# Spacelike initial data for black hole stability

Allen Juntao Fang  Universität Münster, Münster, Deutschland
<EMAIL_ADDRESS> ,  Jérémie Szeftel  CNRS & Laboratoire
Jacques-Louis Lions, Sorbonne Université, Paris, (jeremie-szeftel@sorbonne-
universite.fr)  and  Arthur Touati  Institut des Hautes Etudes Scientifiques,
Bures-sur-Yvette<EMAIL_ADDRESS>
###### Abstract.

We construct initial data suitable for the Kerr stability conjecture, that is,
solutions to the constraint equations on a spacelike hypersurface with
boundary entering the black hole horizon that are arbitrarily decaying
perturbations of a Kerr initial data set. This results from a more general
perturbative construction on any asymptotically flat initial data set with the
topology of ${\mathbb{R}}^{3}\setminus\\{r<1\\}$ enjoying some analyticity
near and at the boundary. In particular, we design a suitable mixed boundary
condition for the elliptic operator of the conformal method in order to
exclude the Killing initial data sets (KIDS).

##  1\. Introduction

The Kerr family of stationary black holes is generally believed to describe
the black holes observed in astrophysics. An important question regarding
whether or not this belief is substantiated is the stability property of this
family. In this article, we construct initial data for the Kerr stability
problem.

###  1.1. The initial value problem in general relativity

The Einstein vacuum equations (EVE) are the defining equations of Einstein’s
theory of general relativity under the assumption of vacuum and are given by

$\mbox{ \bf Ric}({\bf g})=0,$ (1.1)

where ${\bf g}$ is a Lorentzian metric defined on a 4-dimensional manifold
${\mathcal{M}}$, and  Ric denotes the Ricci tensor. As first shown in the
seminal papers [6, 7], there is a well-defined Cauchy problem for (1.1), where
one solves for a maximal globally hyperbolic development $({\mathcal{M}},{\bf
g})$ of a given initial data triplet $(\Sigma,g,k)$ consisting of a
3-dimensional Riemannian manifold $(\Sigma,g)$ and a symmetric two tensor $k$
satisfying the constraint equations. The constraint equations themselves are
given by

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

where $R(g)$ is the scalar curvature of $g$.

###  1.2. The Kerr stability conjecture

Studying the stability properties of black hole solutions to EVE can be
formulated in the context of the above Cauchy problem for EVE. We provide
below a rough statement of the Kerr stability conjecture, and refer the reader
for example to the introduction of [21, 17] for a more detailed discussion.

######  Conjecture 1.1 (Kerr stability conjecture).

If $(\Sigma,g,k)$ is an initial data triplet such that $(g,k)$ is an
appropriately small perturbation of $(g_{m,a},k_{m,a})$ the initial data of a
subextremal Kerr black hole ${\bf g}_{m,a}$ (i.e. $|a|<m$), then the evolution
of $(\Sigma,g,k)$ under (1.1), $({\mathcal{M}},{\bf g})$, has a domain of
outer communication that converges in the appropriate sense to a nearby member
of the Kerr family $({\mathcal{M}}_{m_{f},a_{f}},{\bf g}_{m_{f},a_{f}})$.

In the context of black hole stability, there are generally three regions of
interest in a black hole spacetime, separated by null cones arising from the
initial data, as indicated by the three regions $I$, $II$, and $III$ in Figure
1. We discuss briefly the stability results in each of these regions.

$\mathcal{H}^{+}$$\;\mathscr{I}^{+}$$\mathcal{H}^{-}$$\mathscr{I}^{-}$$\Sigma$$I$$III$$II$$i^{+}$$i^{-}$$i^{0}$
Figure 1. Penrose diagram of the future of initial perturbations of Kerr.

Region $I$ is a compact region of spacetime, and thus, is a local existence
regime. In region $II$, the full subextremal Kerr family was shown to be
stable by [3, 28]. In region $III$, stability has been shown111See also [20,
12] for the stability of Schwarzschild (i.e. $a=0$) for restrictive initial
data. for slowly-rotating Kerr black holes (i.e. $|a|\ll m$) in [17, 19, 18,
21, 27], using initialization222This initialization takes place on the null
cones of Figure 1 where the constraint equations take the form of non-linear
ODEs, as opposed to (1.2) which are non-linear elliptic PDEs. from regions $I$
and $II$, thus proving Conjecture 1.1 in the slowly-rotating case.

###  1.3. First version of the main result

Our goal is to construct $(g,k)$ such that $(\Sigma,g,k)$ is an initial data
triplet satisfying the constraint equations (1.2) and is an arbitrarily
decaying perturbation of the initial data for a given Kerr black hole in order
to satisfy the requirements in [3, 28]. We remark that this will also induce
data that suffices for [12, 20, 21]. We give here a rough version of our main
result, see Theorem 3.3 for the precise statement.

######  Theorem 1.2 (Rough statement).

Let $(\Sigma,g_{m,a},k_{m,a})$ be the spacelike initial data set induced by
the Kerr metric ${\bf g}_{m,a}$, and let $q\in\mathbb{N}^{*}$, $0<\delta<1$
and $\varepsilon>0$ small enough. There exists a codimension $4q^{2}$ family
of perturbations $\left(g_{m,a}+\tilde{g},k_{m,a}+\tilde{k}\right)$ of
$(g_{m,a},k_{m,a})$ such that

$\left\lvert\tilde{g}\right\rvert\sim\varepsilon(1+r)^{-q-\delta},\quad\left\lvert\tilde{k}\right\rvert\sim\varepsilon(1+r)^{-q-\delta-1},$
(1.3)

and $\left(\Sigma,g_{m,a}+\tilde{g},k_{m,a}+\tilde{k}\right)$ satisfies the
constraint equations (1.2).

Theorem 1.2 provides initial data for the works mentioned in Section 1.2:

  * •

In region $II$, [3] requires $q\geq 3$ and [28] requires $q\geq 1$, with
$0<\delta<1$ in both cases.

  * •

In region $III$, [20, 21] require $q=1$ and [12] requires $q=2$, with
$\delta=\left(\frac{1}{2}\right)_{+}$ in both cases.

###  1.4. Construction of initial data sets

There is a rich literature on the constraint equations (1.2) and we refer the
reader for example to [4] for a detailed review. The main two approaches to
produce solutions to these equations are the conformal method and the gluing
method. First introduced in [23], the conformal method proposes a conformal
ansatz for $(g,k)$ and transforms the quasi-linear underdetermined system
(1.2) into a semi-linear elliptic system for a scalar function and a vector
field. This approach has been successful in constructing many interesting
solutions to the constraint equations, including those with large mean
curvature or with apparent horizons [13, 25]. However these results fall short
of constructing perturbations of a given black hole satisfying (1.3). This is
due to the fact that these constructions are typically done within the range
of invertibility of the relevant elliptic operators resulting in perturbations
that decay like $r^{-\delta}$ for $0<\delta<1$, falling short of the decay
rates assumed in the aforementioned stability results, see the discussion
after Theorem 1.2.

The gluing method, first introduced in [10] and later extended in [11, 9],
consists of gluing any asymptotically flat solution of (1.2) to an exact Kerr
initial data set outside a far away annulus. However, critically, these
constructions, viewed as a perturbation of the background black hole metric,
are compactly supported, whereas we aim to construct non-compactly supported
initial data.

######  Remark 1.3.

Gluing results use the Kerr initial data, viewed as a 10-parameter family, to
saturate the 10-dimensional space of linear obstructions coming from the
symmetries of Minkowski. Theorem 1.2 exhibits infinitely many 10-parameter
families with the same asymptotics at main order. As a result, one should be
able to glue asymptotically flat initial data to arbitrarily decaying
perturbations of Kerr.

The authors have previously constructed and parametrized initial data
perturbations of Minkowski space with arbitrary decay [14]. While the overall
methodology is similar, there are two key differences between the current
paper and [14]. We first recall that in [14] the Killing initial data sets
(KIDS) presented a fundamental, linear obstacle to the construction. Since
black hole spacetimes have fewer Killing vectorfields than Minkowski, there
are fewer KIDS to deal with in the present paper than in [14]. In addition,
motivated by the stability of Minkowski, initial data were constructed in [14]
for a manifold without boundary. On the other hand, in the present paper,
initial data are constructed for a manifold with boundary, reflecting the
presence of an event horizon. The presence of the boundary adds flexibility,
as the elliptic problem to be solved does not itself impose boundary
conditions. This will play a crucial role in the ensuing proof, where we take
advantage of this freedom.

###  1.5. Sketch of proof

We now give a brief sketch of the proof of Theorem 1.2. Rather than
considering initial data $(\Sigma,g,k)$ it will be convenient to instead
consider initial data triplets $(\Sigma,g,\pi)$, where
$\pi=k-(\mathrm{tr}_{g}k)g$ will be referred to as the _reduced second
fundamental form_ of $\Sigma$.

Using the conformal method, we consider as an ansatz $(g,\pi)$ of the form

$\displaystyle(g,\pi)=\left((1+\widecheck{u})^{4}\left(g_{m,a}+\widecheck{g}\right),(1+\widecheck{u})^{2}\left(\pi_{m,a}+\widecheck{\pi}+L_{g_{m,a}}\widecheck{X}\right)\right),$
(1.4)

where $L_{g_{m,a}}$ is a modified Lie derivative (see (3.2)), and where
$(\widecheck{g},\widecheck{\pi})$ are the ”seed” of the construction and enjoy
smallness and decay satisfying

$\displaystyle\sup_{k=0,1}\left(r^{k}|\nabla^{k}\widecheck{g}|+r^{k+1}|\nabla^{k}\widecheck{\pi}|\right)\leq\frac{\varepsilon}{(1+r)^{q+\delta}}.$
(1.5)

The constraint equations (1.2) for $(g,\pi)$ then become the following system
of PDEs on $\Sigma$

$\displaystyle
P\left(\widecheck{u},\widecheck{X}\right)=D\Phi[g_{m,a},\pi_{m,a}](\widecheck{g},\widecheck{\pi})+\text{non-
linear terms},$ (1.6)

where $P$ is an elliptic operator depending only on $(g_{m,a},\pi_{m,a})$, and
$D\Phi[g_{m,a},\pi_{m,a}]$ denotes the linearization of the constraint
operator at $(g_{m,a},\pi_{m,a})$. Theorem 1.2 then asks whether there exists
a solution $\left(\widecheck{u},\widecheck{X}\right)$ to (1.6) satisfying

$\displaystyle|\widecheck{u}|+r\left|\nabla\widecheck{X}\right|\lesssim\frac{\varepsilon}{(1+r)^{q+\delta}}.$
(1.7)

We remark that applying the conformal method does not impose boundary
conditions on $\partial\Sigma$ for the solutions
$\left(\widecheck{u},\widecheck{X}\right)$ to (1.6), which will play a
critical role as mentioned above.

