}\lambda^{k-j}S^{(k)}$. This allows us to clearly troncate high-frequency
expansions at a fixed order as in

$\displaystyle S=\sum_{k\leq j-1}\lambda^{k}S^{(k)}+\lambda^{j}S^{(\geq j)}.$

To emphasize the fact that a high-frequency coefficient $S^{(i)}$ of a tensor
$S$ is oscillating we will often write
$S^{(i)}\left(\frac{u_{0}}{\lambda}\right)$ instead of just $S^{(i)}$.

In order to describe concisely the oscillating behaviour of a high-frequency
coefficient $S^{(i)}$ of a tensor $S$, we write for $\mathcal{A}$ a finite
subset of (1.18)

$\displaystyle
S^{(i)}\;\overset{\mathrm{osc}}{\sim}\;\sum_{\mathrm{T}\in\mathcal{A}}\mathrm{T}(\theta)$

if there exists tensors $(S^{(i)}_{\mathrm{T}})_{\mathrm{T}\in\mathcal{A}}$ of
the same type as $S^{(i)}$ such that

$\displaystyle
S^{(i)}=\sum_{\mathrm{T}\in\mathcal{A}}\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)S^{(i)}_{\mathrm{T}}.$

This notation allows us to compute the oscillating behaviour of non-linear
quantities without caring too much about the non-oscillating coefficients
$S^{(i)}_{\mathrm{T}}$. Note that $S^{(i)}\;\overset{\mathrm{osc}}{\sim}\;1$
simply means that $S^{(i)}$ is non-oscillating, i.e does not depend on
$\frac{u_{0}}{\lambda}$.

In terms of derivation, we use the symbol $\theta$ for the derivation with
respect to the $\frac{u_{0}}{\lambda}$ variable. For example, if $f$ is a
scalar function and if $g=\mathrm{T}\left(\frac{u_{0}}{\lambda}\right)f$ then

$\displaystyle\partial_{\theta}g$
$\displaystyle=\mathrm{T}^{\prime}\left(\frac{u_{0}}{\lambda}\right)f.$

###  1.4 Acknowledgment

The author would like to thank Caterina Vâlcu for all the helpful discussions.

##  2 Statement of the results

In this section, we give the assumptions on the background, state our main
result and discuss its main application.

###  2.1 The background

The background metric and 2-tensor $(\bar{g}_{0},K_{0})$ solve (1.5)-(1.7)
with $F_{0}$ and $u_{0}$ two scalar functions defined on $\mathbb{R}^{3}$. The
full background solution is then $(\bar{g}_{0},K_{0},F_{0},u_{0})$ and we make
some assumptions on it.

  * •

Assumptions on $(\bar{g}_{0},K_{0})$. Even though $(\bar{g}_{0},K_{0})$
satisfies non-vacuum constraint equations, the sources are compactly supported
(see (2.2) below) and we assume that $(\bar{g}_{0},K_{0})$ corresponds to
asymptotically Euclidean and highly regular initial data. By this we mean that
there exists $\delta>-\frac{3}{2}$, a large integer $N\geq 10$ and
$\varepsilon>0$ such that

$\displaystyle\left\|\bar{g}_{0}-e\right\|_{H^{N+1}_{\delta}}+\left\|K_{0}\right\|_{H^{N}_{\delta+1}}\leq\varepsilon.$
(2.1)

We denote by $\bar{\mathbf{D}}$ the covariant derivative associated to
$\bar{g}_{0}$.

  * •

Assumptions on $F_{0}$. The density $F_{0}$ is supported in a ball of size
$R>0$ in $\mathbb{R}^{3}$, i.e

$\displaystyle\mathrm{supp}(F_{0})\subset
B_{R}\vcentcolon=\left\\{x\in\mathbb{R}^{3}~{}\middle|~{}|x|\leq R\right\\}.$
(2.2)

It also enjoys some regularity:

$\displaystyle\left\|F_{0}\right\|_{H^{N}}\leq\varepsilon$ (2.3)

where $\varepsilon$ is defined above.

  * •

Assumptions on $u_{0}$. There exists a constant non-zero vector field
$\mathfrak{z}=(\mathfrak{z}_{1},\mathfrak{z}_{2},\mathfrak{z}_{3})$ such that

$\displaystyle\left\|\nabla
u_{0}-\mathfrak{z}\right\|_{H^{N}_{\delta+1}}\leq\varepsilon$ (2.4)

where $\nabla u_{0}=(\partial_{1}u_{0},\partial_{2}u_{0},\partial_{3}u_{0})$
is the euclidean gradient of $u_{0}$. By taking $\varepsilon$ small enough in
(2.4) we can assume that $|\nabla u_{0}|$ is uniformly bounded from below,
which implies that $u_{0}$ has no critical points. Moreover, the level
hypersurfaces of $u_{0}$ defined by

$\displaystyle
P_{0,u}=\left\\{x\in\mathbb{R}^{3}~{}\middle|~{}u_{0}(x)=u\right\\}$

foliates $\mathbb{R}^{3}$ and have the topology of planes thanks to (2.4).
This allows us to define a particular frame at each point of $\mathbb{R}^{3}$.
We define the vector field

$\displaystyle
N_{0}=-\frac{\bar{g}_{0}^{ij}\partial_{i}u_{0}\partial_{j}}{|\nabla
u_{0}|_{\bar{g}_{0}}}.$

It satisfies $\bar{g}_{0}(N_{0},N_{0})=1$ and is orthogonal to the
hypersurface $P_{0,u}$. We consider at each point $x\in\mathbb{R}^{3}$ an
orthonormal basis $(e_{\mathbf{1}},e_{\mathbf{2}})$ of $T_{x}P_{0,u}$ for the
metric $\bar{g}_{0}$. The frame $(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ will
play a crucial role in our construction. While we reserve the usual latin
indices for the coordinates system $(x^{1},x^{2},x^{3})$, i.e
$i,j\in\\{1,2,3\\}$, the bold latin indices are used for the frame
$(e_{\mathbf{1}},e_{\mathbf{2}})$, i.e
$\mathbf{i},\mathbf{j}\in\\{\mathbf{1},\mathbf{2}\\}$.

In this article, we don’t prove that a background solution
$(\bar{g}_{0},K_{0},F_{0},u_{0})$ solving (1.5)-(1.7) and satisfying the above
assumptions exists. We refer to [CBIY00] for the details of how one can solve
the constraint equations with sources in the asymptotically Euclidean setting.

###  2.2 Solving the constraint equations

The following theorem is the main result of this article.

######  Theorem 2.1.

Let $(\bar{g}_{0},K_{0},F_{0},u_{0})$ be the solution of the maximal
constraint equations coupled with a null dust described in Section 2.1, and
let $\varepsilon>0$ be the smallness threshold. There exists
$\varepsilon_{0}=\varepsilon_{0}(\delta,R)>0$ such that if
$0<\varepsilon\leq\varepsilon_{0}$, there exists for all $\lambda\in(0,1]$ a
solution $(\bar{g}_{\lambda},K_{\lambda})$ solution of the vacuum constraint
equations (1.2)-(1.3) on $\mathbb{R}^{3}$ of the form

$\displaystyle\bar{g}_{\lambda}$
$\displaystyle=\bar{g}_{0}+\lambda\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}+\lambda^{2}\left(\sin\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(2,1)}+\cos\left(\frac{2u_{0}}{\lambda}\right)\bar{F}^{(2,2)}\right)+\lambda^{2}\bar{\mathfrak{h}}_{\lambda},$
(2.5) $\displaystyle K_{\lambda}$ $\displaystyle=K^{(0)}_{\lambda}+\lambda
K^{(1)}_{\lambda}+\lambda^{2}K^{(\geq 2)}_{\lambda},$ (2.6)

with

$\displaystyle K^{(0)}_{\lambda}$
$\displaystyle=K_{0}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)},$ (2.7) $\displaystyle K^{(1)}_{\lambda}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)K^{(1,1)}+\sin\left(\frac{2u_{0}}{\lambda}\right)K^{(1,2)}.$
(2.8)

Moreover:

  * (i)

the tensors $\bar{F}^{(1)}$, $\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$ are
supported in $\\{|x|\leq R\\}$ and there exists
$C_{\mathrm{cons}}=C_{\mathrm{cons}}(\delta,R)>0$ such that

$\displaystyle\left\|\bar{F}^{(1)}\right\|_{H^{N}}+\left\|\bar{F}^{(2,1)}\right\|_{H^{N-1}}+\left\|\bar{F}^{(2,2)}\right\|_{H^{N-1}}$
$\displaystyle\leq C_{\mathrm{cons}}\varepsilon,$

  * (ii)

the tensor $\bar{F}^{(1)}$ is $\bar{g}_{0}$-traceless, tangential to $P_{0,u}$
and satisfies

$\displaystyle\left|\bar{F}^{(1)}\right|^{2}_{\bar{g}_{0}}=8F_{0}^{2},$ (2.9)

  * (iii)

$K^{(1,1)}$ and $K^{(1,2)}$ are given by

$\displaystyle K^{(1,1)}$
$\displaystyle=\frac{1}{2}N_{0}\bar{F}^{(1)}_{ij}+\frac{1}{2}\left(\bar{g}_{0}^{k\ell}(K_{0})_{(i\ell}-N_{0}^{\ell}\Gamma(\bar{g}_{0})^{k}_{\ell(i}\right)\bar{F}^{(1)}_{j)k}$
(2.10) $\displaystyle\quad+\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}\left(K_{0}^{k\ell}\partial_{k}u_{0}\partial_{\ell}u_{0}+\bar{g}_{0}^{k\ell}\partial_{k}u_{0}\partial_{\ell}|\nabla
u_{0}|_{\bar{g}_{0}}-\frac{1}{2}\bar{g}_{0}^{k\ell}\partial_{k}\partial_{\ell}u_{0}\right)\bar{F}^{(1)}_{ij}$
$\displaystyle\quad-\frac{1}{2}\left|\nabla
u_{0}\right|_{\bar{g}_{0}}\bar{F}^{(2,1)},$ $\displaystyle K^{(1,2)}$
$\displaystyle=\left|\nabla u_{0}\right|_{\bar{g}_{0}}\bar{F}^{(2,2)},$ (2.11)

  * (iv)

the tensors $\bar{\mathfrak{h}}_{\lambda}$ and $K^{(\geq 2)}_{\lambda}$ belong
to the spaces $H^{5}_{\delta}$ and $H^{4}_{\delta+1}$ respectively and satisfy

$\displaystyle\max_{r\in\llbracket
0,4\rrbracket}\lambda^{r}\left\|\nabla^{r+1}\bar{\mathfrak{h}}_{\lambda}\right\|_{L^{2}_{\delta+r+1}}$
$\displaystyle\leq C_{\mathrm{cons}}\varepsilon,$ (2.12)
$\displaystyle\max_{r\in\llbracket
0,4\rrbracket}\lambda^{r}\left\|\nabla^{r}K_{\lambda}^{(\geq
2)}\right\|_{L^{2}_{\delta+r+1}}$ $\displaystyle\leq
C_{\mathrm{cons}}\varepsilon.$ (2.13)

Note that in Theorem 2.1, the only free data are the background quantities
$(\bar{g}_{0},K_{0},F_{0},u_{0})$. The tensors $\bar{F}^{(1)}$,
$\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$ are obtained in the proof and are
determined by $(\bar{g}_{0},K_{0},F_{0},u_{0})$. The only freedom lies in the
choice of the polarization of $\bar{F}^{(1)}$, see the definition of
$\omega^{(1)}$ in Section 4.1.

######  Remark 2.1.

Note that if the background metric $\bar{g}_{0}$ satisfies a stronger decay
assumption, one can define its ADM mass (see Remark 1.1). In this case, it is
also possible to show the convergence of the ADM mass of $\bar{g}_{\lambda}$
towards the ADM mass of the background when $\lambda$ tends to 0. Indeed, the
high-frequency perturbations in (2.5) are compactly supported and only have an
impact at infinity through the remainder $\bar{\mathfrak{h}}_{\lambda}$, for
which we can show enough control in terms of decay and when $\lambda$ tends to
0.

###  2.3 Application to high-frequency vacuum spacetimes

Even though Theorem 2.1 is formulated in a purely elliptic way, its main
purpose is to provide spacelike initial data for the high-frequency solutions
to (1.1) constructed in [Tou22] in generalised wave gauge. More precisely, in
this article we construct a family of Lorentzian metric
$(g_{\lambda})_{\lambda\in(0,1]}$ of the form

$\displaystyle g_{\lambda}=g_{0}+\lambda
g^{(1)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}g^{(2)}\left(\frac{u_{0}}{\lambda}\right)+O\left(\lambda^{2}\right)$
(2.14)

where the $g^{(i)}$ are periodic and smooth functions of their argument
$\frac{u_{0}}{\lambda}$ and where $(g_{0},F_{0},u_{0})$ is a given solution of
the Einstein-null dust system (1.10)-(1.12) with initial data on $\Sigma_{0}$
given by $(\bar{g}_{0},K_{0},F_{0},u_{0})$ described in Section 2.1. In this
section, let us discuss how it affects the construction of Theorem 2.1.

Since $\bar{g}_{\lambda}$ is the induced metric of $g_{\lambda}$ on
$\Sigma_{0}$, the oscillating behaviours of $g^{(1)}$ and $g^{(2)}$ in [Tou22]
explains the high-frequency expansion of $\bar{g}_{\lambda}$ in (2.5). In
particular, we have $g^{(1)}=\cos\left(\frac{u_{0}}{\lambda}\right)F^{(1)}$
with $F^{(1)}\restriction{\Sigma_{0}}=\bar{F}^{(1)}$. The fact that
$\bar{F}^{(1)}$ is $\bar{g}_{0}$-traceless and tangential to $P_{0,u}$
translates as an initial polarization condition for $F^{(1)}$ equivalent to
$R_{\mu\nu}(g_{\lambda})=O\left(\lambda^{0}\right)$.

######  Remark 2.2.

These conditions correspond exactly to the definition of the TT gauge in the
linearized gravity setting for a plane wave propagating in the $N_{0}$
direction. Note that $N_{0}$ is not a constant vector field, therefore
strictly speaking a plane wave can’t propagate in the $N_{0}$ direction.
However, as (2.1) and (2.4) show, $N_{0}$ is close to the $\mathfrak{z}$
direction and the analogy with the TT gauge of linearized gravity is thus
valid.

In [Tou22], these polarization conditions are propagated by the transport
equation that $F^{(1)}$ must satisfy in the spacetime so that
$R_{\mu\nu}(g_{\lambda})=O\left(\lambda^{1}\right)$, namely

$\displaystyle-2\mathbf{D}_{L_{0}}F^{(1)}+(\Box_{g_{0}}u_{0})F^{(1)}=0,$
(2.15)

where $\mathbf{D}$ is the covariant derivative associated to $g_{0}$ and
$L_{0}=-g_{0}^{\mu\nu}\partial_{\mu}u_{0}\partial_{\nu}$ is the spacetime
gradient of $u_{0}$. Moreover, in [Tou22] we assume without loss of generality
that $\partial_{t}$ is the unit normal to $\Sigma_{0}$ for $g_{0}$ and that
$u_{0}$ solves the eikonal equation (see (1.11)), which thus give
$\partial_{t}u_{0}=|\nabla u_{0}|_{\bar{g}_{0}}$ and $L_{0}=|\nabla
u_{0}|_{\bar{g}_{0}}(\partial_{t}+N_{0})$ on $\Sigma_{0}$. This also implies
that on $\Sigma_{0}$ we have

$\displaystyle\partial_{t}\left(\lambda
g^{(1)}_{ij}\right)=-\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}F^{(1)}_{ij}+\lambda\cos\left(\frac{u_{0}}{\lambda}\right)\partial_{t}F^{(1)}_{ij}$
(2.16)

where $\partial_{t}F^{(1)}_{ij}$ on $\Sigma_{0}$ is directly given by the
first order transport equation (2.15). Note in particular that thanks to our
previous geometric assumption on $\partial_{t}$ and $u_{0}$ on $\Sigma_{0}$
and thanks to the wave gauge condition that $g_{0}$ is also assumed to satisfy
in the spacetime, we can completely rewrite
$\partial_{t}F^{(1)}_{ij}\restriction{\Sigma_{0}}$ given by (2.15) in terms of
$\bar{g}_{0}$, $K_{0}$ and $\bar{F}^{(1)}$. For instance, we can show that

$\displaystyle-\frac{1}{2}\Box_{g_{0}}u_{0}\restriction{\Sigma_{0}}=(K_{0})^{k\ell}\partial_{k}u_{0}\partial_{\ell}u_{0}+\bar{g}_{0}^{k\ell}\partial_{k}u_{0}\partial_{\ell}|\nabla
u_{0}|_{\bar{g}_{0}}-\frac{1}{2}\bar{g}_{0}^{k\ell}\partial_{k}\partial_{\ell}u_{0}.$

A similar treatment can be applied to all the terms in (2.15). Now, on the one
hand, $K_{\lambda}$ must be the second fundamental form
$-\frac{1}{2}\mathcal{L}_{T_{\lambda}}g_{\lambda}$ of $\Sigma_{0}$, where
$T_{\lambda}$ is the unit normal to $\Sigma_{0}$ for $g_{\lambda}$. On the
other hand, we choose data in [Tou22] such that
$T_{\lambda}=\partial_{t}+O\left(\lambda^{2}\right)$. Therefore, the
oscillating parts of $K^{(0)}_{\lambda}$ and $K^{(1)}_{\lambda}$ must be
consistent with (2.16), which explains expressions (2.7) and (2.10).

######  Remark 2.3.

This entanglement between the construction of the data and the evolution
equations in the spacetime is characteristic of the present high-frequency
setting. Indeed, the spacelike constraint equations give two initial data,
roughly $g\restriction{\Sigma_{0}}$ and
$\partial_{t}g\restriction{\Sigma_{0}}$, while the oscillating terms in the
spacetime ansatz satisfy first order transport equations which only require
$g\restriction{\Sigma_{0}}$ and prescribe, in part,
$\partial_{t}g\restriction{\Sigma_{0}}$.

Finally, note that the use of Sobolev spaces instead of Hölder spaces in
Theorem 2.1 also comes from the application to [Tou22], since Sobolev spaces
are more suited to non-linear wave equations. Similarly, the constraint $N\geq
10$ allows us to perform a bootstrap argument for the remainder in [Tou22]
using only $L^{2}-L^{\infty}$ estimates.

##  3 Strategy of proof

In this section, we present the strategy of the proof of Theorem 2.1, which
consists mainly in an adaptation of the conformal method to the high-frequency
setting.

######  Remark 3.1.

From now on and until Section 7 we drop the index $\lambda$ in the solution
$(\bar{g}_{\lambda},K_{\lambda})$ obtained in Theorem 2.1 and simply write
$(\bar{g},K)$.

###  3.1 High-frequency expansion of the parameters

We start by giving and motivating the expansions of the parameters of the
conformal method, i.e the metric $\gamma$, the scalar function $\tau$ and the
TT-tensor $\sigma$.

Since the metric $\bar{g}$ obtained in Theorem 2.1 will ultimately inherit its
properties from the metric $\gamma$ defining the conformal class, $\gamma$
needs needs to fulfill two requirements: it needs to oscillate and to be close
to $\bar{g}_{0}$. Therefore, we define $\gamma$ by

$\displaystyle\gamma$
$\displaystyle=\bar{g}_{0}+\lambda\gamma^{(1)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}\gamma^{(2)}\left(\frac{u_{0}}{\lambda}\right),$
(3.1)

where we recall that the notation
$\gamma^{(i)}\left(\frac{u_{0}}{\lambda}\right)$ is used to emphasize the fact
that $\gamma^{(i)}$ is a linear combination of terms of the form (1.17). See
Section 4.1 for the exact definition of $\gamma^{(1)}$ and $\gamma^{(2)}$.

Similarly, we want $K$ to be close to $K_{0}$ (in a weak sense since it is at
the level of one derivative of the metric, see (1.4)) and $K_{0}$ is assumed
to be $\bar{g}_{0}$-traceless. We thus define $\tau$ to be of order
$\lambda^{1}$, i.e

$\displaystyle\tau$
$\displaystyle=\lambda\tau^{(1)}\left(\frac{u_{0}}{\lambda}\right).$ (3.2)

The fact that the high-frequency perturbations don’t contribute to the mean
curvature at the $\lambda^{0}$ order will be justified in the proof, but this
can already be seen in Theorem 2.1: the contribution at $\lambda^{0}$ order to
$K$ is given by $\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}$ (see (2.7)), and $\bar{F}^{(1)}$ is
$\bar{g}_{0}$-traceless.

The definition of the TT-tensor $\sigma$ requires a special construction
linked to expressions (2.10)-(2.11). Since it involves also the expansion of
the solution $\varphi$ given in the next section, we postpone the discussion
of this special construction to Section 3.3. However, we can already give its
high-frequency expansion:

$\displaystyle\sigma$
$\displaystyle=\sigma^{(0)}\left(\frac{u_{0}}{\lambda}\right)+\lambda\sigma^{(1)}\left(\frac{u_{0}}{\lambda}\right)+\lambda^{2}\left(\sigma^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\widetilde{\sigma}\right).$
(3.3)

Here, $\widetilde{\sigma}$ is a non-oscillating remainder whose role is to
solve the equations defining the class of TT-tensors, that is
$\mathrm{div}_{\gamma}\sigma=0$ and $\mathrm{tr}_{\gamma}\sigma=0$, with
$\gamma$ defined by (3.1).

###  3.2 High-frequency expansion of the solutions

The solutions $(\varphi,W)$ need to solve equations (1.15)-(1.16), where the
parameters defined above appear as coefficients. To understand the expansions
for $(\varphi,W)$, let us look at the effect of the expansions
(3.1)-(3.2)-(3.3) on the equations (1.15)-(1.16).

A standard feature of high-frequency quantities is that they lose one power of
$\lambda$ per derivative. In (1.15), derivatives of the metric $\gamma$ only
appear in its scalar curvature $R(\gamma)$ and we can derive from (3.1) that

$\displaystyle R(\gamma)$ $\displaystyle=\frac{|\nabla
u_{0}|^{2}_{\bar{g}_{0}}}{\lambda}\partial^{2}_{\theta}\left(\gamma^{(1)}_{N_{0}N_{0}}-\mathrm{tr}_{\bar{g}_{0}}\gamma^{(1)}_{ij}\right)+O\left(\lambda^{0}\right).$
(3.4)

Therefore, if $\gamma^{(1)}$ is well-chosen, the RHS of (1.15) is
$O\left(\lambda^{0}\right)$. This implies that the first oscillating term in
$\varphi$ must be at order $\lambda^{2}$, since $\Delta_{\gamma}$ is a second
order elliptic operator. More precisely we choose

$\displaystyle\varphi$
$\displaystyle=1+\lambda^{2}\left(\varphi^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\widetilde{\varphi}\right)+\lambda^{3}\varphi^{(3)}\left(\frac{u_{0}}{\lambda}\right),$
(3.5)

where $\widetilde{\varphi}$ is a non-oscillating remainder. The constant
coefficient in (3.5) ensures that
$\bar{g}_{\lambda}=\bar{g}_{0}+O\left(\lambda\right)$. Even though
$\varphi^{(3)}$ appears after the remainder $\widetilde{\varphi}$ in the
ansatz for $\varphi$, it appears before $\widetilde{\varphi}$ in the hierarchy
obtained from (1.15), again since $\Delta_{\gamma}$ is a second order elliptic
operator.

Similarly, (3.2) and (3.5) imply that
$\varphi^{6}\mathrm{d}\tau=O\left(\lambda^{0}\right)$ and the ellipticity of
$\mathrm{div}_{\gamma}L_{\gamma}$ allows us to choose $W$ of the following
form

$\displaystyle W$
$\displaystyle=\lambda^{2}\left(W^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\widetilde{W}\right)+\lambda^{3}W^{(3)}\left(\frac{u_{0}}{\lambda}\right),$
(3.6)

where $\widetilde{W}$ is a non-oscillating remainder.

As is standard when considering high-frequency expansions, we obtain a
hierarchy of equations. Very schematically, one can say that $\varphi^{(2)}$
and $\varphi^{(3)}$ (resp. $W^{(2)}$ and $W^{(3)}$) will solve the orders
$\lambda^{0}$ and $\lambda^{1}$ of (1.15) (resp. (1.16)), while the non-
oscillating remainder $\widetilde{\varphi}$ (resp. $\widetilde{W}$) will solve
the orders higher than $\lambda^{2}$.

Even though the operators $\Delta_{\gamma}$ and
$\mathrm{div}_{\gamma}L_{\gamma}$ are both second order elliptic operators,
they depend differently on $\gamma$. Indeed, the coefficients of
$\Delta_{\gamma}$ depend on $\gamma$ and $\nabla\gamma$ only, while the
coefficients of $\mathrm{div}_{\gamma}L_{\gamma}$ also depend on
$\nabla^{2}\gamma$. From (3.1), we thus see that
$\mathrm{div}_{\gamma}L_{\gamma}$ loses inherently one power of $\lambda$. The
analysis of the last equations in the hierarchy will therefore differ from the
hamiltonian to the momentum constraint.

###  3.3 The TT-tensor

In this section, we give more details on the definition of $\sigma$, and in
particular the first two terms in its high-frequency expansion (3.3), i.e
$\sigma^{(0)}$ and $\sigma^{(1)}$. Thanks to (3.5) we have
$\varphi=1+O\left(\lambda^{2}\right)$ which implies that

$\displaystyle K^{(0)}$ $\displaystyle=\sigma^{(0)}+(L_{\gamma}W)^{(0)},$
(3.7) $\displaystyle K^{(1)}$
$\displaystyle=\sigma^{(1)}+(L_{\gamma}W)^{(1)}+\frac{1}{3}\bar{g}_{0}\tau^{(1)},$
(3.8)

where we used (1.14), (3.1) and (3.2). As explained in Section 2.3, the
expressions of $K^{(0)}$ and $K^{(1)}$ are actually prescribed the application
of Theorem 2.1 to the definition of initial data for (1.1). Therefore,
(2.7)-(2.11) are used to define $K^{(0)}$ and $K^{(1)}$, while $\sigma^{(0)}$
and $\sigma^{(1)}$ are defined a posteriori such that (3.7)-(3.8) hold. We are
then left with the task of showing that $\sigma^{(0)}+\lambda\sigma^{(1)}$
defines an approximate TT-tensor and of constructing an exact one, with the
help of $\sigma^{(2)}$ and the remainder $\widetilde{\sigma}$ introduced in
(3.3). The full construction of $\sigma$ is the content of Sections 5.3 and
5.4.

This procedure goes against the standard conformal method, where one starts by
defining the parameters, then solve for the $(\varphi,W)$, and finally obtain
$(\bar{g},K)$ via (1.13)-(1.14). This is another characteristic feature of the
high-frequency setting, which originates in the redundancy described in Remark
2.3.

###  3.4 Outline of the proof

We give here the outline of the rest of this article, which proves Theorem
2.1.

  * •

In Section 4 we define the metric $\gamma$ and compute useful high-frequency
expansions such as its scalar curvature $R(\gamma)$ or differential operators
depending on $\gamma$ and appearing in the constraint equations.

  * •

In Section 5 we define $\varphi^{(2)}$, $\varphi^{(3)}$ and $W^{(2)}$ such
that they solve the first orders of the constraint equations and by doing so
we also define the parameter $\tau$. This section is concluded by the
construction of the TT-tensor $\sigma$.

  * •

In Section 6 we fully solve the constraint equations with a fixed point
argument for the remainders $\widetilde{\varphi}$ and $\widetilde{W}$ (we also
define $W^{(3)}$ in the process).

  * •

In Section 7 we conclude the proof of Theorem 2.1 by proving (2.12) and
(2.13).

##  4 High-frequency conformal class

In this section we define the metric $\gamma$ on $\Sigma_{0}$, that is the
prefered member of the conformal class in which we look for $\bar{g}$
according to (1.13).

###  4.1 Definitions and first computations

We choose

$\displaystyle\gamma=\bar{g}_{0}+\lambda\gamma^{(1)}+\lambda^{2}\gamma^{(2)},$
(4.1)

where

$\displaystyle\gamma^{(1)}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)F_{0}\omega^{(1)},$ (4.2)
$\displaystyle\gamma^{(2)}$
$\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)\omega^{(2)}.$ (4.3)

The two symmetric 2-forms $\omega^{(1)}$ and $\omega^{(2)}$ are directly
defined in the frame $(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$.

  * •

Definition of $\omega^{(1)}$. The coefficients of $\omega^{(1)}$ in the frame
$(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ are constants and satisfy

$\displaystyle\omega^{(1)}_{N_{0}i}$ $\displaystyle=0,$ (4.4)
$\displaystyle\omega^{(1)}_{\mathbf{1}\mathbf{1}}+\omega^{(1)}_{\mathbf{2}\mathbf{2}}$
$\displaystyle=0,$ (4.5)
$\displaystyle\left(\omega^{(1)}_{\mathbf{1}\mathbf{1}}\right)^{2}+\left(\omega^{(1)}_{\mathbf{1}\mathbf{2}}\right)^{2}$
$\displaystyle=4.$ (4.6)

  * •

Definition of $\omega^{(2)}$. The 2-form $\omega^{(2)}$ depends on
$\omega^{(1)}$ in the following way: all the coefficients of $\omega^{(2)}$ in
the frame $(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ are set to be zero except
$\omega^{(2)}_{N_{0}\mathbf{1}}$ and $\omega^{(2)}_{N_{0}\mathbf{2}}$ which
are defined by

$\displaystyle\omega^{(2)}_{N_{0}\mathbf{j}}$ $\displaystyle=\frac{1}{|\nabla
u_{0}|_{\bar{g}_{0}}^{2}}(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}F_{0}\omega^{(1)})_{\mathbf{j}}-\frac{1}{\left|\nabla
u_{0}\right|_{\bar{g}_{0}}}F_{0}\omega^{(1)}_{i\mathbf{j}}\left(\mathbf{D}_{N_{0}}N_{0}^{i}-\bar{g}_{0}^{ik}(K_{0})_{N_{0}k}\right).$
(4.7)

This expression will be justified in the proof of Lemma 5.3, where we
construct the TT-tensor of the conformal method.

######  Remark 4.1.

The properties (4.4) and (4.5) and the symmetry of $\omega^{(1)}$ imply that
the tensor $\gamma^{(1)}$ is a linear combination of the tensors

$\displaystyle e_{\mathbf{1}}\otimes e_{\mathbf{1}}-e_{\mathbf{2}}\otimes
e_{\mathbf{2}}\quad\text{and}\quad e_{\mathbf{1}}\otimes
e_{\mathbf{2}}+e_{\mathbf{2}}\otimes e_{\mathbf{1}}.$

This is the equivalent of the 2 degrees of freedom (or polarization) of the TT
gauge in the linearized gravity setting where $g_{0}$ is replaced by the
Minkowskian metric, see Remark 2.2. The choice of the polarization of
$\omega^{(1)}$, i.e the choice of $\omega^{(1)}_{\mathbf{1}\mathbf{1}}$ and
$\omega^{(1)}_{\mathbf{1}\mathbf{2}}$ under the constraint (4.6), is the only
freedom we have in the definition of the tensors $\bar{F}^{(1)}$,
$\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$.

We define

$\displaystyle\bar{F}^{(1)}$ $\displaystyle=F_{0}\omega^{(1)}.$ (4.8)

The following lemma summarizes the important properties of $\bar{F}^{(1)}$.

######  Lemma 4.1.

The symmetric 2-tensor $\bar{F}^{(1)}$ is supported in $B_{R}$,
$\bar{g}_{0}$-traceless and $P_{0,u}$-tangential. Moreover, (2.9) holds and

$\displaystyle\left\|\bar{F}^{(1)}\right\|_{H^{N}}\lesssim\varepsilon,$ (4.9)

with a constant depending only on $\delta$ and $R$.

###### Proof.

The support property is implied by the support property of $F_{0}$. The
property (4.4) implies that $\bar{F}^{(1)}$ is $P_{0,u}$-tangential, which
together with (4.5) implies that $\bar{F}^{(1)}$ is $\bar{g}_{0}$-traceless
since

$\displaystyle\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=\bar{F}^{(1)}_{N_{0}N_{0}}+\bar{F}^{(1)}_{\mathbf{1}\mathbf{1}}+\bar{F}^{(1)}_{\mathbf{2}\mathbf{2}}.$

Moreover, (4.6) and (4.8) imply (2.9). The estimation (4.9) comes from (2.3)
and (2.4), the latter allowing us to estimate the coefficients of
$\omega^{(1)}$ in the usual Euclidean coordinates, i.e $\omega^{(1)}_{ij}$. ∎

The following lemma gives the expansion of the scalar curvature of $\gamma$.
This will be used to solve the Hamitonian constraint (1.15).