To address this question, we study the behavior of the elliptic operator $P$
on weighted Sobolev spaces333Recall that $u\in H^{2}_{\eta}(\Sigma)$ implies
$|u|\lesssim r^{\eta}$ at infinity, see Definition 2.2.
$H^{2}_{-q-\delta}(\Sigma)$. Using qualitative properties of elliptic
operators to understand the decay of their solutions is a classical method,
and we illustrate the basic idea behind with toy model of the scalar Laplacian
on Euclidean space before addressing the full problem at hand. The mapping
properties of the Laplacian

$\Delta:H^{2}_{\delta}(\mathbb{R}^{3})\longrightarrow
L^{2}_{\delta-2}(\mathbb{R}^{3})$ (1.8)

are well studied [26, 2], and in particular, it is known that the Laplacian as
defined in (1.8) is a Fredholm operator, if $\delta\notin\mathbb{Z}$. This
implies, via the Fredholm alternative, that provided $h$ is orthogonal to the
kernel of $\Delta^{*}$, then $\Delta^{-1}h$ is well defined. Thus, given some
$h\in L^{2}_{\delta-2}(\mathbb{R}^{3})$, finding a perturbation $\tilde{h}$
such that $\Delta^{-1}\left(h+\tilde{h}\right)\in
H^{2}_{\delta}(\mathbb{R}^{3})$ reduces to solving for $\tilde{h}$ such that

$\left\langle
h+\tilde{h},w\right\rangle_{L^{2}(\mathbb{R}^{3})}=0,\qquad\text{for all
}w\in\ker(\Delta^{*}).$ (1.9)

In this case of the Laplacian, the kernel of $\Delta^{*}$ can be explicitly
described and it consists just of harmonic polynomials. Thus, a suitable
perturbation $\tilde{h}$ can be explicitly solved for. In particular, this toy
model emphasizes the fact that a full characterization of the cokernel of the
elliptic operator under consideration is crucial.

We now return to giving a sketch of the proof in our setting. It can be
verified that $P$ is also a Fredholm operator acting between weighted Sobolev
spaces. However, characterizing the kernel of $P^{*}$ is more delicate. We
characterize the kernel of $P^{*}$ as perturbations of the harmonic
polynomials. The critical tool will be a good choice of mixed boundary
condition on $\partial\Sigma$ of the form

$\displaystyle(B_{\nu}+F)\left(\widecheck{u},\widecheck{X}\right)=0,$ (1.10)

such that the operator $(P,B_{\nu}+F)$ acting on $H^{2}_{-\delta}(\Sigma)$ for
$0<\delta<1$ will be injective, and where $B_{\nu}$ is a Neumann type boundary
operator defined on $\partial\Sigma$. We will also require that $F$ vanishes
identically on an open subset of $\partial\Sigma$ and is the identity on
another open subset. This condition on $F$ is not necessary to characterize
the kernel of $P^{*}$ but will play a key role in eliminating certain
obstructions in what follows.

To ensure the orthogonality conditions necessary for $P$ to be invertible, we
will modify the ansatz (1.4) to include compactly supported symmetric
2-tensors $(\breve{g},\breve{\pi})$ such that the elliptic system reads

$\displaystyle
P\left(\widecheck{u},\widecheck{X}\right)=D\Phi[g_{m,a},\pi_{m,a}]\left((\widecheck{g},\widecheck{\pi})+(\breve{g},\breve{\pi})\right)+\text{non-
linear terms}.$ (1.11)

The correctors $(\breve{g},\breve{\pi})$ will then be chosen in a $4q^{2}$
dimensional space such as to ensure the following analogue of (1.9):

$\displaystyle\left\langle(\breve{g},\breve{\pi}),D\Phi[g_{m,a},\pi_{m,a}]^{*}(\mathbb{W})\right\rangle=-\left\langle
D\Phi[g_{m,a},\pi_{m,a}](\widecheck{g},\widecheck{\pi}),\mathbb{W}\right\rangle+\text{non-
linear terms}$ (1.12)

for all $\mathbb{W}$ in the cokernel of $(P,B_{\nu}+F)$. The linearization of
the system in (1.12) is solvable if $D\Phi[g_{m,a},\pi_{m,a}]^{*}$ acts
injectively on the cokernel of $(P,B_{\nu}+F)$ restricted to the support of
$(\breve{g},\breve{\pi})$. Now, since the Kerr initial data
$(g_{m,a},\pi_{m,a})$ can be proved to be analytic near and at the boundary,
elements of $D\Phi[g_{m,a},\pi_{m,a}]^{*}$ can also be proved to be analytic
near and at the boundary. Therefore, they cannot satisfy the boundary
condition (1.10) since we assumed that $F$ vanishes in an open set of
$\partial\Sigma$ and is the identity on another one. This shows how
$D\Phi[g_{m,a},\pi_{m,a}]^{*}$ acts injectively on the cokernel of
$(P,B_{\nu}+F)$.

The above procedure illustrates how a careful choice of $F$ allows us to
simultaneously describe the mapping properties of $P$ and also to ensure that
we can appropriately solve for the correctors $(\breve{g},\breve{\pi})$ by
eliminating the KIDS, i.e. the elements of
$\mathrm{ker}(D\Phi[g_{m,a},\pi_{m,a}]^{*})$, which are well-known linear
obstructions to solving (1.12). A straightforward fixed point argument then
concludes the resolution of the non-linear system (1.11).

######  Remark 1.4.

In fact, the only properties of the Kerr initial data set we use are its
analyticity near and at the boundary, and the fact that it solves (1.2). For
this reason, we prove a more general statement on asymptotically flat
solutions of (1.2) which are analytic near and at the boundary, see Theorem
3.1, from which the result for Kerr is then deduced, see Theorem 3.3.

###  1.6. Outline of the article

We conclude this introduction by giving an outline of the article:

  * •

In Section 2, we start with geometric and analytic preliminaries.

  * •

In Section 3, we give a precise statement of our main result Theorem 3.1, from
which we deduce Theorem 1.2.

  * •

In Section 4, we study the properties of the elliptic operator $P$ at the
heart of our construction.

  * •

Finally, in Section 5 we prove Theorem 3.1.

### Acknowledgments

The first author acknowledges support from NSF award DMS-2303241 and support
through Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster:
Dynamics–Geometry–Structure.

##  2\. Preliminaries

###  2.1. Notations and geometry

Throughout the paper, we will use Einstein summation notation, with roman
indices referring to spatial indices $\\{1,2,3\\}$.

######  Definition 2.1.

A triplet $(\Sigma,\bar{g},\bar{\pi})$, where $\Sigma$ is a 3-manifold with
boundary, $\bar{g}$ is a Riemannian metric on $\Sigma$ and $\bar{\pi}$ is a
symmetric 2-tensor is called _asymptotically flat_ if:

  * (i)

$\Sigma$ is smooth, diffeomorphic to ${\mathbb{R}}^{3}\setminus B$ where $B$
is an open ball in ${\mathbb{R}}^{3}$ and there exists a global coordinate
system $(x^{1},x^{2},x^{3})$ on $\Sigma$ such that
$\partial\Sigma=\\{r=r_{0}\\}$ for some $r_{0}>0$ where
$r\vcentcolon=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$,

  * (ii)

$(\bar{g},\bar{\pi})$ is smooth and asymptotically flat, i.e. there exists
constants $C_{k}$ such that for every $k\in\mathbb{N}$ we have

$\displaystyle\sup_{\Sigma}\left(r^{k+1}\left|\nabla^{k}(\bar{g}_{ij}-\delta_{ij})\right|+r^{k+2}\left|\nabla^{k}\bar{\pi}_{ij}\right|\right)\leq
C_{k},$

where
$\nabla=\left(\partial_{x^{1}},\partial_{x^{2}},\partial_{x^{3}}\right)$.

We will also refer to $\Sigma$ satisfying the above conditions as an
asymptotically flat manifold with boundary.

###  2.2. Function spaces

On $(\Sigma,\bar{g},\bar{\pi})$ an asymptotically flat manifold with boundary,
as defined in Definition 2.1, we use the standard $L^{p}(\Sigma)$ spaces for
$1\leq p\leq\infty$, where the integration is defined with respect to the
volume form of $\bar{g}$. Moreover, the boundary $\partial\Sigma$ is a compact
2-manifold and we will use the standard $H^{s}(\partial\Sigma)$ spaces for
$s\in{\mathbb{R}}$. We now define function spaces with weights at infinity.

######  Definition 2.2.

Let $(\Sigma,\bar{g},\bar{\pi})$ be an asymptotically flat manifold with
boundary as defined in Definition 2.1 and let $k\in\mathbb{N}$ and
$\delta\in{\mathbb{R}}$.

  1. (i)

We define the _weighted Sobolev spaces_ $H^{k}_{\delta}(\Sigma)$ to be the set
of distribution on $\Sigma$ for which the following norm is finite

$\left\lVert
u\right\rVert_{H^{k}_{\delta}(\Sigma)}\vcentcolon=\sum_{\left\lvert\alpha\right\rvert\leq
k}\left\lVert(1+r)^{-\delta-\frac{3}{2}+\left\lvert\alpha\right\rvert}\nabla^{\alpha}u\right\rVert_{L^{2}(\Sigma)}.$

We denote $L^{2}_{\delta}(\Sigma)=H^{0}_{\delta}(\Sigma)$.

  2. (ii)

We define the _weighted Hölder spaces_ $C^{k}_{\delta}(\Sigma)$ to be the set
of distribution on $\Sigma$ for which the following norm is finite

$\left\lVert
u\right\rVert_{C^{k}_{\delta}(\Sigma)}\vcentcolon=\sum_{\left\lvert\alpha\right\rvert\leq
k}\left\lVert(1+r)^{-\delta+\left\lvert\alpha\right\rvert}\nabla^{\alpha}u\right\rVert_{L^{\infty}(\Sigma)}.$

  3. (iii)

We extend these definitions to tensors of any type by summing over components
in the global coordinates system $(x^{1},x^{2},x^{3})$ of Definition 2.1.

In the following lemma, we gather standard properties of these spaces. Proofs
of these properties can be found in [5].

######  Lemma 2.3.

Let $k,k_{1},k_{2}\in\mathbb{N}$ and
$\delta,\delta_{1},\delta_{2}\in{\mathbb{R}}$.

  * (i)

If $k\geq 1$ and $\delta^{\prime}<\delta$ then the inclusion
$H^{k}_{\delta^{\prime}}(\Sigma)\subset H^{k-1}_{\delta}(\Sigma)$ is compact.

  * (ii)

If $k\geq 2$ then $H^{k}_{\delta}(\Sigma)\subset C^{0}_{\delta}(\Sigma)$ and
this inclusion is continuous.