######  Lemma 4.2.

We have

$\displaystyle R(\gamma)$ $\displaystyle=R^{(0)}+\lambda
R^{(1)}+\lambda^{2}R^{(\geq 2)},$

with

$\displaystyle R^{(0)}$ $\displaystyle=R(\bar{g}_{0})-|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}-7\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}$ (4.10)
$\displaystyle\quad+\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}\right),$

and

$\displaystyle\left|R^{(1)}\right|+\left|R^{(\geq
2)}\right|\lesssim\left|\gamma^{-2}\partial^{2}\gamma\right|+\left|\gamma^{-3}(\partial\gamma)^{2}\right|.$
(4.11)

Moreover

$\displaystyle
R^{(1)}\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta)+\cos(3\theta).$
(4.12)

###### Proof.

We recall the definition of the scalar curvature

$\displaystyle R(\gamma)$
$\displaystyle=\gamma^{ij}\left(\partial_{k}\Gamma(\gamma)^{k}_{ij}-\partial_{i}\Gamma(\gamma)^{k}_{jk}+\Gamma(\gamma)^{k}_{k\ell}\Gamma(\gamma)^{\ell}_{ij}-\Gamma(\gamma)^{k}_{i\ell}\Gamma(\gamma)^{\ell}_{jk}\right).$

We start by giving an estimation up to second order in $\lambda$ of the
inverse of $\gamma$:

$\displaystyle\gamma^{ij}$
$\displaystyle=\bar{g}_{0}^{ij}-\lambda\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}$
(4.13)
$\displaystyle\quad+\lambda^{2}\left(\cos^{2}\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ik}(\bar{F}^{(1)})^{j\ell}\bar{F}^{(1)}_{k\ell}-\sin\left(\frac{u_{0}}{\lambda}\right)(\omega^{(2)})^{ij}\right)+O\left(\lambda^{3}\right),$

where on the RHS all the inverses are taken with respect to the background
metric $\bar{g}_{0}$. We now expand the Christoffel symbols of $\gamma$, we
obtain

$\displaystyle\Gamma(\gamma)^{k}_{ij}$
$\displaystyle=\Gamma(\bar{g}_{0})^{k}_{ij}+(\tilde{\Gamma}^{(0)})^{k}_{ij}+\lambda(\Gamma^{(1)})^{k}_{ij}+O\left(\lambda^{2}\right),$
(4.14)

with

$\displaystyle(\tilde{\Gamma}^{(0)})^{k}_{ij}$
$\displaystyle=-\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{k\ell}\left(\partial_{(i}u_{0}\bar{F}^{(1)}_{\ell
j)}-\partial_{\ell}u_{0}\bar{F}^{(1)}_{ij}\right),$ (4.15)
$\displaystyle(\Gamma^{(1)})^{k}_{ij}$
$\displaystyle=\frac{1}{2}\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{k\ell}\left(\partial_{(i}u_{0}\omega^{(2)}_{\ell
j)}-\partial_{\ell}u_{0}\omega^{(2)}_{ij}\right)$ (4.16)
$\displaystyle\quad+\frac{1}{4}\sin\left(\frac{2u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{k\ell}\partial_{(i}u_{0}\bar{F}^{(1)}_{\ell
j)}+\cos\left(\frac{u_{0}}{\lambda}\right)\bar{Q}^{k}_{ij},$

where we defined

$\displaystyle\bar{Q}^{k}_{ij}$
$\displaystyle=\frac{1}{2}\bar{g}_{0}^{k\ell}\left(\partial_{(i}\bar{F}^{(1)}_{\ell
j)}-\partial_{\ell}\bar{F}^{(1)}_{ij}\right)-\frac{1}{2}(\bar{F}^{(1)})^{k\ell}\left(\partial_{(i}(\bar{g}_{0})_{\ell
j)}-\partial_{\ell}(\bar{g}_{0})_{ij}\right).$

By using $\bar{F}^{(1)}_{N_{0}i}=0$ and
$\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=0$, we can compute useful contractions
of $\bar{Q}^{k}_{ij}$. We first look at $\bar{Q}^{k}_{jk}$ which vanishes
thanks to $\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=0$:

$\displaystyle\bar{Q}^{k}_{jk}$
$\displaystyle=\frac{1}{2}\left(\bar{g}_{0}^{k\ell}\partial_{j}\bar{F}^{(1)}_{\ell
k}-(\bar{F}^{(1)})^{k\ell}\partial_{j}(\bar{g}_{0})_{\ell
k}\right)=\frac{1}{2}\partial_{j}\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=0.$
(4.17)

Using in addition $\bar{F}^{(1)}_{N_{0}i}=0$ we have:

$\displaystyle\bar{g}_{0}^{ij}(N_{0})_{k}\bar{Q}^{k}_{ij}$
$\displaystyle=\bar{g}_{0}^{ij}N_{0}^{\ell}\left(\partial_{i}\bar{F}^{(1)}_{\ell
j}-\frac{1}{2}\partial_{\ell}\bar{F}^{(1)}_{ij}\right)$
$\displaystyle=\bar{g}_{0}^{ij}\partial_{i}\bar{F}^{(1)}_{N_{0}j}-\frac{1}{2}N_{0}\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}-\bar{g}_{0}^{ij}\bar{F}^{(1)}_{\ell
j}\partial_{i}N_{0}^{\ell}+\frac{1}{2}\bar{F}^{(1)}_{\ell
j}N_{0}\bar{g}_{0}^{\ell j}$ $\displaystyle=\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}\right).$ (4.18)

Since the scalar curvature contains first derivatives of the Christoffel
symbols, $R(\gamma)$ contains a priori a $\lambda^{-1}$ contribution from
$\partial_{\theta}\tilde{\Gamma}^{(0)}$, but thanks to the properties of
$\bar{F}^{(1)}$ we can see that it vanishes. Indeed we have

$\displaystyle
R^{(-1)}=\bar{g}_{0}^{ij}\left(\partial_{k}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{ij}-\partial_{i}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{jk}\right),$

and $\bar{F}^{(1)}_{N_{0}i}=0$ and $\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=0$
imply that

$\displaystyle\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{ij}=0\quad\text{and}\quad(\tilde{\Gamma}^{(0)})^{k}_{jk}=0.$
(4.19)

Let us now look at the $\lambda^{0}$ terms in $R(\gamma)$. Using again (4.19)
we obtain:

$\displaystyle R^{(0)}-R(\bar{g}_{0})$
$\displaystyle=\bar{g}_{0}^{ij}\left(\partial_{k}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{ij}-\partial_{i}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{jk}\right)$
$\displaystyle\quad+\bar{g}_{0}^{ij}\partial_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}-2\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}\Gamma(\bar{g}_{0})^{\ell}_{jk}$
$\displaystyle\quad-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\left(\partial_{k}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{ij}-\partial_{i}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{jk}\right)-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}(\tilde{\Gamma}^{(0)})^{\ell}_{jk}$
$\displaystyle=\bar{g}_{0}^{ij}\left(\partial_{k}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{ij}-\partial_{i}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{jk}\right)$
$\displaystyle\quad+\bar{g}_{0}^{ij}\partial_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}-2\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}\Gamma(\bar{g}_{0})^{\ell}_{jk}$
$\displaystyle\quad-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\partial_{k}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{ij}-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}(\tilde{\Gamma}^{(0)})^{\ell}_{jk},$

where we used $(\bar{F}^{(1)})^{ij}\partial_{i}u_{0}=0$. We set

$\displaystyle I$
$\displaystyle\vcentcolon=\bar{g}_{0}^{ij}\left(\partial_{k}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{ij}-\partial_{i}u_{0}\partial_{\theta}(\Gamma^{(1)})^{k}_{jk}\right),$
$\displaystyle II$
$\displaystyle\vcentcolon=\bar{g}_{0}^{ij}\partial_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}-2\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}\Gamma(\bar{g}_{0})^{\ell}_{jk},$
$\displaystyle III$
$\displaystyle\vcentcolon=-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\partial_{k}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{ij}-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}(\tilde{\Gamma}^{(0)})^{\ell}_{jk},$

and compute further each of these terms. The term $I$ contains all the
contributions from $\Gamma^{(1)}$ and (4.16) implies

$\displaystyle I$ $\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\left(\omega^{(2)}_{\mathbf{1}\mathbf{1}}+\omega^{(2)}_{\mathbf{2}\mathbf{2}}\right)-\frac{1}{2}\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\left|\bar{F}^{(1)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle\quad-\sin\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\left(\partial_{k}u_{0}\bar{Q}^{k}_{ij}-\partial_{i}u_{0}\bar{Q}^{k}_{jk}\right)$
$\displaystyle=-4\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|^{2}_{\bar{g}_{0}}F_{0}^{2}+\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}\right),$

where we used that the diagonal coefficients of $\omega^{(2)}$ in the frame
$(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ vanish in addition to (4.17)-(4.18)
and (2.9). The term $II$ contains all the linear terms in
$\tilde{\Gamma}^{(0)}$. Note that in the notation
$\partial_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}$ the derivative does not hit the
oscillating part of $\tilde{\Gamma}^{(0)}$ (i.e
$\sin\left(\frac{u_{0}}{\lambda}\right)$ in (4.15)). Using (4.19) we obtain
$\bar{g}_{0}^{ij}\partial_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}=-(\tilde{\Gamma}^{(0)})^{k}_{ij}\partial_{k}\bar{g}_{0}^{ij}$
and

$\displaystyle II$
$\displaystyle=-(\tilde{\Gamma}^{(0)})^{k}_{i\ell}\left(\partial_{k}\bar{g}_{0}^{i\ell}+2\bar{g}_{0}^{ij}\Gamma(\bar{g}_{0})^{\ell}_{jk}\right)=0,$

where we used
$(\tilde{\Gamma}^{(0)})^{k}_{i\ell}=(\tilde{\Gamma}^{(0)})^{k}_{\ell i}$. The
term $III$ contains quadratic terms in $\tilde{\Gamma}^{(0)}$. Using (4.15)
and (2.9) we obtain

$\displaystyle III$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\partial_{k}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{ij}-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{i\ell}(\tilde{\Gamma}^{(0)})^{\ell}_{jk}$
$\displaystyle=-\frac{1}{2}\cos^{2}\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\left|\bar{F}^{(1)}\right|^{2}_{\bar{g}_{0}}+\frac{1}{4}\sin^{2}\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\left|\bar{F}^{(1)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle=-|\nabla
u_{0}|^{2}_{\bar{g}_{0}}F_{0}^{2}-3\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|^{2}_{\bar{g}_{0}}F_{0}^{2}.$

We conclude the proof of (4.10) by adding $I$ and $III$.

The proof of (4.12) comes from the schematic formula

$\displaystyle
R(\gamma)=\gamma^{-2}\partial^{2}\gamma+\gamma^{-3}(\partial\gamma)^{2},$

which gives schematically

$\displaystyle R^{(1)}$
$\displaystyle=\left((\gamma^{-1})^{(2)}(\gamma^{-1})^{(0)}+\left((\gamma^{-1})^{(1)}\right)^{2}\right)(\partial^{2}\gamma)^{(-1)}+(\gamma^{-1})^{(1)}(\gamma^{-1})^{(0)}(\partial^{2}\gamma)^{(0)}$
(4.20)
$\displaystyle\quad+\left((\gamma^{-1})^{(0)}\right)^{2}(\partial^{2}\gamma)^{(1)}+(\gamma^{-1})^{(1)}\left((\gamma^{-1})^{(0)}(\partial\gamma)^{(0)}\right)^{2}+\left((\gamma^{-1})^{(0)}\right)^{3}(\partial\gamma)^{(1)}(\gamma^{-1})^{(0)},$

where we recall that our high-frequency notations introduced in Section 1.3.4
give for instance

$\displaystyle(\partial^{2}\gamma)^{(1)}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\partial^{2}\bar{F}^{(1)}+2\cos\left(\frac{u_{0}}{\lambda}\right)\partial\omega^{(2)},$

and

$\displaystyle(\gamma^{-1})^{(1)}$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}.$

Using (4.1) and (4.13) we obtain

$\displaystyle(\gamma^{-1})^{(0)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;1,$
$\displaystyle(\gamma^{-1})^{(1)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta),$
$\displaystyle(\gamma^{-1})^{(2)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;1+\sin(\theta)+\cos(2\theta),$

and

$\displaystyle(\partial\gamma)^{(0)},\;(\partial^{2}\gamma)^{(0)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;1+\sin(\theta),$
$\displaystyle(\partial\gamma)^{(1)},\;(\partial^{2}\gamma)^{(-1)},\;(\partial^{2}\gamma)^{(1)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta).$

Using these oscillating behaviours we can prove by a direct computation that
$R^{(1)}$ defined by (4.20) satisfies (4.12). This concludes the proof of the
lemma. ∎

######  Remark 4.2.

Note that we denote by $\gamma^{-k}$ any product of $k$ coefficients of the
inverse metric $\gamma^{-1}$. This also applies to the background inverse
metric $\bar{g}_{0}^{-1}$.

###  4.2 Expansion of differential operators

The equations (1.15) and (1.16) involve differential operators depending on
the metric $\gamma$. When they are applied to high-frequency quantities, we
need to take into account the expansion (4.1) affecting the coefficients of
the operators as well as the expansion of the quantities themselves. In this
section we compute such expansions for all the operators involved, that is the
Laplace-Beltrami operator, the divergence operator, the conformal Killing
operator and the conformal Laplacian.

In terms of notation, if $P_{\gamma}$ is a differential operator acting on
tensors of any type and whose coefficients depends on $\gamma$, we can
formally obtain an expansion of $P_{\gamma}(T)$ of the form

$\displaystyle P_{\gamma}(T)$
$\displaystyle=\sum_{k}\lambda^{k}P^{[k]}_{\gamma}(T).$

where the previous sum has finite support. The bracket notation $[k]$ thus
plays the same role for differential operators as the parenthesis notation
$(i)$ introduced in Section 1.3.4 for tensors.

Moreover we can mix the two cases, i.e apply differential operators
$P_{\gamma}$ depending on $\gamma$ to oscillating tensors
$T\left(\frac{u_{0}}{\lambda}\right)$. The expansion of $P_{\gamma}(T)$ then
depends on the order of $T\longmapsto P_{\gamma}(T)$ and also
$\gamma\longmapsto P_{\gamma}(T)$. By the order of $\gamma\longmapsto
P_{\gamma}(T)$, we simply mean the top derivative of $\gamma$ appearing in the
coefficients of $P_{\gamma}$. Since both oscillating and non-oscillating terms
appear in the expansions for the parameters and the unknowns of the conformal
method (see Section 3.1), we make a difference between the expansions for
$P_{\gamma}(T)$ when $T$ is non-oscillating and when $T$ is oscillating. Usual
capital letters are used in the first case and bold capital letters in the
second case, i.e

$\displaystyle P_{\gamma}(T)$
$\displaystyle=\sum_{k}\lambda^{k}P^{[k]}_{\gamma}(T)\quad\text{and}\quad
P_{\gamma}\left(T\left(\frac{u_{0}}{\lambda}\right)\right)=\sum_{k}\lambda^{k}\mathbf{P}^{[k]}_{\gamma}(T),$

where the support of the two previous finite sums are a priori different,
depending on $T$ and $P_{\gamma}$. This explains the difference between Lemmas
4.6 and 4.7 below.

####  4.2.1 The Laplace-Beltrami operator

The only differential operator in the hamiltonian constraint (1.15) is the
Laplace-Beltrami operator associated to $\gamma$. If $h$ is a generic
Riemannian metric on $\Sigma_{0}$, we define its Laplace-Beltrami operator by

$\displaystyle\Delta_{h}f=h^{ij}\left(\partial_{i}\partial_{j}f-\Gamma(h)_{ij}^{k}\partial_{k}f\right),$

for $f$ a scalar function. Note that $\Delta_{e}$ is the usual Laplacian
operator on $\mathbb{R}^{3}$ and is denoted by $\Delta$ in this article. Since
$\gamma=O\left(\lambda^{0}\right)$ and
$\partial\gamma=O\left(\lambda^{0}\right)$, if $f$ does not admit a high-
frequency expansion then

$\displaystyle\Delta_{\gamma}f=O\left(\lambda^{0}\right).$

If $f$ admit a high-frequency expansion we have the following lemma.

######  Lemma 4.3.

For $f\left(\frac{u_{0}}{\lambda}\right)$ an oscillating scalar function we
have

$\displaystyle\Delta_{\gamma}\left(f\left(\frac{u_{0}}{\lambda}\right)\right)$
$\displaystyle=\frac{1}{\lambda^{2}}\mathbf{H}^{[-2]}(f)+\frac{1}{\lambda}\mathbf{H}^{[-1]}(f)+\mathbf{H}^{[\geq
0]}(f),$

with

$\displaystyle\mathbf{H}^{[-2]}(f)$ $\displaystyle=|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\partial_{\theta}^{2}f,$ (4.21)
$\displaystyle\mathbf{H}^{[-1]}(f)$
$\displaystyle=2\bar{g}_{0}^{ij}\partial_{i}u_{0}\partial_{j}\partial_{\theta}f+(\tilde{\Delta}_{\bar{g}_{0}}u_{0})\partial_{\theta}f-\bar{g}_{0}^{ij}\Gamma(\bar{g}_{0})_{ij}^{k}\partial_{k}u_{0}\partial_{\theta}f,$
(4.22)

and

$\displaystyle\left|\mathbf{H}^{[\geq
0]}(f)\right|\lesssim\left|\gamma^{-1}\partial^{2}f\right|+\left|\gamma^{-2}\partial\gamma\partial
f\right|.$ (4.23)

###### Proof.

We start with the definition of the Laplace-Beltrami operator :

$\Delta_{\gamma}\left(\varphi\left(\frac{u_{0}}{\lambda}\right)\right)=\gamma^{ij}\partial_{i}\partial_{j}\left(\varphi\left(\frac{u_{0}}{\lambda}\right)\right)-\gamma^{ij}\Gamma(\gamma)_{ij}^{k}\partial_{k}\left(\varphi\left(\frac{u_{0}}{\lambda}\right)\right).$

Using the expansion of the inverse of $\gamma$ and $\bar{F}^{(1)}_{N_{0}i}=0$
we have

$\displaystyle\gamma^{ij}\partial_{i}\partial_{j}\left(\varphi\left(\frac{u_{0}}{\lambda}\right)\right)$
$\displaystyle=\frac{1}{\lambda^{2}}|\nabla
u_{0}|_{\gamma}^{2}\partial_{\theta}^{2}\varphi+\frac{1}{\lambda}\left(2\gamma^{ij}\partial_{i}u_{0}\partial_{j}\partial_{\theta}\varphi+(\tilde{\Delta}_{\gamma}u_{0})\partial_{\theta}\varphi\right)+O\left(\lambda^{0}\right)$
$\displaystyle=\frac{1}{\lambda^{2}}|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\partial_{\theta}^{2}\varphi+\frac{1}{\lambda}\left(2\bar{g}_{0}^{ij}\partial_{i}u_{0}\partial_{j}\partial_{\theta}\varphi+(\tilde{\Delta}_{\bar{g}_{0}}u_{0})\partial_{\theta}\varphi\right)+O\left(\lambda^{0}\right),$

where $\tilde{\Delta}_{h}=h^{ij}\partial_{i}\partial_{j}$. Moreover from the
decomposition of the Christoffel symbols (4.14) and (4.19) we obtain

$\displaystyle\gamma^{ij}\Gamma(\gamma)_{ij}^{k}\partial_{k}\left(\varphi\left(\frac{u_{0}}{\lambda}\right)\right)$
$\displaystyle=\frac{1}{\lambda}\bar{g}_{0}^{ij}\Gamma(\bar{g}_{0})_{ij}^{k}\partial_{k}u_{0}\partial_{\theta}\varphi+O\left(\lambda\right).$

The estimate (4.23) simply comes from the definition of $\Delta_{\gamma}$. ∎

####  4.2.2 The divergence operator

The divergence operator appears in the momentum constraint (1.16) but also in
the definition of the parameter $\sigma$, which in particular needs to be a
divergence free tensor for the metric $\gamma$. Recall that if $h$ is a
Riemannian metric on $\Sigma_{0}$ and if $\mathbf{D}^{(h)}$ denotes the
covariant derivative associated, then
$\mathrm{div}_{h}A=h^{k\ell}\mathbf{D}^{(h)}_{k}A_{\ell}$ and
$(\mathrm{div}_{h}B)_{i}=h^{k\ell}\mathbf{D}^{(h)}_{k}B_{\ell i}$ for $A$ a
1-tensor and $B$ a 2-tensor. The divergence operator only depends on first
derivatives of $\gamma$ which are $O\left(\lambda^{0}\right)$ so we only need
an expansion when the tensor on which it acts is itself oscillating.