  * (iii)

If $k\leq\min(k_{1},k_{2})$, $k<k_{1}+k_{2}-\frac{3}{2}$ and
$\delta>\delta_{1}+\delta_{2}$ then multiplication is a continuous map from
$H^{k_{1}}_{\delta_{1}}(\Sigma)\times H^{k_{2}}_{\delta_{2}}(\Sigma)$ to
$H^{k}_{\delta}(\Sigma)$.

###  2.3. The constraint operator

Using the reduced second fundamental form $\pi$, the constraint equations
(1.2) can be rewritten as

$\displaystyle R(g)+\frac{1}{2}(\mathrm{tr}_{g}\pi)^{2}-|\pi|^{2}_{g}$
$\displaystyle=0,$ (2.1) $\displaystyle\mathrm{div}_{g}\pi$ $\displaystyle=0.$

If $g$ is a Riemannian metric on $\Sigma$ and $\pi$ is a symmetric 2-tensor on
$\Sigma$, we define

$\displaystyle{\mathcal{H}}(g,\pi)$
$\displaystyle\vcentcolon=R(g)+\frac{1}{2}(\mathrm{tr}_{g}\pi)^{2}-|\pi|^{2}_{g},$
$\displaystyle{\mathcal{M}}(g,\pi)$
$\displaystyle\vcentcolon=\mathrm{div}_{g}\pi.$

For these operators, we will use the following notation for their expansion
around some given $(g,\pi)$:

$\displaystyle{\mathcal{H}}(g+h,\pi+\varpi)-{\mathcal{H}}(g,\pi)$
$\displaystyle=D{\mathcal{H}}[g,\pi](h,\varpi)+Q_{\mathcal{H}}[g,\pi](h,\varpi),$
(2.2) $\displaystyle{\mathcal{M}}(g+h,\pi+\varpi)-{\mathcal{M}}(g,\pi)$
$\displaystyle=D{\mathcal{M}}[g,\pi](h,\varpi)+Q_{\mathcal{M}}[g,\pi](h,\varpi),$
(2.3)

where $D{\mathcal{H}}[g,\pi]$ and $D{\mathcal{M}}[g,\pi]$ are the
linearization of ${\mathcal{H}}$ and ${\mathcal{M}}$ and where
$Q_{\mathcal{H}}[g,\pi](h,\varpi)$ and $Q_{\mathcal{H}}[g,\pi](h,\varpi)$
contain all quadratic and higher order terms in $(h,\varpi)$. Note that in
this article, and in particular in the following lemma, we do not distinguish
between coefficients of a metric $g$ and coefficients of its inverse and thus
simply write $g$ in the schematic notation for error terms.

######  Lemma 2.4.

Let $g$ be a Riemannian metric on $\Sigma$ and $\pi$ a symmetric 2-tensor on
$\Sigma$. We have

$\displaystyle D{\mathcal{H}}[g,\pi](h,\varpi)={}$
$\displaystyle-\Delta_{g}\mathrm{tr}_{g}h+D^{i}D^{j}h_{ij}-h^{ij}Ric(g)_{ij}$
(2.4)
$\displaystyle\quad+\mathrm{tr}_{g}\pi\left(\mathrm{tr}_{g}\varpi-\pi^{ij}h_{ij}\right)-2\pi^{ij}\varpi_{ij}+2h^{ij}g^{k\ell}\pi_{ik}\pi_{j\ell},$
$\displaystyle D{\mathcal{M}}[g,\pi](h,\varpi)_{i}={}$
$\displaystyle\mathrm{div}_{g}\varpi_{i}-h^{k\ell}D_{k}\pi_{\ell
i}-\pi_{i}^{j}g^{k\ell}\left(D_{k}h_{\ell
j}-\frac{1}{2}D_{j}h_{k\ell}\right)-\frac{1}{2}\pi^{k\ell}D_{i}h_{k\ell},$
(2.5)

where $D$ is the covariant derivative of $g$ and all inverse are with respect
to $g$. Moreover the terms $Q_{\mathcal{H}}[g,\pi](h,\varpi)$ and
$Q_{\mathcal{M}}[g,\pi](h,\varpi)$ have the following schematic form

$\displaystyle Q_{\mathcal{H}}[g,\pi](h,\varpi)={}$ $\displaystyle
h\nabla^{2}h+(g\nabla g)(h\nabla h)+g^{2}\left((\nabla
h)^{2}+\varpi^{2}\right)+\left((\nabla g)^{2}+\pi^{2}\right)h^{2}$
$\displaystyle+g\pi h\varpi+gh(\nabla h)^{2}+h^{2}\nabla g\nabla
h+h^{2}\left((\nabla h)^{2}+\varpi^{2}\right)+\pi h^{2}\varpi,$ (2.6)
$\displaystyle Q_{\mathcal{M}}[g,\pi](h,\varpi)={}$ $\displaystyle
g^{2}h\nabla\varpi+(g\nabla g)h\varpi+g\pi h\nabla h+h^{2}\pi\nabla g$
$\displaystyle+gh\varpi\nabla h+h^{2}\varpi\nabla g+h^{2}\pi\nabla
h+h^{2}\varpi\nabla h.$ (2.7)

###### Proof.

See [15] for (2.4) and (2.5). For (2.6) and (2.7), we use the schematic
expressions

$\displaystyle{\mathcal{H}}(g,\pi)$ $\displaystyle=g\nabla^{2}g+(g\nabla
g)^{2}+(g\pi)^{2},$ $\displaystyle{\mathcal{M}}(g,\pi)$
$\displaystyle=g\left(\nabla\pi+g\pi\nabla g\right).$

Isolating nonlinear terms in these expressions concludes the proof of the
lemma. ∎

######  Definition 2.5.

The _constraint operator_ is defined to be

$\Phi(g,\pi)\vcentcolon=\left(-{\mathcal{H}}(g,\pi),2{\mathcal{M}}(g,\pi)\right).$
(2.8)

Its linearization around some given $(g,\pi)$ is denoted by $D\Phi[g,\pi]$ and
is given by

$D\Phi[g,\pi](h,\varpi)=\left(-D{\mathcal{H}}[g,\pi](h,\varpi),2D{\mathcal{M}}[g,\pi](h,\varpi)\right).$
(2.9)

Its formal adjoint $D\Phi[g,\pi]^{*}$ for the $L^{2}(\Sigma)$ scalar product
induced by the metric $g$ will play a major role in our construction since
elements of its kernel will be obstructions to solvability. From the formulas
(2.4), (2.5) and (2.8) we find that $(f,X)$ belongs to
$\ker(D\Phi[g,\pi]^{*})$ if and only if

$\displaystyle\mathrm{Hess}_{g}(f)$
$\displaystyle=f\left(Ric(g)_{ij}+\frac{3}{4}\pi_{ij}\mathrm{tr}_{g}\pi-2\pi_{ik}\pi_{j}^{k}+\frac{1}{4}|\pi|^{2}_{g}g_{ij}\right)$
$\displaystyle\quad+\pi_{kj}D_{i}X^{k}+\pi_{ki}D_{j}X^{k}+D_{X}\pi_{ij}-\frac{1}{2}X(\mathrm{tr}_{g}\pi)g_{ij},$
(2.10) $\displaystyle{\mathcal{L}}_{X}g_{ij}$
$\displaystyle=f\left(\pi_{ij}-\frac{1}{2}\mathrm{tr}_{g}\pi g_{ij}\right),$
(2.11)

where ${\mathcal{L}}_{X}g$ is the Lie derivative with respect to $X$.

##  3\. Statement of the main results

###  3.1. The main theorem

We now state the main result of this article, which handles any asymptotically
flat solution of the constraint equations with analyticity near and at the
boundary, see Remark 1.4. We will then apply it to Kerr in Section 3.2.

######  Theorem 3.1.

Let $(\Sigma,\bar{g},\bar{\pi})$ be an asymptotically flat initial data as
defined in Definition 2.1. Assume that there exists $r_{1}>r_{0}$ such that if
$\Omega\vcentcolon=\Sigma\cap\\{r_{0}\leq r<r_{1}\\}$, then the initial data
set $\left(\Omega,\bar{g}_{|_{\Omega}},\bar{\pi}_{|_{\Omega}}\right)$ is
analytic444By this we mean analytic in $\\{r_{0}<r<r_{1}\\}$ and on the sphere
$\\{r=r_{0}\\}$.. Let $q\in\mathbb{N}^{*}$ and $0<\delta<1$. There exists
$\varepsilon_{0}>0$ only depending on $(\Sigma,\bar{g},\bar{\pi})$, $q$ and
$\delta$ such that if $0<\varepsilon\leq\varepsilon_{0}$ then the following
holds: for all $(\widecheck{g},\widecheck{\pi})$ such that

$\displaystyle\left\|\widecheck{g}\right\|_{H^{2}_{-q-\delta}(\Sigma)}+\left\|\widecheck{\pi}\right\|_{H^{1}_{-q-\delta-1}(\Sigma)}\leq\varepsilon,$
(3.1)

there exists a solution $(g,\pi)$ to the constraint equations (2.1) on
$\Sigma$ of the form

$\displaystyle g$ $\displaystyle=u^{4}(\bar{g}+\widecheck{g}+\breve{g}),$
$\displaystyle\pi$
$\displaystyle=u^{2}(\bar{\pi}+\widecheck{\pi}+\breve{\pi}+L_{\bar{g}+\widecheck{g}+\breve{g}}X),$

where

$L_{g}X\vcentcolon={\mathcal{L}}_{X}g-(\mathrm{div}_{g}X)g,$ (3.2)

the scalar function $u$ and the vector field $X$ satisfy

$\displaystyle\left\|u-1\right\|_{H^{2}_{-q-\delta}(\Sigma)}+\left\|X\right\|_{H^{2}_{-q-\delta}(\Sigma)}\leq
C\varepsilon,$

and $(\breve{g},\breve{\pi})$ belong to a vector space of dimension $4q^{2}$
only depending on $(\Sigma,\bar{g},\bar{\pi})$ and composed of pairs of smooth
symmetric 2-tensors compactly supported in $\Sigma\cap\\{r_{0}<r<r_{1}\\}$ and
moreover

$\displaystyle\left\|\breve{g}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}\right\|_{W^{1,\infty}(\Sigma)}\leq
C\varepsilon,$

where $C>0$ only depends on $(\Sigma,\bar{g},\bar{\pi})$.

Theorem 3.1 will be proved in Section 5 relying on the properties of the
linearized operator exhibited in Section 4.