######  Lemma 4.4.

For $A_{ij}\left(\frac{u_{0}}{\lambda}\right)$ an oscillating symmetric
2-tensor we have

$\displaystyle\mathrm{div}_{\gamma}A\left(\frac{u_{0}}{\lambda}\right)_{\ell}$
$\displaystyle=\frac{1}{\lambda}\mathbf{d}^{[-1]}_{\ell}(A)+\mathbf{d}^{[0]}_{\ell}(A)+\lambda\mathbf{d}^{[\geq
1]}_{\ell}(A),$

with

$\displaystyle\mathbf{d}^{[-1]}_{\ell}(A)$ $\displaystyle=-|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\theta}A_{N_{0}\ell},$ (4.24)
$\displaystyle\mathbf{d}^{[0]}_{\ell}(A)$
$\displaystyle=\mathrm{div}_{\bar{g}_{0}}A_{\ell}-(\bar{g}_{0})^{ij}(\tilde{\Gamma}^{(0)})^{a}_{i\ell}A_{aj},$
(4.25)

and

$\displaystyle\left|\mathbf{d}^{[\geq
1]}_{\ell}(A)\right|\lesssim\left|\gamma^{-1}\partial
A\right|+\left|\gamma^{-2}\partial\gamma A\right|.$ (4.26)

Moreover

$\displaystyle\mathbf{d}^{[0]}(A)$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;(1+\sin(\theta))A,$ (4.27)
$\displaystyle\mathbf{d}^{[1]}(A)$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\sin(\theta)\partial_{\theta}A+(\cos(\theta)+\sin(2\theta))A.$
(4.28)

###### Proof.

We start by using the expansion of the Christoffel’s symbols from (4.14):

$\displaystyle\mathbf{D}^{(\gamma)}_{i}A\left(\frac{u_{0}}{\lambda}\right)_{j\ell}$
$\displaystyle=\frac{1}{\lambda}\partial_{i}u_{0}\partial_{\theta}A_{j\ell}+\bar{\mathbf{D}}_{i}A_{j\ell}-(\tilde{\Gamma}^{(0)})^{a}_{i(j}A_{a\ell)}-\lambda(\Gamma^{(1)})^{a}_{i(j}A_{a\ell)}+O\left(\lambda^{2}\right).$

We now use the expansion of the inverse of $\gamma$:

$\displaystyle\mathrm{div}_{\gamma}A\left(\frac{u_{0}}{\lambda}\right)_{\ell}$
$\displaystyle=\gamma^{ij}\mathbf{D}^{(\gamma)}_{i}A\left(\frac{u_{0}}{\lambda}\right)_{j\ell}$
$\displaystyle=-\frac{1}{\lambda}|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\theta}A_{N_{0}\ell}+\mathrm{div}_{\bar{g}_{0}}A_{\ell}-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{a}_{i(j}A_{a\ell)}-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\partial_{i}u_{0}\partial_{\theta}A_{j\ell}$
(4.29)
$\displaystyle\quad+\lambda\left[\cos^{2}\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ik}(\bar{F}^{(1)})^{ja}\bar{F}^{(1)}_{ka}\partial_{i}u_{0}\partial_{\theta}A_{j\ell}-\sin\left(\frac{u_{0}}{\lambda}\right)(\omega^{(2)})^{ij}\partial_{i}u_{0}\partial_{\theta}A_{j\ell}\right.$
$\displaystyle\qquad\qquad\qquad\left.-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{ij}\left(\bar{\mathbf{D}}_{i}A_{j\ell}-(\tilde{\Gamma}^{(0)})^{a}_{i(j}A_{a\ell)}\right)-\bar{g}_{0}^{ij}(\Gamma^{(1)})^{a}_{i(j}A_{a\ell)}\right]$
$\displaystyle\quad+O\left(\lambda^{2}\right).$

This concludes the proof, using $\bar{F}^{(1)}_{N_{0}i}=0$ and
$\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{a}_{ij}=0$. The estimate (4.26) just
comes from the definition of the divergence. The oscillating behaviours (4.27)
and (4.28) can be directly read on (4.29). ∎

####  4.2.3 The conformal Killing operator

In order to compute $L_{\gamma}W$ in the ansatz for $K$ (1.14), we need to
expand the conformal Killing operator defined by

$\displaystyle(L_{h}A)_{ij}=\mathbf{D}^{(h)}_{(i}A_{j)}-\frac{2}{3}(\mathrm{div}_{h}A)h_{ij},$
(4.30)

where $A$ is a 1-tensor. Note that $L_{h}A$ is a symmetric 2-tensor traceless
with respect to $h$. As for the divergence operator, $L_{\gamma}$ only depends
on first derivatives of $\gamma$ so only an expansion of $L_{\gamma}W$ with
$W$ oscillating is necessary.

######  Lemma 4.5.

Let $W\left(\frac{u_{0}}{\lambda}\right)$ be an oscillating 1-form. We have

$L_{\gamma}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{ij}=\frac{1}{\lambda}\mathbf{K}_{ij}^{[-1]}(W)+\mathbf{K}_{ij}^{[0]}(W)+\lambda\mathbf{K}_{ij}^{[\geq
1]}(W),$

where

$\displaystyle\mathbf{K}_{ij}^{[-1]}(W)$
$\displaystyle=\partial_{(i}u_{0}\partial_{\theta}W_{j)}+\frac{2}{3}|\nabla
u_{0}|_{\bar{g}_{0}}(\bar{g}_{0})_{ij}\partial_{\theta}W_{N_{0}},$ (4.31)
$\displaystyle\mathbf{K}_{ij}^{[0]}(W)$
$\displaystyle=\bar{\mathbf{D}}_{(i}W_{j)}-2W_{k}(\tilde{\Gamma}^{(0)})^{k}_{ij}+\frac{2}{3}\cos\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{ij}\partial_{\theta}W_{N_{0}}-\frac{2}{3}\mathrm{div}_{\bar{g}_{0}}W(\bar{g}_{0})_{ij},$
(4.32)

and

$\displaystyle\left|\mathbf{K}_{ij}^{[\geq 1]}(W)\right|\lesssim\left|\partial
W\right|+\left|\gamma^{-1}\partial\gamma W\right|.$ (4.33)

The following hold:

$\displaystyle\mathrm{tr}_{\bar{g}_{0}}\mathbf{K}^{[-1]}(W)$
$\displaystyle=0,$ (4.34)
$\displaystyle\mathrm{tr}_{\bar{g}_{0}}\mathbf{K}^{[0]}(W)$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}\mathbf{K}^{[-1]}(W).$
(4.35)

The following oscillating behaviour holds

$\displaystyle\mathbf{K}^{[0]}(W)$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;(1+\sin(\theta))W+\cos(\theta)\partial_{\theta}W.$
(4.36)

###### Proof.

We start with the covariant derivative of the oscillating 1-form
$W^{(\cdot)}\left(\frac{u_{0}}{\lambda}\right)$:

$\displaystyle\mathbf{D}^{(\gamma)}_{i}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{j}$
$\displaystyle=\frac{1}{\lambda}\partial_{i}u_{0}\partial_{\theta}W_{j}+\bar{\mathbf{D}}_{i}W_{j}\left(\frac{u_{0}}{\lambda}\right)-W_{k}\left((\tilde{\Gamma}^{(0)})^{k}_{ij}+\lambda(\Gamma^{(1)})^{k}_{ij}+\lambda^{2}(\Gamma^{(2)})^{k}_{ij}+\cdots\right),$

which also gives us the divergence

$\displaystyle\mathrm{div}_{\gamma}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)$
$\displaystyle=\gamma^{ij}\mathbf{D}^{(\gamma)}_{i}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{j}$
$\displaystyle=\frac{1}{\lambda}(\bar{g}_{0})^{k\ell}\partial_{k}u_{0}\partial_{\theta}W_{\ell}-\cos\left(\frac{u_{0}}{\lambda}\right)(\bar{F}^{(1)})^{k\ell}\partial_{k}u_{0}\partial_{\theta}W_{\ell}$
$\displaystyle\quad+\mathrm{div}_{\bar{g}_{0}}W-(\bar{g}_{0})^{k\ell}(\tilde{\Gamma}^{(0)})^{a}_{k\ell}W_{a}+O\left(\lambda\right).$

We conclude the proof of (4.31)-(4.32) with

$\displaystyle
L_{\gamma}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{ij}$
$\displaystyle=\mathbf{D}^{(\gamma)}_{(i}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{j)}-\frac{2}{3}\mathrm{div}_{\gamma}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)\gamma_{ij},$

and $\bar{F}^{(1)}_{N_{0}i}=0$ and
$(\bar{g}_{0})^{ij}(\tilde{\Gamma}^{(0)})^{a}_{ij}=0$. The trace identities
(4.34)-(4.35) and the oscillating behaviour (4.36) follows directly from
(4.31)-(4.32). ∎

######  Remark 4.3.

In the previous lemma we used the notation
$\mathrm{tr}_{\bar{F}^{(1)}}\mathbf{K}^{[-1]}(W)$ to denote

$(\bar{F}^{(1)})^{ij}\mathbf{K}^{[-1]}_{ij}(W)=\bar{g}_{0}^{ik}\bar{g}_{0}^{j\ell}\bar{F}^{(1)}_{k\ell}\mathbf{K}^{[-1]}_{ij}(W),$

even though $\bar{F}^{(1)}$ is not a Lorentzian metric.

####  4.2.4 The conformal Laplacian

The conformal Laplacian associated to a Riemannian metric $h$ is the operator
$\mathrm{div}_{h}L_{h}$ acting on 1-tensor. It is the only operator considered
in this article depending on $\partial^{2}\gamma$, i.e second derivatives of
the metric $\gamma$. Indeed $L_{\gamma}$ contains the Christoffel symbols of
$\gamma$ through the covariant derivative, and they are differenciated once by
the divergence. Since $\partial^{2}\gamma=O\left(\lambda^{-1}\right)$, this a
major difference with the Laplace-Beltrami operator or the divergence operator
from the point of view of expanding quantities in powers of $\lambda$. Indeed,
this implies that even if the 1-form $\beta$ is not oscillating, the quantity
$\mathrm{div}_{\gamma}L_{\gamma}\beta$ still loses one power of $\lambda$. If
$\beta$ is oscillating, then $\mathrm{div}_{\gamma}L_{\gamma}\beta$ loses two
powers of $\lambda$ since the conformal Laplacian is a second order operator.
This explains the two following lemmas.

######  Lemma 4.6.

Let $\beta$ be a 1-form. We have

$\displaystyle\mathrm{div}_{\gamma}L_{\gamma}\beta$
$\displaystyle=\frac{1}{\lambda}M^{[-1]}(\beta)+M^{[\geq 0]}(\beta),$

with

$\displaystyle M^{[-1]}_{\ell}(\beta)$ $\displaystyle=|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\bar{F}^{(1)}_{i\ell}\beta_{j},$
(4.37)

and

$\displaystyle\left|M^{[\geq 0]}(\beta)-\mathrm{div}_{e}L_{e}\beta\right|$
$\displaystyle\lesssim\left|\gamma^{-1}-e^{-1}\right||\partial^{2}\beta|+\left|\gamma^{-2}\partial\gamma\right|\left|1+\gamma^{-1}\gamma\right||\partial\beta|$
(4.38)
$\displaystyle\quad+\left|\gamma^{-3}(\partial\gamma)^{2}+\gamma^{-2}(\partial^{2}\gamma)^{(\geq
0)}\right|\left|1+\gamma^{-1}\gamma\right||\beta|.$

###### Proof.

Let $\bar{g}$ be any Riemannian metric on $\mathbb{R}^{3}$, we have

$\displaystyle\mathrm{div}_{\bar{g}}L_{\bar{g}}\beta_{\ell}$
$\displaystyle=\bar{g}^{ij}\left(\partial_{i}\partial_{(j}\beta_{\ell)}-\frac{2}{3}\partial_{\ell}\partial_{i}\beta_{j}\right)+\beta_{k}\left(-2\bar{g}^{ij}\partial_{i}\Gamma(\bar{g})^{k}_{j\ell}+\frac{2}{3}\partial_{\ell}\left(\bar{g}^{ij}\Gamma(\bar{g})^{k}_{ij}\right)\right)$
(4.39)
$\displaystyle\quad-\bar{g}^{ij}\left(2\Gamma(\bar{g})^{k}_{j\ell}\partial_{i}\beta_{k}+\frac{2}{3}\mathrm{div}_{\bar{g}}\beta\partial_{i}\bar{g}_{j\ell}\right)+\frac{2}{3}\left(-\partial_{i}\beta_{j}\partial_{\ell}\bar{g}^{ij}+\bar{g}^{ij}\Gamma(\bar{g})^{k}_{ij}\partial_{\ell}\beta_{k}\right)$
$\displaystyle\quad-\bar{g}^{ij}\Gamma(\bar{g})^{k}_{i(j}L_{\bar{g}}\beta_{k\ell)}.$

In the case of the high-frequency metric $\gamma$, the only terms loosing
$\frac{1}{\lambda}$, i.e contributing to $M^{[-1]}(\beta)$, are the
$\gamma^{-2}\partial^{2}\gamma\beta$. More precisely, it concerns terms
involving $\partial\tilde{\Gamma}^{(0)}$, and since
$\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{k}_{ij}=0$, the only contribution is

$\displaystyle M^{[-1]}_{\ell}(\beta)$
$\displaystyle=-2\beta_{k}\bar{g}^{ij}\partial_{i}u_{0}\partial_{\theta}(\tilde{\Gamma}^{(0)})^{k}_{j\ell}=|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\bar{F}^{(1)}_{i\ell}\beta_{j},$

where we used (4.15) and $\bar{F}^{(1)}_{N_{0}i}=0$. From (4.39) we also get

$\displaystyle\mathrm{div}_{e}L_{e}\beta_{\ell}$
$\displaystyle=e^{ij}\left(\partial_{i}\partial_{(j}\beta_{\ell)}-\frac{2}{3}\partial_{\ell}\partial_{i}\beta_{j}\right),$

which concludes the proof of (4.38). ∎

The estimation (4.38) will allow us to invert the operator $M^{[\geq 0]}$,
since we know how to invert $\mathrm{div}_{e}L_{e}$ (see Proposition 1.2) and
since we gain a smallness constant $\varepsilon$ in front of
$\partial^{2}\beta$ thanks to (2.1) and (4.13). We now state the final lemma
of this section, which will allow us to construct high-frequency solutions of
the momentum constraint. The proof is left to the reader since it follows from
Lemmas 4.4 and 4.5.

######  Lemma 4.7.

Let $W\left(\frac{u_{0}}{\lambda}\right)$ be an oscillating 1-form. We have

$\displaystyle\mathrm{div}_{\gamma}L_{\gamma}\left(W\left(\frac{u_{0}}{\lambda}\right)\right)_{\ell}$
$\displaystyle=\frac{1}{\lambda^{2}}\mathbf{M}^{[-2]}_{\ell}(W)+\frac{1}{\lambda}\mathbf{M}^{[-1]}_{\ell}(W)+\mathbf{M}^{[\geq
0]}_{\ell}(W),$

with

$\displaystyle\mathbf{M}^{[-2]}_{\ell}(W)$
$\displaystyle=\mathbf{d}^{[-1]}_{\ell}\left(\mathbf{K}^{[-1]}(W)\right),$
(4.40) $\displaystyle\mathbf{M}^{[-1]}_{\ell}(W)$
$\displaystyle=\mathbf{d}^{[-1]}_{\ell}\left(\mathbf{K}^{[0]}(W)\right)+\mathbf{d}^{[0]}_{\ell}\left(\mathbf{K}^{[-1]}(W)\right).$
(4.41)

The following oscillating behaviour holds

$\displaystyle\mathbf{M}^{[-1]}(W)$
$\displaystyle\sim\cos(\theta)W+\cos(\theta)\partial^{2}_{\theta}W+(1+\sin(\theta))\partial_{\theta}W.$
(4.42)

From (4.24), (4.31) and (4.40) we obtain

$\displaystyle\mathbf{M}^{[-2]}_{\ell}(W)$ $\displaystyle=|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\left(\partial^{2}_{\theta}W_{\ell}+\frac{1}{3}(N_{0})_{\ell}\partial^{2}_{\theta}W_{N_{0}}\right).$
(4.43)

In order to solve the momentum constraint, we need to invert this operator.
This is done in the following simple lemma.