###  3.2. Spacelike initial data for Kerr stability

In this section we present the main application of Theorem 3.1, i.e. the
construction of initial data for the Kerr stability conjecture. Let $m$ and
$a$ be two real numbers representing the mass and angular momentum of the Kerr
black hole respectively. Let
$\mathcal{M}_{m,a}:={\mathbb{R}}\times(r_{+},+\infty)\times(0,\pi)\times\mathbb{S}^{1}$
with $r_{+}=m+\sqrt{m^{2}-a^{2}}$. Then, the Kerr metric in Boyer-Lindquist
coordinates $(t,r,\theta,\phi)$ is given by

${\bf
g}_{m,a}=-\frac{|q|^{2}\Delta}{\Sigma^{2}}\mathrm{d}t^{2}+\frac{\Sigma^{2}\sin^{2}\theta}{|q|^{2}}\left(\mathrm{d}\phi-\frac{2amr}{\Sigma^{2}}\mathrm{d}t\right)^{2}+\frac{|q|^{2}}{\Delta}\mathrm{d}r^{2}+|q|^{2}\mathrm{d}\theta^{2},$
(3.3)

where

$\displaystyle\Delta$ $\displaystyle\vcentcolon=r^{2}+a^{2}-2mr,\quad
q\vcentcolon=r+ia\cos\theta,\quad\Sigma^{2}\vcentcolon=(r^{2}+a^{2})^{2}-a^{2}\sin^{2}\theta\Delta,$

which is well-defined for555As usual, the coordinate singularities at the
north and south poles of the spheres can be resolved by considering for
example $(x_{\pm},y_{\pm})=(\sin\theta\cos\phi,\sin\theta\cos\phi)$ as
coordinates on neighborhoods of the north and south pole respectively.
$(t,r,\theta,\phi)\in\mathcal{M}_{m,a}$.

It is well known that the singularity of the Kerr metric at the event horizon
$\\{r=r_{+}\\}$ is a coordinate singularity, and that the metric can in fact
be smoothly extended past the event horizon. We can use this fact to construct
a spacelike hypersurface of Kerr that enters into the black hole interior.

######  Lemma 3.2.

There exist coordinates $\tau$ and $\varphi$ such that

  1. (i)

in the coordinates $(\tau,r,\theta,\varphi)$ the Kerr metric ${\bf g}_{m,a}$
is smooth on

$\displaystyle{\mathbb{R}}\times[r_{+}(1-\delta_{*}),+\infty)\times(0,\pi)\times\mathbb{S}^{1}$

and analytic in particular on

$\displaystyle{\mathbb{R}}\times[r_{+}(1-\delta_{*}),3m)\times(0,\pi)\times\mathbb{S}^{1}$

for any $0<\delta_{*}<1$,

  2. (ii)

the hypersurface

$\displaystyle\Sigma_{m,a}\vcentcolon=\\{\tau=0\\}\times[r_{+}(1-\delta_{*}),+\infty)\times(0,\pi)\times\mathbb{S}^{1}$

is uniformly spacelike.

###### Proof.

We recall the expression of the canonical ingoing principal null frame for the
metric ${\bf g}_{m,a}$, see for example Section 2.7.2 in [21]:

$\displaystyle e_{3}$
$\displaystyle\vcentcolon=\frac{r^{2}+a^{2}}{\Delta}\partial_{t}-\partial_{r}+\frac{a}{\Delta}\partial_{\phi},$
$\displaystyle e_{4}$
$\displaystyle\vcentcolon=\frac{r^{2}+a^{2}}{|q|^{2}}\partial_{t}+\frac{\Delta}{|q|^{2}}\partial_{r}+\frac{a}{|q|^{2}}\partial_{\phi},$
(3.4) $\displaystyle e_{1}$
$\displaystyle\vcentcolon=\frac{1}{|q|}\partial_{\theta},$ $\displaystyle
e_{2}$
$\displaystyle\vcentcolon=\frac{a\sin\theta}{|q|}\partial_{t}+\frac{1}{|q|\sin\theta}\partial_{\phi}.$

The new coordinates $\tau$ and $\varphi$ are defined in terms of the Boyer-
Lindquist coordinates by

$\tau=t+f(r),\quad\varphi=\phi+h(r),$ (3.5)

where

$\displaystyle h^{\prime}(r)$ $\displaystyle=\frac{a}{\Delta},\quad
f^{\prime}(r)=\chi(r)\frac{r^{2}+a^{2}}{\Delta}\sqrt{1-\frac{2m^{2}\Delta}{(r^{2}+a^{2})^{2}}},$

with $f(4m)=0$ and where $0\leq\chi\leq 1$ is smooth and such that $\chi=1$
for $r<3m$ and $\chi=0$ for $r>4m$. From (3.4) and (3.5) we first obtain

$\displaystyle e_{3}(r)=-1,\quad e_{4}(r)=\frac{\Delta}{|q|^{2}},\quad
e_{1}(r)=e_{2}(r)=0,$ $\displaystyle e_{1}(\theta)=\frac{1}{|q|},\quad
e_{3}(\theta)=e_{4}(\theta)=e_{2}(\theta)=0,$ $\displaystyle
e_{4}(\varphi)=\frac{2a}{|q|^{2}},\quad
e_{2}(\varphi)=\frac{1}{|q|\sin\theta},\quad e_{3}(\varphi)=e_{1}(\varphi)=0,$
$\displaystyle
e_{3}(\tau)=\frac{r^{2}+a^{2}}{\Delta}\left(1-\chi(r)\sqrt{1-\frac{2m^{2}\Delta}{(r^{2}+a^{2})^{2}}}\right),\quad
e_{4}(\tau)=\frac{r^{2}+a^{2}}{|q|^{2}}\left(1+\chi(r)\sqrt{1-\frac{2m^{2}\Delta}{(r^{2}+a^{2})^{2}}}\right),$
$\displaystyle e_{2}(\tau)=\frac{a\sin\theta}{|q|},\quad e_{1}(\tau)=0.$

Together with the formula for the inverse metric

$\displaystyle{\bf g}^{{\alpha}{\beta}}_{m,a}$
$\displaystyle=-\frac{1}{2}e_{3}(x^{\alpha})e_{4}(x^{\beta})-\frac{1}{2}e_{3}(x^{\beta})e_{4}(x^{\alpha})+e_{1}(x^{\alpha})e_{1}(x^{\beta})+e_{2}(x^{\alpha})e_{2}(x^{\beta})$

from Lemma 4.1.3 in [21], this allows us to obtain the expression of the
inverse Kerr metric in the new coordinates system $(\tau,r,\theta,\varphi)$:

$\displaystyle{\bf g}_{m,a}^{rr}$
$\displaystyle=\frac{\Delta}{|q|^{2}},\quad{\bf
g}_{m,a}^{r\varphi}=\frac{a}{|q|^{2}},\quad{\bf
g}_{m,a}^{\varphi\varphi}=\frac{1}{|q|^{2}\sin^{2}\theta},\quad{\bf
g}_{m,a}^{\theta\theta}=\frac{1}{|q|^{2}},$ $\displaystyle{\bf
g}_{m,a}^{\tau\tau}$
$\displaystyle=\frac{1}{|q|^{2}}\left((\chi^{2}-1)\frac{(r^{2}+a^{2})^{2}}{\Delta}-2m^{2}\chi^{2}+a^{2}\sin^{2}\theta\right),$
$\displaystyle{\bf g}_{m,a}^{\tau r}$
$\displaystyle=\frac{r^{2}+a^{2}}{|q|^{2}}\chi\sqrt{1-\frac{2m^{2}\Delta}{(r^{2}+a^{2})^{2}}},$
$\displaystyle{\bf g}_{m,a}^{\tau\varphi}$
$\displaystyle=\frac{a}{|q|^{2}}\left(1+\frac{r^{2}+a^{2}}{\Delta}\left(\chi\sqrt{1-\frac{2m^{2}\Delta}{(r^{2}+a^{2})^{2}}}-1\right)\right),$
$\displaystyle{\bf g}_{m,a}^{r\theta}$ $\displaystyle={\bf
g}_{m,a}^{\theta\varphi}={\bf g}_{m,a}^{\tau\theta}=0.$

We see on the above formulas that the inverse Kerr metric is smooth on the
manifold
${\mathbb{R}}\times[r_{+}(1-\delta_{*}),+\infty)\times(0,\pi)\times\mathbb{S}^{1}$.
Moreover, if $r<3m$ then $\chi=1$ and the above formulas show that the inverse
Kerr metric is actually analytic, which shows the first part of the lemma. We
also easily check that there exists a constant $c_{m,a}>0$ only depending on
$m$ and $a$ such that ${\bf g}_{m,a}^{\tau\tau}\leq-c_{m,a}$ which shows the
second part of the lemma. ∎

Lemma 3.2 implies that the initial data set $(\Sigma_{m,a},g_{m,a},\pi_{m,a})$
satisfies the assumptions of Theorem 3.1 and we immediately deduce the
following statement (which is the precise version of Theorem 1.2).

######  Theorem 3.3.

Let $0\leq|a|\leq m$ and $(\Sigma_{m,a},g_{m,a},\pi_{m,a})$ be the
hypersurface and initial data set constructed in Lemma 3.2. Let
$q\in\mathbb{N}^{*}$ and $0<\delta<1$. There exists
$\varepsilon_{0}=\varepsilon_{0}(m,a,q,\delta)>0$ such that if
$0<\varepsilon\leq\varepsilon_{0}$ then the following holds: for all
$(\widecheck{g},\widecheck{\pi})$ with

$\displaystyle\left\|\widecheck{g}\right\|_{H^{2}_{-q-\delta}(\Sigma_{m,a})}+\left\|\widecheck{\pi}\right\|_{H^{1}_{-q-\delta}(\Sigma_{m,a})}\leq\varepsilon,$

there exists a solution $(g,\pi)$ of the constraint equations (2.1) on
$\Sigma_{m,a}$ of the form

$\displaystyle g$
$\displaystyle=u^{4}\left(g_{m,a}+\widecheck{g}+\breve{g}\right),$
$\displaystyle\pi$
$\displaystyle=u^{2}\left(\pi_{m,a}+\widecheck{\pi}+\breve{\pi}+L_{g_{m,a}+\widecheck{g}+\breve{g}}X\right),$

where the correctors $(\breve{g},\breve{\pi})$ belong to a vector space of
dimension $4q^{2}$ only depending on $(m,a)$ and composed of pairs of smooth
symmetric 2-tensors compactly supported and where $u$, $X$ and
$(\breve{g},\breve{\pi})$ satisfy

$\displaystyle\left\|u-1\right\|_{H^{2}_{-q-\delta}(\Sigma_{m,a})}+\left\|X\right\|_{H^{2}_{-q-\delta}(\Sigma_{m,a})}+\left\|\breve{g}\right\|_{W^{2,\infty}(\Sigma_{m,a})}+\left\|\breve{\pi}\right\|_{W^{1,\infty}(\Sigma_{m,a})}\lesssim\varepsilon.$

######  Remark 3.4.

We remark that Theorem 3.3 is true even in the extremal case where $|a|=m$.
However, it is unclear if the data we construct for $|a|=m$ are relevant to
black hole stability, in particular seeing as extremal black holes are known
to exhibit certain horizon linear instabilities [1, 16].

##  4\. Properties of the linearized operator

Our first task is to develop a spectral theoretic framework for the following
elliptic operator on $\Sigma$ acting on pairs $\mathbb{U}=(u,X)$ composed of a
scalar function $u$ and a vector field $X$:

$\displaystyle P(\mathbb{U})$
$\displaystyle\vcentcolon=\left(-8\Delta_{\bar{g}}u-2\bar{\pi}^{ij}{\mathcal{L}}_{X}\bar{g}_{ij}+\mathrm{tr}_{\bar{g}}\bar{\pi}\mathrm{div}_{\bar{g}}X,-2\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}-8\bar{g}^{j\ell}\bar{\pi}_{i\ell}\partial_{j}u+4\mathrm{tr}_{\bar{g}}\bar{\pi}\partial_{i}u\right).$
(4.1)

This operator will naturally appear after we formulate the constraint
equations following the conformal ansatz proposed in [11], see Section 5.1.
For any $\mathbb{U}=(u,X)$, we will also define666Directly integrating by
parts $\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}$ would lead to the boundary term
$L_{\bar{g}}X(\nu,\cdot)$. Here we instead first expand
$\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}=\Delta^{(1)}_{\bar{g}}X_{i}+Ric(\bar{g})(\partial_{i},X)$
and then integrate by parts to obtain the boundary term $D_{\nu}X$.

$\displaystyle P^{*}(\mathbb{U})$
$\displaystyle\vcentcolon=\left(-8\Delta_{\bar{g}}u+4\bar{\pi}^{ij}{\mathcal{L}}_{X}\bar{g}_{ij}-4\mathrm{div}_{\bar{g}}(\mathrm{tr}_{\bar{g}}\bar{\pi}X),-2\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}+4\bar{\pi}_{i}^{j}\partial_{j}u-\partial_{i}(\mathrm{tr}_{\bar{g}}\bar{\pi}u)\right),$
$\displaystyle B_{\nu}(\mathbb{U})$
$\displaystyle\vcentcolon=\left(-8\partial_{\nu}u+\mathrm{tr}_{\bar{g}}\bar{\pi}\bar{g}(X,\nu)-4\bar{\pi}(X,\nu),-2D_{\nu}X\right),$
$\displaystyle B_{\nu}^{*}(\mathbb{U})$
$\displaystyle\vcentcolon=\left(-8\partial_{\nu}u-4\mathrm{tr}_{\bar{g}}\bar{\pi}\bar{g}(X,\nu)+8\bar{\pi}(X,\nu),-2D_{\nu}X\right),$

with $D$ being the covariant derivative associated with $\bar{g}$ so that

$\displaystyle\int_{\Sigma}\left(P(\mathbb{U})\cdot\mathbb{V}-\mathbb{U}\cdot
P^{*}(\mathbb{V})\right)+\int_{\partial\Sigma}\left(B_{\nu}(\mathbb{U})\cdot\mathbb{V}-\mathbb{U}\cdot
B_{\nu}^{*}(\mathbb{V})\right)=0,$ (4.2)

where $\mathbb{U}\cdot\mathbb{V}=uv+\bar{g}(X,Y)$ for $\mathbb{U}=(u,X)$ and
$\mathbb{V}=(v,Y)$, and where $\nu$ denotes the inward pointing unit normal to
$\partial\Sigma$ with respect to $\bar{g}$.

We will consider perturbations of the boundary operators $B_{\nu}$ and
$B_{\nu}^{*}$. For $b$ a scalar function, $\omega$ a 1-form, $Z$ a vector
field and $\xi$ a $(1,1)$-tensor, we define $F=(b,\omega,Z,\xi)$ so that $F$
acts on the space of pairs $\mathbb{U}=(u,X)$ of functions and vector fields
through the formula

$\displaystyle F(\mathbb{U})=\left(bu+\omega(X),uZ+\xi(X)\right).$

We define the space of such smooth $F$ restricted to $\partial\Sigma$ by
${\mathcal{E}}(\partial\Sigma)$. If $F\in{\mathcal{E}}(\partial\Sigma)$ we
define its transpose $F^{*}\in{\mathcal{E}}(\partial\Sigma)$ by the formula
$F^{*}=\left(b,Z^{\flat},\omega^{\sharp},\xi^{\natural}\right)$, where
$\xi^{\natural}$ is defined in coordinates by
$(\xi^{\natural})_{i}^{\;\;j}=\bar{g}^{j\ell}\xi_{\ell}^{\;\;k}\bar{g}_{ki}$.
This definition is such that

$\displaystyle F(\mathbb{U})\cdot\mathbb{V}=\mathbb{U}\cdot
F^{*}(\mathbb{V}).$ (4.3)

Also, we define $\mathrm{Id}\in{\mathcal{E}}(\partial\Sigma)$ by
$\mathrm{Id}=(1,0,0,\delta)$ so that $\mathrm{Id}(\mathbb{U})=\mathbb{U}$.

######  Definition 4.1.

For $F\in{\mathcal{E}}(\partial\Sigma)$ and $\delta\in{\mathbb{R}}$ we define

$\displaystyle P_{F,\delta}$
$\displaystyle\vcentcolon=\left((P,B_{\nu}+F):H^{2}_{\delta}(\Sigma)\longrightarrow
L^{2}_{\delta-2}(\Sigma)\times H^{\frac{1}{2}}(\partial\Sigma)\right),$ (4.4)
$\displaystyle P^{*}_{F,\delta}$
$\displaystyle\vcentcolon=\left((P^{*},B^{*}_{\nu}+F^{*}):H^{2}_{\delta}(\Sigma)\longrightarrow
L^{2}_{\delta-2}(\Sigma)\times H^{\frac{1}{2}}(\partial\Sigma)\right).$ (4.5)

Note that (4.2) and (4.3) imply

$\left\langle
P(\mathbb{U}),\mathbb{V}\right\rangle_{L^{2}(\Sigma)}+\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbb{V}\right\rangle_{L^{2}(\partial\Sigma)}=\left\langle\mathbb{U},P^{*}(\mathbb{V})\right\rangle_{L^{2}(\Sigma)}+\left\langle\mathbb{U},(B_{\nu}^{*}+F^{*})(\mathbb{V})\right\rangle_{L^{2}(\partial\Sigma)}.$
(4.6)

###  4.1. Basic properties

We gather in the following proposition general facts about the operators (4.4)
and (4.5), which follow from $P_{f,\delta}$ and $P^{*}_{f,\delta}$ being
compact perturbations of the Laplace-Beltrami operator (acting on both scalar
functions and vector fields) with Neumann boundary conditions.

######  Proposition 4.2.

If $\delta\notin\mathbb{Z}$ and $F\in{\mathcal{E}}(\partial\Sigma)$, then:

  1. (i)

There exists $C_{F,\delta}>0$ such that for all $\mathbb{U}\in
H^{2}_{\delta}(\Sigma)$ we have

$\begin{split}\left\|\mathbb{U}\right\|_{H^{2}_{\delta}(\Sigma)}&\leq
C_{F,\delta}\left(\left\|P(\mathbb{U})\right\|_{L^{2}_{\delta-2}(\Sigma)}+\left\|(B_{\nu}+F)(\mathbb{U})\right\|_{H^{\frac{1}{2}}(\partial\Sigma)}+\left\|\mathbb{U}\right\|_{L^{2}_{\delta}(\Sigma)}\right),\\\
\left\|\mathbb{U}\right\|_{H^{2}_{\delta}(\Sigma)}&\leq
C_{F,\delta}\left(\left\|P^{*}(\mathbb{U})\right\|_{L^{2}_{\delta-2}(\Sigma)}+\left\|(B^{*}_{\nu}+F^{*})(\mathbb{U})\right\|_{H^{\frac{1}{2}}(\partial\Sigma)}+\left\|\mathbb{U}\right\|_{L^{2}_{\delta}(\Sigma)}\right).\end{split}$
(4.7)

  2. (ii)

The operator $P_{F,\delta}$ and $P^{*}_{F,\delta}$ are Fredholm and if
$-1<\delta<0$ their index is zero.

###### Proof.

We consider the rescaled Neumann Laplace-Beltrami operator of $\overline{g}$
given by

$\begin{split}\Delta_{N}:H^{2}_{\delta}(\Sigma)&\longrightarrow
L^{2}_{\delta-2}(\Sigma)\times H^{\frac{1}{2}}(\partial\Sigma),\\\
(u,X)&\longmapsto\left(\left(-8\Delta_{\bar{g}}u,-2\Delta_{\bar{g}}^{(1)}X\right),(-8\partial_{\nu}u,-2D_{\nu}X)\right),\end{split}$

where $\Delta_{\bar{g}}^{(1)}$ is the vectorial Laplace-Beltrami operators.
Since $\Delta_{\bar{g}}^{(1)}$ is a compact perturbation of $\Delta_{\bar{g}}$
acting on components of vector fields in a given coordinate system, we deduce
from Proposition 4.13 in [24] the following estimate

$\displaystyle\left\|(u,X)\right\|_{H^{2}_{\delta}(\Sigma)}$
$\displaystyle\lesssim\left\|\Delta_{N}(u,X)\right\|_{L^{2}_{\delta-2}(\Sigma)\times
H^{\frac{1}{2}}(\partial\Sigma)}+\left\|(u,X)\right\|_{L^{2}_{\delta}(\Sigma)}.$
(4.8)

Since both $P_{F,\delta}$ and $P^{*}_{F,\delta}$ are compact perturbations of
$\Delta_{N}$, we deduce (4.7) from (4.8) using interpolation to control lower
order terms. Using again the fact that $P_{F,\delta}$ and $P^{*}_{F,\delta}$
are compact perturbations of $\Delta_{N}$, the second part of the lemma
follows from the fact that $\Delta_{N}$ is Fredholm if
$\delta\notin\mathbb{Z}$ and is of index 0 if $-1<\delta<0$, see Propositions
1 and 6 in [25]. ∎

The adjoint $(P_{F,\delta})^{*}$ of $P_{F,\delta}$ is defined on the space
$L^{2}_{-1-\delta}(\Sigma)\times H^{-\frac{1}{2}}(\partial\Sigma)$ by the
relation

$\displaystyle\left\langle
P(\mathbb{U}),\mathbb{V}\right\rangle_{L^{2}(\Sigma)}+\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbbm{h}\right\rangle_{L^{2}(\partial\Sigma)}$
$\displaystyle=\left\langle\mathbb{U},(P_{F,\delta})^{*}(\mathbb{V},\mathbbm{h})\right\rangle_{L^{2}(\Sigma)}.$
(4.9)

The adjoint $\left(P^{*}_{F,\delta}\right)^{*}$ of $P^{*}_{F,\delta}$ is
defined similarly. In order to characterize the mapping properties of
$P_{F,\delta}$ and $P_{F,\delta}^{*}$, we will use the following lemma that
characterizes the kernels of their adjoints.