######  Lemma 4.8.

If $\alpha$ and $\beta$ are 1-forms such that

$\displaystyle\alpha_{\ell}+\frac{1}{3}(N_{0})_{\ell}\alpha_{N_{0}}=\beta_{\ell},$
(4.44)

then

$\displaystyle\alpha_{\ell}=\beta_{\ell}-\frac{1}{4}(N_{0})_{\ell}\beta_{N_{0}}.$

###### Proof.

We contract (4.44) with the vector field $N_{0}$ to obtain

$\displaystyle\frac{4}{3}\alpha_{N_{0}}=\beta_{N_{0}}.$

Inserting this into (4.44) concludes the proof. ∎

##  5 Approximate solution to the constraint equations

In this section, we construct an approximate solution to the constraint
equations (1.15) and (1.16) by solving the $\lambda^{0}$ and $\lambda^{1}$
Hamiltonian levels and the $\lambda^{0}$ momentum level. In the process we
will define the parameter $\tau$ and in Sections 5.3 and 5.4 we construct the
parameter $\sigma$.

###  5.1 The approximate Hamiltonian constraint

We solve the first two levels of the Hamiltonian constraint by choosing
$\varphi^{(2)}$ and $\varphi^{(3)}$. At those levels, (1.15) decouples from
(1.16) since we replace $(\sigma+L_{\gamma}W)^{(\leq 1)}$ by
$\left(K-\frac{1}{3}\tau\gamma\right)^{(\leq 1)}$, where $K^{(\leq 1)}$ and
$\tau^{(\leq 1)}$ will be defined along the way.

####  5.1.1 The $\lambda^{0}$ Hamiltonian level

We want $\varphi^{(2)}$ to solve the $\lambda^{0}$ Hamiltonian level. Using
the expansion of the Laplace-Beltrami operator $\Delta_{\gamma}$ (see Lemma
4.3), this is equivalent to the following equation:

$\displaystyle 8|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\partial^{2}_{\theta}\varphi^{(2)}=R^{(0)}+\frac{2}{3}\left(\tau^{(0)}\right)^{2}-\left|K^{(0)}-\frac{1}{3}\tau^{(0)}\bar{g}_{0}\right|^{2}_{\bar{g}_{0}}.$
(5.1)

where $K^{(0)}$ is defined by

$\displaystyle K^{(0)}$
$\displaystyle=K_{0}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)},$ (5.2)

following the discussion of Section 3.3. As the parameter $\tau$ is the trace
with respect to $\gamma$ of $K$, we have

$\displaystyle\tau^{(0)}$ $\displaystyle=\mathrm{tr}_{\bar{g}_{0}}K^{(0)}$
$\displaystyle=\mathrm{tr}_{\bar{g}_{0}}K_{0}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}$ $\displaystyle=0,$
(5.3)

where we used (1.7) and Lemma 4.1. Therefore, (5.1) rewrites

$\displaystyle 8|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\partial^{2}_{\theta}\varphi^{(2)}=R^{(0)}-\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}.$
(5.4)

The LHS of (5.4) being a derivative with respect to $\theta$, we need the RHS
to be purely oscillating. Since this shows how the creation of non-oscillating
terms is absorbed by the background constraint equations, we state this in a
separate lemma.

######  Lemma 5.1.

We have

$\displaystyle R^{(0)}-\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}-K_{0}^{\ell j}\right)-6\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}.$

###### Proof.

Let us first expand $\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}$ using (5.2) and
(2.9):

$\displaystyle\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle=|K_{0}|^{2}_{\bar{g}_{0}}+|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}-\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}+\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\left|\bar{F}^{(1)}\cdot K_{0}\right|_{\bar{g}_{0}}.$

We use now (4.10) to compute the full RHS of (5.4):

$\displaystyle R^{(0)}-\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle=R(\bar{g}_{0})-|K_{0}|^{2}_{\bar{g}_{0}}-2|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}$
$\displaystyle\quad+\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\left(\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}\right)-\left|\bar{F}^{(1)}\cdot K_{0}\right|_{\bar{g}_{0}}\right)$
$\displaystyle\quad-6\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2}.$

Using the background Hamiltonian constraint (1.5) to cancel the non-
oscillating term, we are left with

$\displaystyle R^{(0)}-\left|K^{(0)}\right|^{2}_{\bar{g}_{0}}$
$\displaystyle=\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\left(\bar{F}^{(1)}_{\ell
j}\left(-\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}+\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}\right)-\left|\bar{F}^{(1)}\cdot K_{0}\right|_{\bar{g}_{0}}\right)$
$\displaystyle\quad-6\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}^{2}F_{0}^{2},$

which concludes the proof. ∎

This lemma allows us to solve (5.4) by simply setting

$\displaystyle\varphi^{(2)}$ $\displaystyle=\frac{1}{8|\nabla
u_{0}|_{\bar{g}_{0}}}\sin\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}_{\ell
j}\left(\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}-\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}+K_{0}^{\ell
j}\right)+\frac{3}{16}\cos\left(\frac{2u_{0}}{\lambda}\right)F_{0}^{2}.$ (5.5)

Now that $\varphi^{(2)}$ is defined, we can set

$\displaystyle\bar{F}^{(2,1)}$ $\displaystyle=\frac{\bar{F}^{(1)}_{\ell
j}}{2|\nabla
u_{0}|_{\bar{g}_{0}}}\left(\bar{g}_{0}^{ij}\partial_{i}N_{0}^{\ell}-\frac{1}{2}N_{0}\bar{g}_{0}^{\ell
j}+K_{0}^{\ell j}\right)\bar{g}_{0}+\omega^{(2)},$ (5.6)
$\displaystyle\bar{F}^{(2,2)}$
$\displaystyle=\frac{3}{4}F_{0}^{2}\bar{g}_{0}.$ (5.7)

These definitions are consistent with (1.13) and the following lemma
summarizes the properties of $\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$:

######  Lemma 5.2.

The symmetric 2-tensors $\bar{F}^{(2,1)}$ and $\bar{F}^{(2,2)}$ are supported
in $B_{R}$ and the following holds

$\displaystyle\left\|\bar{F}^{(2,1)}\right\|_{H^{N-1}}+\left\|\bar{F}^{(2,2)}\right\|_{H^{N-1}}$
$\displaystyle\lesssim\varepsilon,$ (5.8)

with a constant depending only on $\delta$ and $R$.

###### Proof.

The support property follows from the support properties of $F_{0}$,
$\bar{F}^{(1)}$ and $\omega^{(2)}$. The estimate (5.8) follows from the
definitions (5.6) and (5.7) and the estimates (2.1), (2.3) and (4.9) (recall
also (4.7) for the definition of $\omega^{(2)}$). ∎

####  5.1.2 The $\lambda^{1}$ Hamiltonian level

We now turn to the $\lambda^{1}$ Hamiltonian level, which is solved thanks to
$\varphi^{(3)}$. More precisely, since $\tau=O\left(\lambda\right)$ we need
$\varphi^{(3)}$ to satisfy

$\displaystyle 8|\nabla
u_{0}|_{\bar{g}_{0}}^{2}\partial_{\theta}^{2}\varphi^{(3)}$
$\displaystyle=-8\mathbf{H}^{[-1]}\left(\varphi^{(2)}\right)+R^{(1)}$ (5.9)
$\displaystyle\quad-\left|K^{(0)}\cdot\left(K^{(1)}-\frac{1}{3}\tau^{(1)}\bar{g}_{0}\right)\right|_{\bar{g}_{0}}+\cos\left(\frac{u_{0}}{\lambda}\right)\left|K^{(0)}\right|^{2}_{\bar{F}^{(1)}},$

where $R^{(1)}$ is defined in Lemma 4.2 and
$\mathbf{H}^{[-1]}\left(\varphi^{(2)}\right)$ in Lemma 4.3. In the RHS of
(5.9), it remains to define $K^{(1)}$ and $\tau^{(1)}$. Following the
discussion of Section 3.3, we define

$\displaystyle K^{(1)}_{ij}$
$\displaystyle=-\frac{1}{2}\cos\left(\frac{u_{0}}{\lambda}\right)\bigg{(}-N_{0}\bar{F}^{(1)}_{ij}+\left(-\bar{g}_{0}^{k\ell}(K_{0})_{(i\ell}+N_{0}^{\ell}\Gamma(g_{0})^{k}_{\ell(i}\right)\bar{F}^{(1)}_{j)k}$
(5.10) $\displaystyle\hskip 85.35826pt+\frac{1}{|\nabla
u_{0}|_{\bar{g}_{0}}}\left(-(K_{0})^{k\ell}\partial_{k}u_{0}\partial_{\ell}u_{0}-\bar{g}_{0}^{k\ell}\partial_{k}u_{0}\partial_{\ell}|\nabla
u_{0}|_{\bar{g}_{0}}+\frac{1}{2}\bar{g}_{0}^{k\ell}\partial_{k}\partial_{\ell}u_{0}\right)\bar{F}^{(1)}_{ij}\bigg{)}$
$\displaystyle\quad-\frac{1}{2}\left|\nabla
u_{0}\right|_{\bar{g}_{0}}\left(\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(2,1)}_{ij}-2\sin\left(\frac{2u_{0}}{\lambda}\right)\bar{F}^{(2,2)}_{ij}\right).$

Since $\tau$ is the trace with respect to $\gamma$ of $K$ we set

$\displaystyle\tau^{(1)}$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\left|\bar{F}^{(1)}\cdot
K_{0}\right|_{\bar{g}_{0}}+\mathrm{tr}_{\bar{g}_{0}}K^{(1)},$ (5.11)

where we used the expansion of the inverse of $\gamma$ given by (4.13). This
actually fully defines the parameter $\tau$, i.e

$\displaystyle\tau=\lambda\tau^{(1)}.$ (5.12)

As above, we need to show that the RHS of (5.9) is purely oscillating in order
to find $\varphi^{(3)}$ solution of this equation. From (5.2), (5.10) and
(5.11) we obtain

$\displaystyle
K^{(0)}\;\overset{\mathrm{osc}}{\sim}\;1+\sin(\theta)\quad\text{and}\quad
K^{(1)},\;\tau^{(1)}\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta),$

which together with (4.12), (4.22) and (5.5) gives

$\displaystyle\text{RHS of \eqref{eq ffi
3}}\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta)+\cos(3\theta).$

This shows that we can solve (5.9) by setting

$\displaystyle\varphi^{(3)}$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\varphi^{(3,1)}+\sin\left(\frac{2u_{0}}{\lambda}\right)\varphi^{(3,2)}+\cos\left(\frac{3u_{0}}{\lambda}\right)\varphi^{(3,3)},$
(5.13)

for some $\varphi^{(3,i)}$ supported in $B_{R}$ and satisfying

$\displaystyle\sum_{i=1,2,3}\left\|\varphi^{(3,i)}\right\|_{H^{N-3}}$
$\displaystyle\lesssim\varepsilon.$ (5.14)

The estimate (5.14) follows from (4.11) and (4.7). Since the ansatz
constructed in this article is an order 2 ansatz, the term $\varphi^{(3)}$
will be ultimately hidden in the remainder $\bar{\mathfrak{h}}_{\lambda}$ (see
(2.5) in Theorem 2.1) and thus we don’t need a precise expression of
$\varphi^{(3,i)}$.

###  5.2 The approximate momentum constraint

In this section we solve the first level of the momentum constraint. Since
$\tau=\lambda\tau^{(1)}$ we have
$\mathrm{d}\tau=\frac{1}{\lambda}\mathrm{d}u_{0}\partial_{\theta}\tau^{(1)}+\lambda\mathrm{d}\tau^{(1)}$
where the derivatives in $\mathrm{d}\tau^{(1)}$ don’t hit the oscillating
parts of $\tau^{(1)}$. Therefore the first momentum level corresponds to the
$\lambda^{0}$ level of (1.16), solved by $W^{(2)}$.

######  Remark 5.1.

This is where the assumption $\mathrm{tr}_{\bar{g}_{0}}K_{0}=0$ simplifies our
construction. If $\mathrm{tr}_{\bar{g}_{0}}K_{0}\neq 0$ then $\tau$ needs to
include a non-oscillating $\lambda^{0}$ term in $\tau^{(0)}$. This would
require a non-oscillating term in $W$ at the $\lambda^{0}$ order to absorb it

$\displaystyle W=w^{(0)}+O\left(\lambda\right),$

where we use a lowercase letter to emphasize that $w^{(0)}$ is non-
oscillating. However, as explained in Section 4.2.4, this term would produce a
$\lambda^{-1}$ term in $\mathrm{div}_{\gamma}L_{\gamma}W$ (precisely
$M^{[-1]}(w^{(0)})$, see Lemma 4.6) and thus would require an oscillating term
$W^{(1)}$ in $W$ to absorb it. We make the assumption
$\mathrm{tr}_{\bar{g}_{0}}K_{0}=0$ precisely to avoid these technicalities.

More precisely, $W^{(2)}$ needs to solve

$\displaystyle\mathbf{M}^{[-2]}_{\ell}(W^{(2)})$
$\displaystyle=\frac{2}{3}\partial_{\ell}u_{0}\partial_{\theta}\tau^{(1)}.$
(5.15)

where we used the expansion obtained in Lemma 4.7. Thanks to Lemma 4.8 this
equation rewrites as

$\displaystyle\partial_{\theta}^{2}W^{(2)}_{\ell}$
$\displaystyle=-\frac{1}{2|\nabla
u_{0}|_{\bar{g}_{0}}}(N_{0})_{\ell}\partial_{\theta}\tau^{(1)}.$ (5.16)

Since the RHS of this equation is a $\partial_{\theta}$ derivative it is
purely oscillating and we can integrate twice to obtain $W^{(2)}$. Using
(5.11) this also gives

$\displaystyle W^{(2)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\sin(\theta)+\cos(2\theta).$
(5.17)

Note that as opposed to the Hamiltonian constraint, we don’t solve the
$\lambda^{1}$ momentum level here. Indeed, as Lemma 4.6 shows, the operator
$\mathrm{div}_{\gamma}L_{\gamma}$ loses one $\lambda$ power even when applied
to a non-oscillating field like $\widetilde{W}$ (the remainder in (3.6)).
Therefore the $\lambda^{1}$ momentum level also involves $\widetilde{W}$ and
the equation for $W^{(3)}$ is coupled with the remainder in the ansatz (3.6),
as opposed to $\varphi^{(3)}$ which is not coupled to $\widetilde{\varphi}$
(the remainder in (3.5)), since
$\Delta_{\gamma}\widetilde{\varphi}=O\left(\lambda^{0}\right)$.

###  5.3 An almost TT-tensor

In this section we define the first terms in the expansion of the parameter
$\sigma$ of the conformal method, i.e $\sigma^{(0)}$ and $\sigma^{(1)}$. As
explained in Section 3.3, the definition of the first orders of $\sigma$
follows simply from our constraint

$\displaystyle\left(\sigma+L_{\gamma}W\right)^{(\leq
1)}=\left(K-\frac{1}{3}\tau\gamma\right)^{(\leq 1)}.$ (5.18)

Since $W$ is given by
$W=\lambda^{2}\left(W^{(2)}+\widetilde{W}\right)+O\left(\lambda^{3}\right)$
(with $W^{(2)}$ defined by (5.16) and $\widetilde{W}$ non-oscillating), we use
Lemma 4.5 to compute $(L_{\gamma}W)^{(\leq 1)}$ and (5.18) forces us to define

$\displaystyle\sigma^{(0)}$ $\displaystyle=K^{(0)},$ (5.19)
$\displaystyle\sigma^{(1)}$
$\displaystyle=K^{(1)}-\frac{1}{3}\tau^{(1)}\bar{g}_{0}-\mathbf{K}^{[-1]}(W^{(2)}),$
(5.20)

where $K^{(0)}$, $K^{(1)}$ and $\tau^{(1)}$ are given by (5.2), (5.10) and
(5.11) respectively and $\mathbf{K}^{[-1]}(W^{(2)})$ is defined in Lemma 4.5.

In this section, we prove that $\sigma^{(0)}+\lambda\sigma^{(1)}$ is almost a
TT-tensor, that is

$\displaystyle\mathrm{tr}_{\gamma}\left(\sigma^{(0)}+\lambda\sigma^{(1)}\right)=O\left(\lambda^{2}\right)\quad\text{and}\quad\mathrm{div}_{\gamma}\left(\sigma^{(0)}+\lambda\sigma^{(1)}\right)=O\left(\lambda\right).$
(5.21)

Using the expansion of $\gamma^{ij}$ given by (4.13) we have

$\displaystyle\mathrm{tr}_{\gamma}\left(\sigma^{(0)}+\lambda\sigma^{(1)}\right)$
$\displaystyle=\mathrm{tr}_{\bar{g}_{0}}\sigma^{(0)}+\lambda\left(-\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}\sigma^{(0)}+\mathrm{tr}_{\bar{g}_{0}}\sigma^{(1)}\right)+O\left(\lambda^{2}\right).$

Using now the expansion of the divergence operator $\mathrm{div}_{\gamma}$
obtained in Lemma 4.4 we have

$\displaystyle\mathrm{div}_{\gamma}\left(\sigma^{(0)}+\lambda\sigma^{(1)}\right)$
$\displaystyle=\frac{1}{\lambda}\mathbf{d}^{[-1]}(\sigma^{(0)})+\mathbf{d}^{[0]}(\sigma^{(0)})+\mathbf{d}^{[-1]}(\sigma^{(1)})+O\left(\lambda\right).$

The following lemma, which shows that $\sigma^{(0)}+\lambda\sigma^{(1)}$ is
almost a TT-tensor, is crucial since it validates a posteriori our whole
approximate construction.