######  Lemma 4.3.

If $\delta\notin\mathbb{Z}$ and $F\in{\mathcal{E}}(\partial\Sigma)$ then

$\displaystyle\ker\left(\left(P_{F,\delta}\right)^{*}\right)$
$\displaystyle=\left\\{\left(\mathbb{V},\mathbb{V}_{|_{\partial\Sigma}}\right)~{}\middle|~{}\mathbb{V}\in\ker\left(P^{*}_{F,-1-\delta}\right)\right\\},$
(4.10) $\displaystyle\ker\left(\left(P^{*}_{F,\delta}\right)^{*}\right)$
$\displaystyle=\left\\{\left(\mathbb{V},\mathbb{V}_{|_{\partial\Sigma}}\right)~{}\middle|~{}\mathbb{V}\in\ker\left(P_{F,-1-\delta}\right)\right\\}.$
(4.11)

###### Proof.

We will only prove (4.10) since (4.11) follows by a similar argument. Let
$(\mathbb{V},\mathbbm{h})\in L^{2}_{-1-\delta}(\Sigma)\times
H^{-\frac{1}{2}}(\partial\Sigma)$. From (4.6) and (4.9), we have that

$\left\langle\mathbb{U},(P_{F,\delta})^{*}(\mathbb{V},\mathbbm{h})\right\rangle_{L^{2}(\Sigma)}=\left\langle\mathbb{U},P^{*}(\mathbb{V})\right\rangle_{L^{2}(\Sigma)}+\left\langle\mathbb{U},(B_{\nu}^{*}+F^{*})(\mathbb{V})\right\rangle_{L^{2}(\partial\Sigma)}+\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbbm{h}-\mathbb{V}\right\rangle_{L^{2}(\partial\Sigma)}$

for all $\mathbb{U}\in H^{2}_{\delta}(\Sigma)$. Then it is immediately clear
that

$\left\\{\left(\mathbb{V},\mathbb{V}_{|_{\partial\Sigma}}\right)~{}\middle|~{}\mathbb{V}\in\ker\left(P^{*}_{F,-1-\delta}\right)\right\\}\subset\ker\left(\left(P_{F,\delta}\right)^{*}\right).$

We now prove the reverse inclusion. If
$\left(P_{F,\delta}\right)^{*}(\mathbb{V},\mathbbm{h})=0$, then

$\left\langle\mathbb{U},P^{*}(\mathbb{V})\right\rangle_{L^{2}(\Sigma)}+\left\langle\mathbb{U},(B_{\nu}^{*}+F)(\mathbb{V})\right\rangle_{L^{2}(\partial\Sigma)}+\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbbm{h}-\mathbb{V}\right\rangle_{L^{2}(\partial\Sigma)}=0$
(4.12)

for all $\mathbb{U}\in H^{2}_{\delta}(\Sigma)$. Taking $\mathbb{U}$ compactly
supported away from $\partial\Sigma$, the boundary terms disappear and we get
$P^{*}(\mathbb{V})=0$ on $\Sigma$. Since $P^{*}(\mathbb{V})=0$, then (4.12)
becomes

$\left\langle\mathbb{U},(B_{\nu}^{*}+F^{*})(\mathbb{V})\right\rangle_{L^{2}(\partial\Sigma)}+\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbbm{h}-\mathbb{V}\right\rangle_{L^{2}(\partial\Sigma)}=0,$

and this should hold for every $\mathbb{U}\in H^{2}_{\delta}(\Sigma)$. In
particular, it holds for every $\mathbb{U}\in H^{2}_{\delta}(\Sigma)$ such
that $(B_{\nu}+F)(\mathbb{U})=0$ on $\partial\Sigma$, thus implying that
$(B_{\nu}^{*}+F^{*})(\mathbb{V})=0$ on $\partial\Sigma$. Thus, using the
elliptic regularity estimate (4.7) we infer from $P^{*}(\mathbb{V})=0$ and
$(B_{\nu}^{*}+F^{*})(\mathbb{V})=0$ that $\mathbb{V}\in
H^{2}_{-1-\delta}(\Sigma)$. Therefore, for every $\mathbb{U}\in
H^{2}_{\delta}(\Sigma)$ we have

$\left\langle(B_{\nu}+F)(\mathbb{U}),\mathbbm{h}-\mathbb{V}\right\rangle_{L^{2}(\partial\Sigma)}=0,$

which implies $\mathbbm{h}=\mathbb{V}$ on $\partial\Sigma$. Thus we have shown
that

$\displaystyle\ker\left(\left(P_{F,\delta}\right)^{*}\right)\subset\left\\{\left(\mathbb{V},\mathbb{V}_{|_{\partial\Sigma}}\right)~{}\middle|~{}\mathbb{V}\in\ker\left(P^{*}_{F,-1-\delta}\right)\right\\}.$

This concludes the proof of (4.10). ∎

In the next lemma, we give a unique continuation result from the boundary for
$P$ and $P^{*}$.

######  Lemma 4.4.

Let $\Gamma$ be an open subset of $\partial\Sigma$. If $\mathbb{U}\in
H^{2}_{loc}(\Sigma)$ satisfies $P(\mathbb{U})=0$ (resp. $P^{*}(\mathbb{U})=0$)
on $\Sigma$ and $B_{\nu}(\mathbb{U})=\mathbb{U}=0$ (resp.
$B^{*}_{\nu}(\mathbb{U})=\mathbb{U}=0$) on $\Gamma$, then $\mathbb{U}=0$ on
$\Sigma$.

###### Proof.

The proof follows along the same lines as the proof of Theorem 9.8 in [22],
which is the analogue of Lemma 4.4 in the scalar case. Since $P$ and $P^{*}$
are diagonal operator at leading order, we can sum the four Carleman’s
estimates given by the proof of Theorem 9.8 in [22] and absorb the lower order
coupling terms between the four components of $\mathbb{U}$. ∎

###  4.2. A special boundary condition

In this section, we will construct a distinguished boundary condition so that
$P_{F,\delta}$ as defined in (4.1) has good mapping properties for any fixed
$\delta\in(0,1)$.

######  Lemma 4.5.

Let $F_{0}\in{\mathcal{E}}(\partial\Sigma)$. There exists $\eta(F_{0})>0$ only
depending on $F_{0}$ such that for all $F\in{\mathcal{E}}(\partial\Sigma)$ on
$\partial\Sigma$,

$\displaystyle\left\|F-F_{0}\right\|_{H^{\frac{1}{2}}(\partial\Sigma)}\leq\eta(F_{0})\Longrightarrow\dim(\ker(P_{F,-\delta}))\leq\dim(\ker(P_{F_{0},-\delta})).$

###### Proof.

Lemma 4.5 follows immediately from Proposition 1.11 in [2] and the fact that

$\displaystyle\left\|P_{F,-\delta}-P_{F_{0},-\delta}\right\|_{\mathrm{op}}\lesssim\left\|F-F_{0}\right\|_{H^{\frac{1}{2}}(\partial\Sigma)},$

where $\left\lVert\cdot\right\rVert_{\mathrm{op}}$ refers to the standard
operator norm. ∎

In the next lemma, we show how to lower the dimension of the kernel in a
constructive way.

######  Lemma 4.6.

Let $\mathcal{W}$ be an open subset of $\partial\Sigma$ such that
$\partial\Sigma\setminus\mathcal{W}$ contains an open set. Let
$F_{0}\in{\mathcal{E}}(\partial\Sigma)$ such that
$\dim(\ker(P_{F_{0},-\delta}))>0$. Then there exists
$F_{1}\in{\mathcal{E}}(\partial\Sigma)$ such that

$\dim(\ker(P_{F_{1},-\delta}))<\dim(\ker(P_{F_{0},-\delta})),\quad\text{and}\quad(F_{1}-F_{0})_{|_{\mathcal{W}}}=0.$
(4.13)

###### Proof.

Using Proposition 4.2, the Fredholm index of $P_{F_{0},-\delta}$ is zero for
$\delta\in(0,1)$. Thus, the assumption that $\dim(\ker(P_{F_{0},-\delta}))>0$
implies that $\dim\left(\ker\left((P_{F_{0},-\delta})^{*}\right)\right)>0$.
According to (4.10) this implies that
$\dim\left(\ker\left(P_{F_{0},-1+\delta}^{*}\right)\right)>0$. Now consider
$(\mathbb{V}_{0},\mathbb{V}_{0}^{*})\in\ker(P_{F_{0},-\delta})\times\ker(P_{F_{0},-1+\delta}^{*})$
such that $\mathbb{V}_{0},\mathbb{V}_{0}^{*}$ are both non-zero. Since they
satisfy $(P,B_{\nu}+F_{0})(\mathbb{V}_{0})=0$ and
$(P^{*},B^{*}_{\nu}+F_{0}^{*})(\mathbb{V}_{0}^{*})=0$, unique continuation for
the operators $P$ and $P^{*}$ (recall Lemma 4.4) implies that

$\displaystyle\\{\mathbb{V}_{0}\neq 0\\}\cap\\{\mathbb{V}_{0}^{*}\neq
0\\}\cap\partial\Sigma$

is a dense open subset of $\partial\Sigma$. Thus, there exists
$H_{0}\in{\mathcal{E}}(\partial\Sigma)$ such that $H_{0|_{\mathcal{W}}}=0$ and

$\displaystyle\int_{\partial\Sigma}H_{0}(\mathbb{V}_{0})\cdot\mathbb{V}_{0}^{*}=1.$
(4.14)

For $\eta>0$, we define

$\displaystyle
F_{\eta}\vcentcolon=F_{0}+\eta\frac{H_{0}}{\left\|H_{0}\right\|_{H^{\frac{3}{2}}(\partial\Sigma)}}.$

We will show that there exists some $\eta$ sufficiently small such that
$\dim\left(\ker(P_{F_{\eta},-\delta})\right)<\dim\left(\ker(P_{F_{0},-\delta})\right)$.
For $\eta\leq\eta(F_{0})$, Lemma 4.5 already implies that
$\dim(\ker(P_{F_{\eta},-\delta}))\leq\dim(\ker(P_{F_{0},-\delta}))$. Now
assume for the sake of contradiction that
$\dim(\ker(P_{F_{\eta},-\delta}))=\dim(\ker(P_{F_{0},-\delta}))$. From
Proposition 1.12 in [2] there exists $0<\eta^{\prime}(F_{0})\leq\eta(F_{0})$
and $C(F_{0})>0$ such that if $\eta\leq\eta^{\prime}(F_{0})$ then