######  Lemma 5.3.

We have

$\displaystyle\mathrm{tr}_{\bar{g}_{0}}\sigma^{(0)}$ $\displaystyle=0,$ (5.22)
$\displaystyle-\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}\sigma^{(0)}+\mathrm{tr}_{\bar{g}_{0}}\sigma^{(1)}$
$\displaystyle=0,$ (5.23)

and

$\displaystyle\mathbf{d}^{[-1]}(\sigma^{(0)})$ $\displaystyle=0,$ (5.24)
$\displaystyle\mathbf{d}^{[0]}(\sigma^{(0)})+\mathbf{d}^{[-1]}(\sigma^{(1)})$
$\displaystyle=0.$ (5.25)

###### Proof.

We start with the trace identities. Since $\sigma^{(0)}=K^{(0)}$, (5.22)
follows from (1.7) and $\mathrm{tr}_{\bar{g}_{0}}\bar{F}^{(1)}=0$. Moreover,
from (5.19) and (5.20) we have

$\displaystyle-\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}\sigma^{(0)}+\mathrm{tr}_{\bar{g}_{0}}\sigma^{(1)}$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}K^{(0)}+\mathrm{tr}_{\bar{g}_{0}}K^{(1)}-\tau^{(1)}$
$\displaystyle\quad-\mathrm{tr}_{\bar{g}_{0}}\mathbf{K}^{[-1]}(W^{(2)})$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\mathrm{tr}_{\bar{F}^{(1)}}K^{(0)}+\mathrm{tr}_{\bar{g}_{0}}K^{(1)}-\tau^{(1)}$
$\displaystyle=0,$

where we use (4.34) and (5.11). This proves (5.23).

We now look at the divergence identities. From (5.19), (5.2) and (4.24) we
obtain

$\displaystyle\mathbf{d}^{[-1]}_{\ell}(\sigma^{(0)})$
$\displaystyle=\frac{1}{2}|\nabla
u_{0}|^{2}_{\bar{g}_{0}}\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}_{N_{0}\ell}=0,$

which proves (5.24). We now compute the two parts of (5.25). Using (4.25) and
(5.19) we obtain

$\displaystyle\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})$
$\displaystyle=\mathrm{div}_{\bar{g}_{0}}K^{(0)}_{\ell}-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{a}_{i\ell}K^{(0)}_{aj}$
$\displaystyle=(\mathrm{div}_{\bar{g}_{0}}K_{0})_{\ell}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)\left(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}\right)_{\ell}$
$\displaystyle\quad-\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{a}_{i\ell}(K_{0})_{aj}-\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\bar{g}_{0}^{ij}(\tilde{\Gamma}^{(0)})^{a}_{i\ell}\bar{F}^{(1)}_{aj}.$

We use (4.15) and (2.9) to rewrite the second line in the previous expression
and obtain

$\displaystyle\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})$
$\displaystyle=(\mathrm{div}_{\bar{g}_{0}}K_{0})_{\ell}+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)\left(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}\right)_{\ell}$
$\displaystyle\quad+\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)\partial_{\ell}u_{0}\left|\bar{F}^{(1)}\cdot
K_{0}\right|_{\bar{g}_{0}}+2\sin^{2}\left(\frac{u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\ell}u_{0}F_{0}^{2}.$

Using the background momentum constraint (1.6), we see that the non-
oscillating terms cancel each other and we are left with

$\displaystyle\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})$
$\displaystyle=\frac{1}{2}\sin\left(\frac{u_{0}}{\lambda}\right)\left(\left(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}\right)_{\ell}+\partial_{\ell}u_{0}\left|\bar{F}^{(1)}\cdot
K_{0}\right|_{\bar{g}_{0}}\right)$ (5.26)
$\displaystyle\quad-\cos\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\ell}u_{0}F_{0}^{2}.$

We now look at $\mathbf{d}^{[-1]}_{\ell}(\sigma^{(1)})$. Using (4.24) and
(5.20) we obtain

$\displaystyle\mathbf{d}^{[-1]}_{\ell}(\sigma^{(1)})$ $\displaystyle=-|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\theta}K^{(1)}_{N_{0}\ell}-\frac{1}{3}\partial_{\ell}u_{0}\partial_{\theta}\tau^{(1)}-\mathbf{M}_{\ell}^{[-2]}(W^{(2)})$
$\displaystyle=\partial_{\theta}\left(-|\nabla
u_{0}|_{\bar{g}_{0}}K^{(1)}_{N_{0}\ell}-\partial_{\ell}u_{0}\tau^{(1)}\right),$

where we use the equation satisfied by $W^{(2)}$ (see (5.15)). We use (5.10),
(5.6) and (5.7) to obtain

$\displaystyle K^{(1)}_{N_{0}\ell}$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\left(\frac{\bar{F}^{(1)}_{k\ell}}{2}\left(\mathbf{D}_{N_{0}}N_{0}^{k}-\bar{g}_{0}^{kj}(K_{0})_{N_{0}j}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+\frac{\bar{F}^{(1)}_{jk}}{4}\left(\bar{g}_{0}^{ik}\partial_{i}N_{0}^{j}-\frac{1}{2}N_{0}\bar{g}_{0}^{jk}+K_{0}^{jk}\right)(N_{0})_{\ell}+\frac{1}{2}\left|\nabla
u_{0}\right|_{\bar{g}_{0}}\omega^{(2)}_{N_{0}\ell}\right)$
$\displaystyle\quad+\frac{3}{4}\sin\left(\frac{2u_{0}}{\lambda}\right)\left|\nabla
u_{0}\right|_{\bar{g}_{0}}F_{0}^{2}(N_{0})_{\ell}.$

Using (5.11) and $\mathrm{tr}_{\bar{g}_{0}}\omega^{(2)}=0$ we also obtain

$\displaystyle\tau^{(1)}$
$\displaystyle=-\frac{3}{4}\cos\left(\frac{u_{0}}{\lambda}\right)\bar{F}^{(1)}_{kj}\left(\bar{g}_{0}^{ij}\partial_{i}N_{0}^{k}-\frac{1}{2}N_{0}\bar{g}_{0}^{jk}+K_{0}^{jk}\right)+\frac{1}{4}\sin\left(\frac{2u_{0}}{\lambda}\right)|\nabla
u_{0}|_{\bar{g}_{0}}F_{0}^{2}.$ (5.27)

This gives

$\displaystyle-|\nabla u_{0}|_{\bar{g}_{0}}$ $\displaystyle
K^{(1)}_{N_{0}\ell}-\partial_{\ell}u_{0}\tau^{(1)}$ $\displaystyle=|\nabla
u_{0}|_{\bar{g}_{0}}\cos\left(\frac{u_{0}}{\lambda}\right)\left(\frac{\bar{F}^{(1)}_{k\ell}}{2}\left(\mathbf{D}_{N_{0}}N_{0}^{k}-\bar{g}_{0}^{kj}(K_{0})_{N_{0}j}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad-\frac{\bar{F}^{(1)}_{jk}}{2}\left(\bar{g}_{0}^{ik}\partial_{i}N_{0}^{j}-\frac{1}{2}N_{0}\bar{g}_{0}^{jk}+K_{0}^{jk}\right)(N_{0})_{\ell}+\frac{1}{2}\left|\nabla
u_{0}\right|_{\bar{g}_{0}}\omega^{(2)}_{N_{0}\ell}\right)$
$\displaystyle\quad+\frac{1}{2}\sin\left(\frac{2u_{0}}{\lambda}\right)\left|\nabla
u_{0}\right|_{\bar{g}_{0}}F_{0}^{2}\partial_{\ell}u_{0}.$

Adding this to the expression of $\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})$ given
by (5.26) we notice that the terms oscillating like $2\theta$ cancel (see
Remark 5.2 below) and we obtain

$\displaystyle\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})+\mathbf{d}^{[-1]}_{\ell}(\sigma^{(1)})$
$\displaystyle\quad=\sin\left(\frac{u_{0}}{\lambda}\right)\left[\frac{1}{2}\left(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}\right)_{\ell}-\frac{|\nabla
u_{0}|_{\bar{g}_{0}}}{2}\bar{F}^{(1)}_{k\ell}\left(\mathbf{D}_{N_{0}}N_{0}^{k}-\bar{g}_{0}^{kj}(K_{0})_{N_{0}j}\right)\right.$
$\displaystyle\quad\quad\left.\qquad\qquad\qquad\qquad-\frac{1}{2}\left|\nabla
u_{0}\right|^{2}_{\bar{g}_{0}}\omega^{(2)}_{N_{0}\ell}-\frac{1}{2}\partial_{\ell}u_{0}\bar{F}^{(1)}_{kj}\bar{g}_{0}^{ij}\partial_{i}N_{0}^{k}+\frac{1}{4}\partial_{\ell}u_{0}\bar{F}^{(1)}_{kj}N_{0}\bar{g}_{0}^{kj}\right].$

Using $\bar{F}^{(1)}_{N_{0}i}=0$ we can compute the divergence of $|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}$ and using in addition that
$\omega^{(2)}_{N_{0}N_{0}}=0$ we obtain

$\displaystyle\mathbf{d}^{[0]}_{N_{0}}(\sigma^{(0)})+\mathbf{d}^{[-1]}_{N_{0}}(\sigma^{(1)})$
$\displaystyle=0.$

The tangential components of
$\mathbf{d}^{[0]}(\sigma^{(0)})+\mathbf{d}^{[-1]}(\sigma^{(1)})$ are given by

$\displaystyle\mathbf{d}^{[0]}_{\mathbf{j}}(\sigma^{(0)})+\mathbf{d}^{[-1]}_{\mathbf{j}}(\sigma^{(1)})$
$\displaystyle\quad=\sin\left(\frac{u_{0}}{\lambda}\right)\left[\frac{1}{2}\left(\mathrm{div}_{\bar{g}_{0}}|\nabla
u_{0}|_{\bar{g}_{0}}\bar{F}^{(1)}\right)_{\mathbf{j}}-\frac{|\nabla
u_{0}|_{\bar{g}_{0}}}{2}\bar{F}^{(1)}_{k\mathbf{j}}\left(\mathbf{D}_{N_{0}}N_{0}^{k}-\bar{g}_{0}^{kj}(K_{0})_{N_{0}j}\right)-\frac{1}{2}\left|\nabla
u_{0}\right|^{2}_{\bar{g}_{0}}\omega^{(2)}_{N_{0}\mathbf{j}}\right].$

This previous expression vanishes thanks to the choice of $\omega^{(2)}$ made
in (4.7). This concludes the proof of (5.25).

∎

######  Remark 5.2.

The cancellation of the $\cos\left(\frac{2u_{0}}{\lambda}\right)$ terms coming
from $\mathbf{d}^{[0]}_{\ell}(\sigma^{(0)})$ and
$\mathbf{d}^{[-1]}_{\ell}(\sigma^{(1)})$ in their sum seems to be linked to
the weak polarized null condition satisfied by the semi-linear terms in the
Einstein equations, which involved products of derivatives of the metric (see
Section 3.1.3 in [Tou22] for the definition of this condition). Indeed, these
terms correspond to terms of the form $\Gamma K$, $\Gamma\Gamma$ or $KK$ with
$\Gamma$ the Christoffel symbols and $K$ the second fundamental form, i.e
terms of the form $\partial g\partial g$.

###  5.4 An exact TT-tensor

In the next section, we are going to solve completely the constraint
equations, i.e solve for the remainders in the high-frequency ansatz
(3.5)-(3.6). We will thus need the full expression of the parameters of the
conformal method. While $\gamma$ and $\tau$ are already fully defined,
$\sigma$ is only partially known yet and is only an almost TT-tensor, as it
was shown in the previous section. In this section we finish the construction
of $\sigma$. We choose the following ansatz

$\displaystyle\sigma$
$\displaystyle=\sigma^{(0)}+\lambda\sigma^{(1)}+\lambda^{2}\left(\sigma^{(2)}+L_{\gamma}Y\right)+\frac{\lambda^{3}}{3}\mathfrak{f}\gamma.$
(5.28)

In this expression, $\sigma^{(0)}$ and $\sigma^{(1)}$ are given by (5.19) and
(5.20) and $\sigma^{(2)}$, $Y$ and $\mathfrak{f}$ are yet to be defined such
that

$\mathrm{tr}_{\gamma}\sigma=0\quad\text{and}\quad\mathrm{div}_{\gamma}\sigma=0.$
(5.29)

Let us explain the ansatz (5.28). Since we need to satisfy the compatibility
with the spacetime ansatz (5.18), we can’t modifiy the order $\lambda^{0}$ and
$\lambda^{1}$ of $\sigma$. Thus, a non-oscillating remainder can only appear
at the order $\lambda^{2}$. However, such a remainder would not be able to
solve the $\lambda^{1}$ level of $\mathrm{div}_{\gamma}\sigma=0$ (recall
(5.21)). Therefore we need to add an oscillating field at the order
$\lambda^{2}$, i.e $\sigma^{(2)}$. This field will also be able to solve the
$\lambda^{2}$ level of $\mathrm{tr}_{\gamma}\sigma=0$. Finally the remainder
is chosen of the form $L_{\gamma}Y+\frac{\lambda}{3}\mathfrak{f}\gamma$, where
the vector field $Y$ ensures $\mathrm{div}_{\gamma}\sigma=0$ and the scalar
function $\mathfrak{f}$ ensures $\mathrm{tr}_{\gamma}\sigma=0$.

We now derive the equations for $\sigma^{(2)}$, $Y$ and $\mathfrak{f}$, which
illustrates the above discussion. Thanks to Lemma 5.3 the equations (5.29)
rewrite as

$\displaystyle\lambda^{2}\left(\mathrm{tr}_{\gamma^{(\geq
2)}}\sigma^{(0)}+\mathrm{tr}_{\gamma^{(\geq
1)}}\sigma^{(1)}+\mathrm{tr}_{\gamma}\sigma^{(2)}\right)+\lambda^{3}\mathfrak{f}$
$\displaystyle=0,$

and

$\displaystyle\lambda\left(\mathbf{d}^{[1]}(\sigma^{(0)})+\mathbf{d}^{[0]}(\sigma^{(1)})+\mathbf{d}^{[-1]}(\sigma^{(2)})+M^{[-1]}(Y)\right)\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle+\lambda^{2}\left(\mathbf{d}^{[\geq
2]}(\sigma^{(0)})+\mathbf{d}^{[\geq 1]}(\sigma^{(1)})+\mathbf{d}^{[\geq
0]}(\sigma^{(2)})+M^{[\geq
0]}(Y)+\frac{\lambda}{3}\mathrm{d}\mathfrak{f}\right)$ $\displaystyle=0,$

where $\mathrm{d}\mathfrak{f}$ also includes derivatives of the oscillating
parts of $\mathfrak{f}$, which implies in particular that
$\mathrm{d}\mathfrak{f}=O\left(\lambda^{-1}\right)$. In order to solve these
two equations, we want $\sigma^{(2)}$, $Y$ and $\mathfrak{f}$ to satisfy the
following coupled system (recall the expression of $\mathbf{d}^{[-1]}$ given
by (4.24)):

$\displaystyle\mathrm{tr}_{\bar{g}_{0}}\sigma^{(2)}$
$\displaystyle=-\mathrm{tr}_{\gamma^{(2)}}\sigma^{(0)}-\mathrm{tr}_{\gamma^{(1)}}\sigma^{(1)},$
(5.30) $\displaystyle\mathfrak{f}$ $\displaystyle=-\mathrm{tr}_{\gamma^{(\geq
1)}}\sigma^{(2)}-\mathrm{tr}_{\gamma^{(\geq
2)}}\sigma^{(1)}-\mathrm{tr}_{\gamma^{(\geq 3)}}\sigma^{(0)},$ (5.31)
$\displaystyle-|\nabla
u_{0}|_{\bar{g}_{0}}\partial_{\theta}\sigma^{(2)}_{N_{0}\ell}$
$\displaystyle=-M^{[-1]}_{\ell}(Y)-\mathbf{d}^{[1]}_{\ell}(\sigma^{(0)})-\mathbf{d}^{[0]}_{\ell}(\sigma^{(1)}),$
(5.32) $\displaystyle M^{[\geq 0]}(Y)$ $\displaystyle=-\mathbf{d}^{[\geq
2]}(\sigma^{(0)})-\mathbf{d}^{[\geq 1]}(\sigma^{(1)})-\mathbf{d}^{[\geq
0]}(\sigma^{(2)})-\frac{\lambda}{3}\mathrm{d}\mathfrak{f}.$ (5.33)

Equations (5.30) and (5.31) ensure $\mathrm{tr}_{\gamma}\sigma=0$ while (5.32)
and (5.33) ensure $\mathrm{div}_{\gamma}\sigma=0$. The rest of this section is
devoted to the resolution of the system (5.30)-(5.31)-(5.32)-(5.33). It
presents a triangular structure, despite the term $M^{[-1]}(Y)$ in (5.32).

####  5.4.1 Definition of $\sigma^{(2)}$ and $\mathfrak{f}$

We start by solving the non-differential equations of the previous system,
that is (5.30)-(5.31)-(5.32). The first step is to show that the RHS of (5.32)
is purely oscillating, which is a necessary condition since the LHS is a
$\partial_{\theta}$ derivative. Thanks to (4.37), the first term in the RHS is
oscillating, and the next lemma deals with the last two.

######  Lemma 5.4.

The following oscillating behaviour holds

$\displaystyle\mathbf{d}^{[1]}(\sigma^{(0)})+\mathbf{d}^{[0]}(\sigma^{(1)})\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta)+\cos(3\theta).$

###### Proof.

From (4.27) we have

$\mathbf{d}^{[0]}(\sigma^{(1)})\;\overset{\mathrm{osc}}{\sim}\;(1+\sin(\theta))\sigma^{(1)},$

and from (5.20) we have

$\displaystyle\sigma^{(1)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;K^{(1)}+\cos(\theta)+\tau^{(1)}+\mathbf{K}^{[-1]}(W^{(2)})$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta)+\partial_{\theta}W^{(2)},$

where we used (5.10) and (5.11). Now using (5.17) we conclude that

$\displaystyle\mathbf{d}^{[0]}(\sigma^{(1)})\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta).$

Now, from (4.28) we have

$\mathbf{d}^{[1]}(\sigma^{(0)})\;\overset{\mathrm{osc}}{\sim}\;\sin(\theta)\partial_{\theta}\sigma^{(0)}+(\cos(\theta)+\sin(2\theta))\sigma^{(0)},$

and from (5.19) and (5.2) we have

$\displaystyle\sigma^{(0)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;1+\sin(\theta).$

This concludes the proof of the lemma. ∎

We have shown that the RHS of (5.32) is purely oscillating, which allows us to
formally integrate this equation in $\theta$ and obtain $\sigma^{(2)}$. More
precisely, thanks to $\bar{F}^{(1)}_{iN_{0}}=0$, (4.37) implies that
$M_{N_{0}}^{[-1]}(Y)=0$. Therefore, (5.32) gives us
$\sigma^{(2)}_{N_{0}N_{0}}$ as a function of lower order terms in the
construction, i.e $\sigma^{(0)}$ and $\sigma^{(1)}$. Then, (5.30) gives us
$\sigma^{(2)}_{\mathbf{1}\mathbf{1}}$ and
$\sigma^{(2)}_{\mathbf{2}\mathbf{2}}$ as a function of
$\sigma^{(2)}_{N_{0}N_{0}}$, $\sigma^{(0)}$ and $\sigma^{(1)}$. All together,
the diagonal components of $\sigma^{(2)}$ in the frame
$(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ are functions of $\sigma^{(0)}$ and
$\sigma^{(1)}$ satisfying

$\displaystyle\left|\sigma^{(2)}_{N_{0}N_{0}}\right|+\left|\sigma^{(2)}_{\mathbf{1}\mathbf{1}}\right|+\left|\sigma^{(2)}_{\mathbf{2}\mathbf{2}}\right|\lesssim\left|\mathbf{d}^{[1]}(\sigma^{(0)})\right|+\left|\mathbf{d}^{[0]}(\sigma^{(1)})\right|+\left|(\gamma^{-1})^{(2)}\sigma^{(0)}\right|+\left|(\gamma^{-1})^{(1)}\sigma^{(1)}\right|,$
(5.34)

with a high-frequency behaviour, meaning that we lose one power of $\lambda$
for each derivatives.

######  Remark 5.3.

Note that we don’t impose conditions on $\sigma^{(2)}_{\mathbf{1}\mathbf{1}}$
and $\sigma^{(2)}_{\mathbf{2}\mathbf{2}}$ separately but only on their sum.

The other components of $\sigma^{(2)}$ in the frame
$(N_{0},e_{\mathbf{1}},e_{\mathbf{2}})$ as well as the scalar function
$\mathfrak{f}$ depends on the vector field $Y$, which is yet to be defined.
The equation (5.32) allows us to define $\sigma^{(2)}_{N_{0}\mathbf{i}}$ as a
linear function of $Y$ and of $\sigma^{(0)}$ and $\sigma^{(1)}$. More
precisely we obtain

$\displaystyle\left|\sigma^{(2)}_{N_{0}\mathbf{1}}(Y)\right|+\left|\sigma^{(2)}_{N_{0}\mathbf{2}}(Y)\right|\lesssim\left|\bar{F}^{(1)}Y\right|+\left|\mathbf{d}^{[1]}(\sigma^{(0)})\right|+\left|\mathbf{d}^{[0]}(\sigma^{(1)})\right|,$
(5.35)

with a high-frequency behaviour. Since this component doesn’t appear in the
equations $\sigma^{(2)}$ needs to solve, we set
$\sigma^{(2)}_{\mathbf{1}\mathbf{2}}=0$.

The scalar function $\mathfrak{f}$ is actually already defined by (5.31), but
as $\sigma^{(2)}$ is a function of $Y$, so is $\mathfrak{f}$. Therefore,
thanks to (5.34) and (5.35) $\mathfrak{f}$ satisfies

$\displaystyle|\mathfrak{f}(Y)|$
$\displaystyle\lesssim\left|(\gamma^{-1})^{(\geq
1)}\sigma^{(2)}\right|+\left|(\gamma^{-1})^{(\geq
2)}\sigma^{(1)}\right|+\left|(\gamma^{-1})^{(\geq 3)}\sigma^{(0)}\right|$
$\displaystyle\lesssim\left|(\gamma^{-1})^{(\geq
1)}\bar{F}^{(1)}Y\right|+\left|(\gamma^{-1})^{(\geq
1)}\mathbf{d}^{[1]}(\sigma^{(0)})\right|+\left|(\gamma^{-1})^{(\geq
1)}\mathbf{d}^{[0]}(\sigma^{(1)})\right|$
$\displaystyle\quad+\left|(\gamma^{-1})^{(\geq
1)}(\gamma^{-1})^{(2)}\sigma^{(0)}\right|+\left|(\gamma^{-1})^{(\geq
1)}(\gamma^{-1})^{(1)}\sigma^{(1)}\right|$
$\displaystyle\quad+\left|(\gamma^{-1})^{(\geq
2)}\sigma^{(1)}\right|+\left|(\gamma^{-1})^{(\geq 3)}\sigma^{(0)}\right|,$
(5.36)

with a high-frequency behaviour.

####  5.4.2 Solving for $Y$

To conclude the construction of $\sigma$, it remains to solve (5.33). As
explained after Lemma 4.6, this is done by actually ”replacing” the operator
$M^{[\geq 0]}$ by $\mathrm{div}_{e}L_{e}$ together with a fixed point
argument. More precisely, we define a map $\mathbf{\Psi}$

$\displaystyle\mathbf{\Psi}:\mathcal{B}\longrightarrow\mathcal{B}$

such that $\mathbf{\Psi}(Y)$ is the solution of

$\displaystyle\mathrm{div}_{e}L_{e}\mathbf{\Psi}(Y)$
$\displaystyle=\mathrm{div}_{e}L_{e}Y-M^{[\geq 0]}(Y)-\mathbf{d}^{[\geq
2]}(\sigma^{(0)})$ (5.37) $\displaystyle\quad-\mathbf{d}^{[\geq
1]}(\sigma^{(1)})-\mathbf{d}^{[\geq
0]}(\sigma^{(2)}(Y))-\frac{\lambda}{3}\mathrm{d}\mathfrak{f}(Y),$

and where

$\displaystyle\mathcal{B}=\left\\{Z\in
H^{2}_{\delta}~{}\middle|~{}\left\|Z\right\|_{H^{2}_{\delta}}\leq
C_{1}\varepsilon\right\\}.$ (5.38)

with $C_{1}>0$ to be chosen later. Note that any fixed point of
$\mathbf{\Psi}$ is a solution of (5.33). In order to prove the existence of a
fixed point, we need to show that $\mathbf{\Psi}$ is well-defined and is a
contraction.

######  Proposition 5.1.

If $C_{1}$ is large enough and $\varepsilon$ is small enough, then the map
$\mathbf{\Psi}$ is well-defined and is a contraction.

###### Proof.

Let $Y\in\mathcal{B}$. We bound the RHS of (5.37) in $L^{2}_{\delta+2}$:

$\displaystyle\left\|\text{RHS of \eqref{eq
psi(Y)}}\right\|_{L^{2}_{\delta+2}}$
$\displaystyle\lesssim\left\|\mathrm{div}_{e}L_{e}Y-M^{[\geq
0]}(Y)\right\|_{L^{2}_{\delta+2}}+\left\|\mathbf{d}^{[\geq
0]}(\sigma^{(2)}(Y))\right\|_{L^{2}_{\delta+2}}+\lambda\left\|\mathrm{d}[\mathfrak{f}(Y)]\right\|_{L^{2}_{\delta+2}}$
$\displaystyle\quad+\left\|\mathbf{d}^{[\geq
2]}(\sigma^{(0)})+\mathbf{d}^{[\geq 1]}(\sigma^{(1)})\right\|_{L^{2}}$
$\displaystyle=\vcentcolon A+B+C+D,$

where we omitted the weights for the last term since it is compactly
supported. For $A$, (4.38) gives

$\displaystyle A$
$\displaystyle\lesssim\left\|(\gamma^{-1}-e^{-1})\partial^{2}Y\right\|_{L^{2}_{\delta+2}}+\left\|\partial\gamma\partial
Y\right\|_{L^{2}_{\delta+2}}+\left\|(\partial\gamma)^{2}Y\right\|_{L^{2}_{\delta+2}}+\left\|(\partial^{2}\gamma)^{(\geq
0)}Y\right\|_{L^{2}_{\delta+2}},$

where we also used the fact that the coefficients of $\gamma$ are bounded. We
bound all the metric terms in $L^{\infty}$ using the background regularity
(2.1) and (4.9). More precisely, we have

$\displaystyle\left\|\gamma^{-1}-e^{-1}\right\|_{L^{\infty}}$
$\displaystyle\lesssim\left\|\bar{g}_{0}-e\right\|_{L^{\infty}}+\lambda\left\|\bar{F}^{(1)}\right\|_{L^{\infty}}+\lambda^{2}\left\|\omega^{(2)}\right\|_{L^{\infty}},$
(5.39) $\displaystyle\left\|\partial\gamma\right\|_{L^{\infty}}$
$\displaystyle\lesssim\left\|\partial\bar{g}_{0}\right\|_{L^{\infty}}+\left\|\bar{F}^{(1)}\right\|_{L^{\infty}}$
(5.40)
$\displaystyle\quad+\lambda\left(\left\|\partial\bar{F}^{(1)}\right\|_{L^{\infty}}+\left\|\omega^{(2)}\right\|_{L^{\infty}}\right)+\lambda^{2}\left\|\partial\omega^{(2)}\right\|_{L^{\infty}},$
$\displaystyle\left\|(\partial^{2}\gamma)^{(\geq 0)}\right\|_{L^{\infty}}$
$\displaystyle\lesssim\left\|\partial^{2}\bar{g}_{0}\right\|_{L^{\infty}}+\left\|\partial\bar{F}^{(1)}\right\|_{L^{\infty}}+\left\|\omega^{(2)}\right\|_{L^{\infty}}$
(5.41)
$\displaystyle\quad+\lambda\left(\left\|\partial^{2}\bar{F}^{(1)}\right\|_{L^{\infty}}+\left\|\partial\omega^{(2)}\right\|_{L^{\infty}}\right)+\lambda^{2}\left\|\partial^{2}\omega^{(2)}\right\|_{L^{\infty}}.$

Using (2.1), (4.9), (4.7) and $\left\|Y\right\|_{H^{2}_{\delta}}\leq
C_{1}\varepsilon$, the estimates (5.39)-(5.40)-(5.41) then imply that
$A\lesssim C_{1}\varepsilon^{2}$.

The term $D$ depends only on previous terms of the construction so (4.26)
simply gives $D\lesssim\varepsilon$. The maps $Y\longmapsto\sigma^{(2)}(Y)$
and $Y\longmapsto\mathfrak{f}(Y)$ are affine with coefficients as $D$ (see
(5.34), (5.35) and (5.36)) so a combination of the two previous arguments
gives $B+C\lesssim C_{1}\varepsilon^{2}+\varepsilon$. Note that for $C$ we
need to compensate the loss in power of $\lambda$ when differenciating the
oscillating parts of $\mathfrak{f}$ with the $\lambda$ in front.

We have proved that $A+B+C+D\lesssim C_{1}\varepsilon^{2}+\varepsilon$. In
particular, this allows us to use the second part of Proposition 1.2 to prove
that there exists a unique $\mathbf{\Psi}(Y)\in H^{2}_{\delta}$ solving
(5.37). Moreover we have

$\displaystyle\left\|\mathbf{\Psi}(Y)\right\|_{H^{2}_{\delta}}\lesssim
C_{1}\varepsilon^{2}+\varepsilon.$

Therefore, taking $C_{1}$ large compared to the numerical constant appearing
in these estimates and $\varepsilon$ small compared to 1 proves that
$\mathbf{\Psi}$ is well-defined and maps $\mathcal{B}$ to itself.

To prove that $\mathbf{\Psi}$ is a contraction, we consider $Y_{a}$ and
$Y_{b}$ two elements of $\mathcal{B}$. By substracting the equations satisfied
by $\mathbf{\Psi}(Y_{a})$ and $\mathbf{\Psi}(Y_{b})$ we obtain the equation
for their difference

$\displaystyle\mathrm{div}_{e}L_{e}\left(\mathbf{\Psi}(Y_{a})-\mathbf{\Psi}(Y_{b})\right)$
$\displaystyle=\left(\mathrm{div}_{e}L_{e}-M^{[\geq 0]}\right)(Y_{a}-Y_{b})$
(5.42) $\displaystyle\quad-\mathbf{d}^{[\geq
0]}\left(\sigma^{(2)}(Y_{a})-\sigma^{(2)}(Y_{b})\right)-\frac{\lambda}{3}\mathrm{d}\left(\mathfrak{f}(Y_{a})-\mathfrak{f}(Y_{b})\right).$

Using again the fact that $Y\longmapsto\sigma^{(2)}(Y)$ and
$Y\longmapsto\mathfrak{f}(Y)$ are affine and using (4.38) for
$\mathrm{div}_{e}L_{e}-M^{[\geq 0]}$, we can prove that

$\displaystyle\left\|\text{RHS of \eqref{eq diff Ya-
Yb}}\right\|_{L^{2}_{\delta+2}}$
$\displaystyle\lesssim\varepsilon\left\|Y_{a}-Y_{b}\right\|_{H^{2}_{\delta}}.$

Therefore, taking $\varepsilon$ small enough ensures that $\mathbf{\Psi}$ is a
contraction. ∎

Thanks to this proposition, the Banach fixed point theorem implies the
existence of $Y\in\mathcal{B}$ solving (5.33), and therefore
$(\sigma^{(2)}(Y),\mathfrak{f}(Y),Y)$ solves (5.30)-(5.31)-(5.32)-(5.33).

We can also prove that $Y$ enjoys higher regularity. Indeed we can bound the
RHS of (5.37) in higher order Sobolev spaces and use elliptic estimates for
$\mathrm{div}_{e}L_{e}$ as in the previous proposition. The worse term in
(5.37) is given by $\nabla\sigma^{(1)}$ which is bounded in $H^{N-2}$ (see
(5.20), (5.10) and (5.8)) and in terms of decay the worse term is
$\nabla\sigma^{(0)}$ (see (5.19), (5.2) and (2.1)). Therefore we obtain a
solution $Y$ of (5.33) such that $Y\in H^{N}_{\delta}$ and

$\displaystyle\left\|Y\right\|_{H^{k+2}_{\delta}}\lesssim\frac{\varepsilon}{\lambda^{k}},$
(5.43)

for $k\in\llbracket 0,N-2\rrbracket$, where the loss of $\lambda$ powers is
due to the high-frequency character of each term in (5.37). We summarize what
we know on the parameter $\sigma$ in the following corollary.

######  Corollary 5.1.

The tensor $\sigma$ defined by (5.28) is a TT-tensor for the metric $\gamma$,
belongs to $H^{N-1}_{\delta+1}$ and satisfies

$\displaystyle\max_{k\in\llbracket
0,N-3\rrbracket}\lambda^{k+2}\left\|\sigma\right\|_{H^{k+2}_{\delta+1}}+\max_{k\in\llbracket
0,N-3\rrbracket}\lambda^{k}\left\|\nabla^{k}\sigma\right\|_{L^{\infty}}\lesssim\varepsilon.$
(5.44)

###### Proof.

The oscillating terms $\sigma^{(0)}$ and $\sigma^{(1)}$ lose one $\lambda$
power for each derivatives and we can estimate the actual tensors by
estimating (5.19)-(5.20) directly in weighted Sobolev spaces or in
$L^{\infty}$ using Sobolev embeddings of Proposition 1.1. Moreover, we can
neglect $\sigma^{(2)}(Y)$ and $\mathfrak{f}(Y)$ and focus on $L_{\gamma}Y$ in
(5.28) which rewrites broadly as a $\nabla Y$ term since we can put the
$\gamma$ and $\partial\gamma$ term in $L^{\infty}$. Therefore, the estimate
(5.44) follows directly from (5.43) and Sobolev embeddings. ∎

##  6 Exact solution to the constraint equations

We are now ready to solve the constraint equations (1.15) and (1.16). The
parameters of these equations are $\gamma$, $\tau$ and $\sigma$. The metric
$\gamma$ and the scalar function $\tau$ are fully known thanks to Section 4.1
and (5.11). The TT-tensor $\sigma$ has been defined in Sections 5.4 and 5.3.
Recall that the solutions of (1.15)-(1.16) are of the form

$\displaystyle W$
$\displaystyle=\lambda^{2}\left(W^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\widetilde{W}\right)+\lambda^{3}W^{(3)}\left(\frac{u_{0}}{\lambda}\right),$
$\displaystyle\varphi$
$\displaystyle=1+\lambda^{2}\left(\varphi^{(2)}\left(\frac{u_{0}}{\lambda}\right)+\widetilde{\varphi}\right)+\lambda^{3}\varphi^{(3)}\left(\frac{u_{0}}{\lambda}\right),$

where $\varphi^{(2)}$, $\varphi^{(3)}$ and $W^{(2)}$ are defined in Sections
5.1.1, 5.1.2 and 5.2 respectively. Therefore, it remains to construct
$\widetilde{\varphi}$, $\widetilde{W}$ and $W^{(3)}$.