$\displaystyle\left\|\mathbb{U}-\ker(P_{F_{\eta},-\delta})\right\|_{H^{2}_{-\delta}(\Sigma)}\leq
C(F_{0})\left(\left\|P(\mathbb{U})\right\|_{L^{2}_{-\delta-2}(\Sigma)}+\left\|(B_{\nu}+F_{\eta})(\mathbb{U})\right\|_{H^{\frac{1}{2}}(\partial\Sigma)}\right),$

for all $\mathbb{U}\in H^{2}_{-\delta}(\Sigma)$. We apply this to
$\mathbb{V}_{0}\in\ker(P_{F_{0},-\delta})$ as chosen above. Since
$\ker(P_{F_{\eta},-\delta})$ is finite-dimensional, there exists some
$\mathbb{U}_{\eta}\in\ker(P_{F_{\eta},-\delta})$ such that

$\displaystyle\left\|\mathbb{V}_{0}-\mathbb{U}_{\eta}\right\|_{H^{2}_{-\delta}(\Sigma)}\lesssim
C(F_{0})\left\|\mathbb{V}_{0}\right\|_{H^{2}_{-\delta}(\Sigma)}\eta,$ (4.15)

where we have used the continuous embeddings
$H^{2}_{-\delta}(\Sigma)\xhookrightarrow{}H^{\frac{3}{2}}(\partial\Sigma)\xhookrightarrow{}L^{\infty}(\partial\Sigma)$.
Then using (4.2), the fact that
$\mathbb{V}_{0}^{*}\in\ker(P^{*}_{F_{0},-1+\delta})$, and that
$\mathbb{U}_{\eta}\in\ker(P_{F_{\eta},-\delta})$, we compute

$\displaystyle\frac{\left\|H_{0}\right\|_{H^{\frac{3}{2}}(\partial\Sigma)}}{\eta}\int_{\Sigma}\left(P(\mathbb{U}_{\eta})\cdot\mathbb{V}_{0}^{*}-\mathbb{U}_{\eta}\cdot
P^{*}(\mathbb{V}_{0}^{*})\right)=-\int_{\partial\Sigma}H_{0}(\mathbb{U}_{\eta})\cdot\mathbb{V}_{0}^{*}.$

Since $P(\mathbb{U}_{\eta})=P^{*}(\mathbb{V}_{0}^{*})=0$ we have proved that

$\displaystyle\int_{\partial\Sigma}H_{0}(\mathbb{U}_{\eta})\cdot\mathbb{V}_{0}^{*}=0.$

However (4.14) and (4.15) imply

$\displaystyle\left|\int_{\partial\Sigma}H_{0}(\mathbb{U}_{\eta})\cdot\mathbb{V}_{0}^{*}-1\right|$
$\displaystyle\leq
CC(F_{0})\left\|H_{0}\right\|_{H^{\frac{3}{2}}(\partial\Sigma)}\left\|\mathbb{V}_{0}^{*}\right\|_{H^{2}_{-\delta}(\Sigma)}\left\|\mathbb{V}_{0}\right\|_{H^{2}_{-\delta}(\Sigma)}\eta,$

for some universal constant $C>0$. Choosing

$\displaystyle\eta_{1}(F_{0})\vcentcolon=\min\left(\eta^{\prime}(F_{0}),\frac{1}{2CC(F_{0})\left\|H_{0}\right\|_{H^{\frac{3}{2}}(\partial\Sigma)}\left\|\mathbb{V}_{0}^{*}\right\|_{H^{2}_{-\delta}(\Sigma)}\left\|\mathbb{V}_{0}\right\|_{H^{2}_{-\delta}(\Sigma)}}\right)$

and taking $0<\eta\leq\eta_{1}(F_{0})$ implies a contradiction. This proves
that $\dim(\ker(P_{F_{\eta},-\delta}))<\dim(\ker(P_{F_{0},-\delta}))$ and
concludes the proof of the lemma, after setting $F_{1}\vcentcolon=F_{\eta}$. ∎

We can iterate the result of Lemma 4.6 to obtain the following corollary.

######  Corollary 4.7.

Let $\mathcal{U}$ and $\mathcal{V}$ two open subsets of $\partial\Sigma$ with
$\overline{\mathcal{U}}\cap\overline{\mathcal{V}}=\emptyset$. There exists
$F_{\delta}\in{\mathcal{E}}(\partial\Sigma)$ such that
$F_{\delta|_{\mathcal{V}}}=\mathrm{Id}$, $F_{\delta|_{\mathcal{U}}}=0$ and

$\displaystyle\ker(P_{F_{\delta},-\delta})$ $\displaystyle=\\{0\\},$ (4.16)
$\displaystyle\ker(P^{*}_{F_{\delta},-1+\delta})$ $\displaystyle=\\{0\\}.$
(4.17)

###### Proof.

First, note that (4.16) implies (4.17). Indeed, from (4.10), we have that
$\dim\left(\ker\left(P^{*}_{F_{\delta},-1+\delta}\right)\right)=\dim\left(\ker\left(\left(P_{F_{\delta},-\delta}\right)^{*}\right)\right)$.
In view of the fact that $P_{F_{\delta},-\delta}$ has Fredholm index zero from
Proposition 4.2 this becomes

$\displaystyle\dim\left(\ker\left(P^{*}_{F_{\delta},-1+\delta}\right)\right)=\dim\left(\ker\left(P_{F_{\delta},-\delta}\right)\right).$

Therefore, it only remains to construct $F_{\delta}$ satisfying the
requirements of the corollary and such that (4.16) holds. Since
$\overline{\mathcal{U}}\cap\overline{\mathcal{V}}=\emptyset$, we may consider
$\chi_{b}$ a smooth function on $\partial\Sigma$ such that
$\chi_{b|_{\mathcal{U}}}=0$ and $\chi_{b|_{\mathcal{V}}}=1$ and set
$F_{0}\vcentcolon=\chi_{b}\mathrm{Id}$. Let
$n_{0}\vcentcolon=\dim(\ker(P_{F_{0},-\delta}))$. If $n_{0}=0$, the lemma is
proved. If $n_{0}>0$, we construct iteratively a finite family of matrices
$(F_{0},F_{1},\dots,F_{n_{*}})$ in the following way:

  1. (1)

We apply Lemma 4.6 to $F_{0}$ and
$\mathcal{W}\vcentcolon=\mathcal{U}\cup\mathcal{V}$ (since
$\overline{\mathcal{U}}\cap\overline{\mathcal{V}}=\emptyset$,
$\partial\Sigma\setminus\mathcal{W}$ does indeed contain an open set), which
gives the existence of $F_{1}\in{\mathcal{E}}(\partial\Sigma)$ with
$\dim(\ker(P_{F_{1},-\delta}))<n_{0}$ and $F_{1|_{\mathcal{W}}}=0$.

  2. (2)

If $(F_{0},F_{1},\dots,F_{j})$ is constructed, then we distinguish two cases:

    1. (a)

If $\dim(\ker(P_{F_{j},-\delta}))=0$, then set $n_{*}=j$ and the iteration
ends.

    2. (b)

Otherwise, $\dim(\ker(P_{F_{j},-\delta}))>0$, and Lemma 4.6 again implies the
existence of $F_{j+1}\in{\mathcal{E}}(\partial\Sigma)$ such that
$\dim(\ker(P_{F_{j+1},-\delta}))<\dim(\ker(P_{F_{j},-\delta}))$ and
$(F_{j+1}-F_{j})_{|_{\mathcal{W}}}=0$.

Since $\left(\dim(\ker(P_{F_{j},-\delta}))\right)_{j}$ is a decreasing
sequence of integers and since $\dim(\ker(P_{F_{0},-\delta}))=n_{0}$, we have
that $n_{*}\leq n_{0}$. We thus have a finite family
$(F_{0},F_{1},\dots,F_{n_{*}})$ in ${\mathcal{E}}(\partial\Sigma)$ with
$\ker(P_{F_{n_{*}},-\delta})=\\{0\\}$. Moreover, we can prove by induction
that $(F_{n_{*}}-F_{0})_{|_{\mathcal{U}\cup\mathcal{V}}}=0$. Since
$F_{0|_{\mathcal{V}}}=\mathrm{Id}$ and $F_{0|_{\mathcal{U}}}=0$, this proves
that $F_{\delta}\vcentcolon=F_{n_{*}}$ does satisfy the requirements of the
lemma. ∎

###  4.3. Adapted harmonic polynomials

We recall the definition of harmonic polynomials and vectorial harmonic
polynomials in Euclidean space.

######  Definition 4.8.

We identify the global coordinates of ${\mathbb{R}}^{3}$ and $\Sigma$ and
consider the real spherical harmonics $Y_{j,\ell}$ for $j\geq 0$ and
$-j\leq\ell\leq j$ with an arbitrary normalization.

  1. (i)

For $j\geq 1$ and $-(j-1)\leq\ell\leq j-1$, we define scalar functions

$\displaystyle w_{j,\ell}$ $\displaystyle\vcentcolon=r^{j-1}Y_{j-1,\ell},$

and if in addition $k=1,2,3$, we define vector fields

$\displaystyle W_{j,\ell,k}$
$\displaystyle\vcentcolon=w_{j,\ell}\partial_{k}.$

  2. (ii)

For $j\geq 1$, $-j\leq\ell\leq j$ and ${\alpha}=0,1,2,3$, we define pairs
consisting of a scalar function and a vector field

$\mathbb{W}_{j,\ell,{\alpha}}^{(e)}\vcentcolon=\left\\{\begin{aligned}
&(w_{j,\ell},0),\quad\text{if ${\alpha}=0$,}\\\
&(0,W_{j,\ell,{\alpha}}),\quad\text{if ${\alpha}=1,2,3$,}\end{aligned}\right.$

######  Remark 4.9.

Note that for each $q\in\mathbb{N}^{*}$ the family

$\displaystyle\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)_{1\leq j\leq
q,-(j-1)\leq\ell\leq j-1,{\alpha}=0,1,2,3}$

is a basis of homogeneous polynomials of degree $1\leq j\leq q$ moreover
satisfying $\Delta_{e}\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)=0$ where
$\Delta_{e}$ denotes here the Euclidean Laplacian.

Using the harmonic polynomials of Definition 4.8, we construct a family of
harmonic polynomials adapted to $P_{F,\delta}$.