###  6.1 System for the remainders

The construction of Sections 5.1.1, 5.1.2 and 5.2 ensures that the constraint
equations (1.15) and (1.16) are partly solved. More precisely, it remains to
solve the $\lambda^{\geq 2}$ levels of (1.15) and the $\lambda^{\geq 1}$
levels of (1.16). In this section, we compute the exact equations this gives
for $\widetilde{\varphi}$, $\widetilde{W}$ and $W^{(3)}$.

####  6.1.1 Definition of $W^{(3)}(\widetilde{W})$

The purpose of the oscillating vector field $W^{(3)}$ is to solve the
$\lambda^{1}$ momentum level. However, since the conformal Laplacian
$\mathrm{div}_{\gamma}L_{\gamma}$ loses one power of $\lambda$ even when
applied to a non-oscillating field such as $\widetilde{W}$ (see Lemma 4.6),
the latter is a source term in the equation for $W^{(3)}$. This explains why
$W^{(3)}$ is considered as part of the remainders, when $\varphi^{(3)}$ is
not.

We define $W^{(3)}$ as a function of $\widetilde{W}$. The $\lambda^{1}$
momentum level writes

$\displaystyle\mathbf{M}_{\ell}^{[-2]}(W^{(3)})+M^{[-1]}_{\ell}(\widetilde{W})+\mathbf{M}_{\ell}^{[-1]}(W^{(2)})=\frac{2}{3}\partial_{\ell}\tau^{(1)}.$
(6.1)

Thanks to (4.43) and (4.37) this is equivalent to

$\displaystyle\partial^{2}_{\theta}W^{(3)}_{\ell}+\frac{1}{3}(N_{0})_{\ell}\partial^{2}_{\theta}W^{(3)}_{N_{0}}=-\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\bar{F}^{(1)}_{i\ell}\widetilde{W}_{j}+\frac{1}{|\nabla
u_{0}|^{2}_{\bar{g}_{0}}}\left(\frac{2}{3}\partial_{\ell}\tau^{(1)}-\mathbf{M}_{\ell}^{[-1]}(W^{(2)})\right).$

Lemma 4.8 then gives

$\displaystyle\partial^{2}_{\theta}W^{(3)}_{\ell}$
$\displaystyle=-\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\bar{F}^{(1)}_{i\ell}\widetilde{W}_{j}$
(6.2) $\displaystyle\quad+\frac{1}{|\nabla
u_{0}|^{2}_{\bar{g}_{0}}}\left[\frac{2}{3}\partial_{\ell}\tau^{(1)}-\mathbf{M}_{\ell}^{[-1]}(W^{(2)})-\frac{1}{4}(N_{0})_{\ell}\left(\frac{2}{3}N_{0}\tau^{(1)}-\mathbf{M}_{N_{0}}^{[-1]}(W^{(2)})\right)\right].$

Let us check that the RHS of this equation is purely oscillating. Since
$\tau^{(1)}$ is purely oscillating (see (5.27)), we only need to check
$\mathbf{M}_{\ell}^{[-1]}(W^{(2)})$. For this we use first (4.42) and then
(5.17), this gives

$\displaystyle\mathbf{M}^{[-1]}(W^{(2)})$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)W^{(2)}+\cos(\theta)\partial^{2}_{\theta}W^{(2)}+(1+\sin(\theta))\partial_{\theta}W^{(2)}$
$\displaystyle\;\overset{\mathrm{osc}}{\sim}\;\cos(\theta)+\sin(2\theta)+\cos(3\theta).$

The RHS of (6.2) is thus purely oscillating and we can integrate it twice with
respect to $\theta$. We obtain

$\displaystyle W^{(3)}_{\ell}(\widetilde{W})$
$\displaystyle=\cos\left(\frac{u_{0}}{\lambda}\right)\bar{g}_{0}^{ij}\bar{F}^{(1)}_{i\ell}\widetilde{W}_{j}+W^{(3,rest)}_{\ell},$
(6.3)

where $W^{(3,rest)}$ satisfies

$\displaystyle\left|W^{(3,rest)}\right|$
$\displaystyle\lesssim\left|\partial\tau^{(1)}\right|+\left|\mathbf{M}^{[-1]}(W^{(2)})\right|,$
(6.4)

with a high-frequency behaviour.

####  6.1.2 The system for $\widetilde{W}$ and $\widetilde{\varphi}$

In this section, we will expand in the most concise way the non-linearities
involved in the equations for $\widetilde{\varphi}$ and $\widetilde{W}$. We
start with the equation for $\widetilde{W}$, which, if we drop the vectorial
notation, writes

$\displaystyle M^{[\geq 0]}(\widetilde{W})$
$\displaystyle=-\mathbf{M}^{[\geq-1]}(W^{(3)}(\widetilde{W}))-\mathbf{M}^{[\geq
0]}(W^{(2)})+\frac{2}{3}(\varphi^{6}\partial\tau)^{(\geq 2)},$ (6.5)

The following lemma expands the non-linearity in (6.5).

######  Lemma 6.1.

We have

$\displaystyle\frac{2}{3}(\varphi^{6}\partial_{\ell}\tau)^{(\geq 2)}$
$\displaystyle=\mathbf{a}_{0}+\sum_{k=1}^{6}\lambda^{2(k-1)}\mathbf{a}_{k}\widetilde{\varphi}^{k},$
(6.6)

where for $k\in\llbracket 0,6\rrbracket$, $\mathbf{a}_{k}$ is supported in
$B_{R}$ and

$\displaystyle\max_{i\in\llbracket
0,N-5\rrbracket}\lambda^{i}\left\|\nabla^{i}\mathbf{a}_{k}\right\|_{L^{\infty}}\lesssim\varepsilon.$
(6.7)

###### Proof.

Recall that $\tau=\lambda\tau^{(1)}$ implies
$\partial\tau=O\left(\lambda^{0}\right)$, thus we only need to expand

$\displaystyle\left(1+\lambda^{2}\left(\varphi^{(2)}+\widetilde{\varphi}\right)+\lambda^{3}\varphi^{(3)}\right)^{6},$

and only keep the terms of order $\lambda^{2}$ or more, which only excludes
the term 1. The coefficient $\mathbf{a}_{0}$ in (6.6) contains all the terms
where $\widetilde{\varphi}$ doesn’t appear, it is thus a polynomial in
$\varphi^{(2)}$ and $\varphi^{(3)}$ with no constant term and multiplied by
$\partial\tau$. Therefore, $\mathbf{a}_{0}$ shares the same support property
as $\varphi^{(2)}$ and $\varphi^{(3)}$ and the estimate (6.7) follows from
(5.5), (5.14) and (5.11). If $k\in\llbracket 0,6\rrbracket$, the same
reasoning applies but $\mathbf{a}_{k}$ is now a polynomial in $\varphi^{(2)}$
and $\varphi^{(3)}$ with a constant term. But as this polynomial is still
multiplied by $\partial\tau$, the support property and the estimate still
hold. ∎

The equation for $\widetilde{\varphi}$ writes

$\displaystyle 8\Delta_{\gamma}\widetilde{\varphi}$
$\displaystyle=-8\sum_{i=2}^{3}\mathbf{H}^{[\geq 2-i]}(\varphi^{(i)})+R^{(\geq
2)}+R(\gamma)\left(\varphi^{(2)}+\widetilde{\varphi}+\lambda\varphi^{(3)}\right)$
(6.8) $\displaystyle\quad+\frac{2}{3}(\tau^{2}\varphi^{5})^{(\geq
2)}-\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\varphi^{-7}\right)^{(\geq
2)}.$

The next two lemmas expand the non-linearities in (6.8).

######  Lemma 6.2.

We have

$\displaystyle\frac{2}{3}(\tau^{2}\varphi^{5})^{(\geq
2)}=\mathbf{b}_{0}+\sum_{k=1}^{5}\lambda^{2k}\mathbf{b}_{k}\widetilde{\varphi}^{k},$

where for $k\in\llbracket 0,5\rrbracket$, $\mathbf{b}_{k}$ is supported in
$B_{R}$ and

$\displaystyle\max_{i\in\llbracket
0,N-5\rrbracket}\lambda^{i}\left\|\nabla^{i}\mathbf{b}_{k}\right\|_{L^{\infty}}\lesssim\varepsilon.$
(6.9)

The proof of Lemma 6.2 is left to the reader since it is very similar to the
one of Lemma 6.1. We now expand the non-linearities with a negative power of
$\varphi$.

######  Lemma 6.3.

There exists a universal constant $C_{emb}>0$ such that if
$\left\|\widetilde{\varphi}\right\|_{H^{2}_{\sigma}}<C_{emb}^{-1}$ and if
$\varepsilon$ is small enough, then we have

$\displaystyle\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\varphi^{-7}\right)^{(\geq
2)}$
$\displaystyle=\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\right)^{(\geq
2)}+\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\left(\mathbf{c}_{0}+\sum_{k\geq
1}\mathbf{c}_{k}\lambda^{2(k-1)}\widetilde{\varphi}^{k}\right),$

where the $\mathbf{c}_{k}$ satisfy:

  * •

$\mathbf{c}_{0}$ is supported in $B_{R}$ and we have

$\displaystyle\max_{i\in\llbracket
0,N-5\rrbracket}\lambda^{i}\left\|\nabla^{i}\mathbf{c}_{0}\right\|_{L^{\infty}}\lesssim\varepsilon,$
(6.10)

  * •

if $k\geq 1$, we have

$\displaystyle\max_{i\in\llbracket
0,N-5\rrbracket}\lambda^{i}\left\|\nabla^{i}\mathbf{c}_{k}\right\|_{L^{\infty}}\lesssim
1.$ (6.11)

###### Proof.

The constant $C_{emb}$ is the one appearing in the embedding
$H^{2}_{\delta}\xhookrightarrow{}L^{\infty}$ (see Proposition 1.1), i.e

$\displaystyle\left\|u\right\|_{L^{\infty}}\leq
C_{emb}\left\|u\right\|_{H^{2}_{\delta}},$

for all $u\in H^{2}_{\delta}$. Now if $\varepsilon$ is small enough and if
$\left\|\widetilde{\varphi}\right\|_{H^{2}_{\delta}}<C_{emb}^{-1}$, we have

$\displaystyle\left\|\varphi^{(2)}+\widetilde{\varphi}+\lambda\varphi^{(3)}\right\|_{L^{\infty}}\leq
1.$

This allows us to expand
$\varphi^{-7}=\left(1+\lambda^{2}\left(\varphi^{(2)}+\widetilde{\varphi}+\lambda\varphi^{(3)}\right)\right)^{-7}$.
Indeed there exists a sequence $(\mathbf{c}_{k})_{k\in\mathbb{N}}$ such that

$\displaystyle\varphi^{-7}=1+\lambda^{2}\left(\mathbf{c}_{0}+\sum_{k\geq
1}\mathbf{c}_{k}\lambda^{2(k-1)}\widetilde{\varphi}^{k}\right)$

where $\mathbf{c}_{0}$ is a polynomial in $\varphi^{(2)}$ and $\varphi^{(3)}$
with no constant term and $\mathbf{c}_{k}$ for $k\geq 1$ is a polynomial in
$\varphi^{(2)}$ and $\varphi^{(3)}$ with a constant term bounded but not
compactly supported. This justifies the estimates (6.10) and (6.11).
Therefore, we have

$\displaystyle\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\varphi^{-7}\right)^{(\geq
2)}$
$\displaystyle=\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\right)^{(\geq
2)}+\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\left(\mathbf{c}_{0}+\sum_{k\geq
1}\mathbf{c}_{k}\lambda^{2(k-1)}\widetilde{\varphi}^{k}\right),$

which concludes the proof. ∎

Putting Lemmas 6.1, 6.2 and 6.3 together, we obtain the final form of the
system solved by $\widetilde{\varphi}$ and $\widetilde{W}$:

$\displaystyle M^{[\geq 0]}(\widetilde{W})$
$\displaystyle=-\mathbf{M}^{[\geq-1]}(W^{(3)}(\widetilde{W}))+\sum_{k=1}^{6}\lambda^{2(k-1)}\mathbf{a}_{k}\widetilde{\varphi}^{k}+\mathcal{R}_{\mathrm{mom}},$
(6.12) $\displaystyle 8\Delta_{\gamma}\widetilde{\varphi}$
$\displaystyle=R(\gamma)\widetilde{\varphi}+\sum_{k=1}^{5}\lambda^{2k}\mathbf{b}_{k}\widetilde{\varphi}^{k}$
(6.13)
$\displaystyle\quad-\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\right)^{(\geq
2)}-\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\left(\mathbf{c}_{0}+\sum_{k\geq
1}\mathbf{c}_{k}\lambda^{2(k-1)}\widetilde{\varphi}^{k}\right)+\mathcal{R}_{\mathrm{Ham}},$

where we define the following remainders

$\displaystyle\mathcal{R}_{\mathrm{mom}}$ $\displaystyle=-\mathbf{M}^{[\geq
0]}(W^{(2)})+\mathbf{a}_{0},$ (6.14) $\displaystyle\mathcal{R}_{\mathrm{Ham}}$
$\displaystyle=-8\sum_{i=2}^{3}\mathbf{H}^{[\geq 2-i]}(\varphi^{(i)})+R^{(\geq
2)}+R(\gamma)\left(\varphi^{(2)}+\lambda\varphi^{(3)}\right)+\mathbf{b}_{0}.$
(6.15)

###  6.2 Fixed point argument

In this section, we solve (6.12) and (6.13) by a fixed point argument. As in
Section 5.4.2, the idea is to replace the operators depending on $\gamma$ by
their Euclidean equivalent and use the smallness of $\gamma-e$. We introduce
the map $\mathbf{\Phi}$

$\displaystyle\mathbf{\Phi}$
$\displaystyle:\mathcal{B}\times\mathcal{B}\longrightarrow\mathcal{B}\times\mathcal{B}$
$\displaystyle\quad(\widetilde{\varphi},\widetilde{W})\longmapsto\left(\mathbf{\Phi}_{1}(\widetilde{\varphi}),\mathbf{\Phi}_{2}(\widetilde{W})\right),$

such that $\mathbf{\Phi}_{1}(\widetilde{\varphi})$ and
$\mathbf{\Phi}_{2}(\widetilde{W})$ are solutions of the coupled system

$\displaystyle\mathrm{div}_{e}L_{e}\mathbf{\Phi}_{2}(\widetilde{W})$
$\displaystyle=\mathrm{div}_{e}L_{e}(\widetilde{W})-M^{[\geq
0]}(\widetilde{W})$ (6.16)
$\displaystyle\quad-\mathbf{M}^{[\geq-1]}(W^{(3)}(\widetilde{W}))+\sum_{k=1}^{6}\lambda^{2(k-1)}\mathbf{a}_{k}\widetilde{\varphi}^{k}+\mathcal{R}_{\mathrm{mom}},$
$\displaystyle 8\Delta\mathbf{\Phi}_{1}(\widetilde{\varphi})$
$\displaystyle=8\Delta\widetilde{\varphi}-8\Delta_{\gamma}\widetilde{\varphi}+R(\gamma)\widetilde{\varphi}+\sum_{k=1}^{5}\lambda^{2k}\mathbf{b}_{k}\widetilde{\varphi}^{k}$
(6.17)
$\displaystyle\quad-\left(\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\right)^{(\geq
2)}-\left|\sigma+L_{\gamma}W\right|^{2}_{\gamma}\left(\mathbf{c}_{0}+\sum_{k\geq
1}\mathbf{c}_{k}\lambda^{2(k-1)}\widetilde{\varphi}^{k}\right)+\mathcal{R}_{\mathrm{Ham}},$

and where $\mathcal{B}$ is defined in (5.38). Note that a fixed point of
$\mathbf{\Phi}$ solves (6.12) and (6.13). In order to apply the Banach fixed
point theorem and prove the existence of a fixed point, we prove in the next
proposition that $\mathbf{\Phi}$ is well-defined and is a contraction.

######  Proposition 6.1.

If $C_{1}$ is large enough and $\varepsilon$ is small enough, then
$\mathbf{\Phi}$ is well-defined and is a contraction.

###### Proof.

Let $(\widetilde{\varphi},\widetilde{W})\in\mathcal{B}\times\mathcal{B}$. We
start by estimating the $L^{2}_{\delta+2}$ norm of the RHS of (6.16):

$\displaystyle\left\|\text{RHS of \eqref{eq Phi
tW}}\right\|_{L^{2}_{\delta+2}}$
$\displaystyle\lesssim\left\|\mathrm{div}_{e}L_{e}(\widetilde{W})-M^{[\geq
0]}(\widetilde{W})\right\|_{L^{2}_{\delta+2}}+\left\|\mathbf{M}^{[\geq-1]}(W^{(3)}(\widetilde{W}))\right\|_{L^{2}_{\delta+2}}$
$\displaystyle\quad+\left\|\sum_{k=1}^{6}\lambda^{2(k-1)}\mathbf{a}_{k}\widetilde{\varphi}^{k}\right\|_{L^{2}}+\left\|\mathcal{R}_{\mathrm{mom}}\right\|_{L^{2}}$
$\displaystyle=\vcentcolon A+B+C+D,$

where we omitted the weights for the last two terms since they are compactly
supported. As in the proof of Proposition 5.1, we obtain $A\lesssim
C_{1}\varepsilon^{2}$. For $B$, we note that the operator
$\mathbf{M}^{[\geq-1]}$ is linear and has bounded coefficients and involves up
to two derivatives of $W^{(3)}(\widetilde{W})$, recall Lemma 4.7. Moreover,
thanks to (6.3) $W^{(3)}(\widetilde{W})$ is compactly supported so we obtain
$B\lesssim\left\|W^{(3)}(\widetilde{W})\right\|_{H^{2}}$. Using (4.9) and