######  Lemma 4.10.

Let $j\geq 1$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$. There exists
$\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}\in H^{2}_{j-2+\delta}(\Sigma)$ such
that

$\displaystyle\mathbb{W}_{j,\ell,{\alpha}}$
$\displaystyle\vcentcolon=\mathbb{W}^{(e)}_{j,\ell,{\alpha}}+\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}$

satisfies

$\left\\{\begin{aligned}
P^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right)&=0,\quad\text{on $\Sigma$,}\\\
(B_{\nu}^{*}+F_{\delta}^{*})\left(\mathbb{W}_{j,\ell,{\alpha}}\right)&=0,\quad\text{on
$\partial\Sigma$.}\end{aligned}\right.$

###### Proof.

It suffices to solve for $\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}\in
H^{2}_{j-2+\delta}(\Sigma)$ such that

$\left\\{\begin{aligned}
P^{*}\left(\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}\right)&=-P^{*}\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right),\quad\text{on
$\Sigma$,}\\\
(B_{\nu}^{*}+F_{\delta}^{*})\left(\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}\right)&=-(B_{\nu}^{*}+F_{\delta}^{*})\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right),\quad\text{on
$\partial\Sigma$.}\end{aligned}\right.$

Now, observe that $P^{*}$ is asymptotic to the Euclidean Laplacian in view of
Definition 2.1. As a result, since $\mathbb{W}_{j,\ell,\alpha}^{(e)}\in
H^{4}_{j-1+\delta}(\Sigma)$, we have that

$P^{*}\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)\in
H^{2}_{j-4+\delta}(\Sigma),$

since $\Delta_{e}\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)=0$ in view of
Remark 4.9. We can thus rewrite the above system for
$\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}$ as

$\displaystyle
P_{F_{\delta},j-2+\delta}^{*}\left(\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}\right)=-\left(P^{*}\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right),(B^{*}_{\nu}+F_{\delta}^{*})\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)\right).$

Since $j\geq 1$ we have
$\ker(P_{F_{\delta},1-j-\delta})\subset\ker(P_{F_{\delta},-\delta})=\\{0\\}$,
where we used (4.16). According to (4.11) this implies
$\ker\left(\left(P^{*}_{F_{\delta},j-2+\delta}\right)^{*}\right)=\\{0\\}$, and
hence that $P^{*}_{F_{\delta},j-2+\delta}$ is surjective. Therefore we can
always solve for $\widetilde{\mathbb{W}}_{j,\ell,{\alpha}}$. ∎

Using the adapted harmonic polynomials constructed in Lemma 4.10, we are now
able to specify the orthogonality conditions necessary to invert
$P_{F_{\delta},-q-\delta}$.

######  Lemma 4.11.

Let $q\in\mathbb{N}^{*}$, $0<\delta<1$ and let $\mathbb{Y}\in
L^{2}_{-q-\delta-2}(\Sigma)$. Then
$(\mathbb{Y},0)\in\mathrm{im}(P_{F_{\delta},-q-\delta})$ if and only if

$\displaystyle\left\langle\mathbb{W}_{j,\ell,{\alpha}},\mathbb{Y}\right\rangle_{L^{2}(\Sigma)}=0,\quad\text{for
all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}.$

###### Proof.

We start by proving that

$\displaystyle\ker\left(P^{*}_{F_{\delta},q-1+\delta}\right)=\mathrm{span}\big{(}\mathbb{W}_{j,\ell,{\alpha}},\text{$1\leq
j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}\big{)},$ (4.18)

for all $q\geq 1$. The inclusion from right to left is immediate by definition
of $\mathbb{W}_{j,\ell,{\alpha}}$ in the Lemma 4.10 and the fact that $j\leq
q$. For the inclusion from left to right we proceed by induction on $q\geq 1$:

  * •

For $q=1$, let $\mathbb{W}\in\ker\left(P^{*}_{F_{\delta},\delta}\right)$.
Since $P^{*}$ is asymptotic to the Euclidean Laplacian $\Delta$, Theorem 1.17
in [2] and Remark 4.9 imply that there exists scalars $\lambda^{\alpha}$ such
that $\mathbb{W}-\lambda^{\alpha}\mathbb{W}^{(e)}_{1,0,{\alpha}}\in
H^{2}_{-1+\delta}(\Sigma)$. Lemma 4.10 thus implies that
$\mathbb{W}-\lambda^{\alpha}\mathbb{W}_{1,0,{\alpha}}\in\ker\left(P^{*}_{F_{\delta},-1+\delta}\right)$.
Thanks to (4.17), we obtain
$\mathbb{W}=\lambda^{\alpha}\mathbb{W}_{1,0,{\alpha}}$. This proves (4.18) for
$q=1$.

  * •

Assume that we have

$\hskip
28.45274pt\ker\left(P^{*}_{F_{\delta},q-1+\delta}\right)=\mathrm{span}\big{(}\mathbb{W}_{j,\ell,{\alpha}},\text{$1\leq
j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}\big{)}$ (4.19)

for some $q\geq 1$ and let
$\mathbb{W}\in\ker\left(P^{*}_{F_{\delta},q+\delta}\right)$. From Theorem 1.17
in [2] and Remark 4.9 we again obtain the existence of scalars
$\lambda^{\ell,{\alpha}}$ such that
$\mathbb{W}-\lambda^{\ell,{\alpha}}\mathbb{W}^{(e)}_{q+1,\ell,{\alpha}}\in
H^{2}_{q-1+\delta}(\Sigma)$. Lemma 4.10 thus implies that
$\mathbb{W}-\lambda^{\ell,{\alpha}}\mathbb{W}_{q+1,\ell,{\alpha}}\in\ker\left(P^{*}_{F_{\delta},q-1+\delta}\right)$.
From (4.19) this gives

$\displaystyle\mathbb{W}-\lambda^{\ell,{\alpha}}\mathbb{W}_{q+1,\ell,{\alpha}}\in\mathrm{span}\big{(}\mathbb{W}_{j,\ell,{\alpha}},\text{$1\leq
j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}\big{)}$

and thus

$\displaystyle\mathbb{W}\in\mathrm{span}\big{(}\mathbb{W}_{j,\ell,{\alpha}},\text{$1\leq
j\leq q+1$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}\big{)}.$

This concludes the proof of (4.18). Now, by definition of the adjoint,
$(\mathbb{Y},0)\in\mathrm{im}(P_{F_{\delta},-q-\delta})$ if and only if
$\left\langle\mathbb{Y},\mathbb{V}\right\rangle_{L^{2}(\Sigma)}=0$ for all
$\mathbb{V}$ such that
$(\mathbb{V},\mathbbm{h})\in\ker((P_{F_{\delta},-q-\delta})^{*})$ for some
$\mathbbm{h}$. Thanks to (4.10) we obtain

$\displaystyle(\mathbb{Y},0)\in\mathrm{im}(P_{F_{\delta},-q-\delta})$
$\displaystyle\iff\left\langle\mathbb{Y},\mathbb{V}\right\rangle_{L^{2}(\Sigma)}=0,\quad\text{for
all $\mathbb{V}\in\ker\left(P^{*}_{F_{\delta},q-1+\delta}\right)$}.$

Using (4.18) concludes the proof of Lemma 4.11. ∎

###  4.4. Conclusion

We summarize the properties of the operator $P$ in (4.1) proved so far.

######  Proposition 4.12.

Let $q\in\mathbb{N}^{*}$ and $0<\delta<1$. There exists
$F_{\delta}\in{\mathcal{E}}(\partial\Sigma)$ and a family

$(\mathbb{W}_{j,\ell,{\alpha}})_{1\leq j\leq q,-(j-1)\leq\ell\leq
j-1,{\alpha}=0,1,2,3}$ (4.20)

with the following properties

  1. (i)

there exists $\mathcal{U}$ and $\mathcal{V}$ open subsets of $\partial\Sigma$
such that $F_{\delta|_{\mathcal{V}}}=\mathrm{Id}$ and
$F_{\delta|_{\mathcal{U}}}=0$,

  2. (ii)

the family (4.20) is linearly independent and each
$\mathbb{W}_{j,\ell,{\alpha}}$ satisfies the estimate
$\left|\mathbb{W}_{j,\ell,{\alpha}}\right|\lesssim r^{j-1}$ and the system

$\left\\{\begin{aligned}
P^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right)&=0,\quad\text{on $\Sigma$,}\\\
(B_{\nu}^{*}+F_{\delta}^{*})\left(\mathbb{W}_{j,\ell,{\alpha}}\right)&=0,\quad\text{on
$\partial\Sigma$,}\end{aligned}\right.$

  3. (iii)

if $\mathbb{Y}\in L^{2}_{-q-\delta-2}(\Sigma)$ satisfies

$\displaystyle\left\langle\mathbb{W}_{j,\ell,{\alpha}},\mathbb{Y}\right\rangle_{L^{2}(\Sigma)}=0,\quad\text{for
all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$},$ (4.21)

then there exists a unique $\mathbb{U}\in H^{2}_{-q-\delta}(\Sigma)$ with
$\left\|\mathbb{U}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\lesssim\left\|\mathbb{Y}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
such that

$\left\\{\begin{aligned} P\left(\mathbb{U}\right)&=\mathbb{Y},\quad\text{on
$\Sigma$,}\\\ (B_{\nu}+F_{\delta})\left(\mathbb{U}\right)&=0,\quad\text{on
$\partial\Sigma$.}\end{aligned}\right.$

###### Proof.

We consider $F_{\delta}$ as constructed in Corollary 4.7. The family (4.20)
has been constructed in Lemma 4.10 and its linear independence follows from
the linear independence of the family

$\displaystyle\left(\mathbb{W}^{(e)}_{j,\ell,{\alpha}}\right)_{1\leq j\leq
q,-(j-1)\leq\ell\leq j-1,{\alpha}=0,1,2,3}$

and the asymptotics
$\mathbb{W}_{j,\ell,{\alpha}}-\mathbb{W}^{(e)}_{j,\ell,{\alpha}}=O\left(r^{j-2+\delta}\right)$
at infinity. If $\mathbb{Y}\in L^{2}_{-q-\delta-2}(\Sigma)$ satisfies (4.21)
then Lemma 4.11 implies that
$(\mathbb{Y},0)\in\mathrm{im}(P_{F_{\delta},-q-\delta})$, so that there exists
$\mathbb{U}\in H^{2}_{-q-\delta}(\Sigma)$ solving
$P_{F_{\delta},-q-\delta}(\mathbb{U})=(\mathbb{Y},0)$. Uniqueness follows from
the choice of $F_{\delta}$ (Corollary 4.7) which implies that
$\ker(P_{F_{\delta},-q-\delta})\subset\ker(P_{F_{\delta},-\delta})=\\{0\\}$.
Finally, the estimate
$\left\|\mathbb{U}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\lesssim\left\|\mathbb{Y}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
follows from (1.31) in [2]. ∎

##  5\. Proof of the main theorem

This section is devoted to the proof of Theorem 3.1.

###  5.1. Conformal formulation of the constraint equations

Let $(\Sigma,\bar{g},\bar{\pi})$ an asymptotically flat initial data set
satisfying the regularity assumptions of Theorem 3.1, and let
$(\widecheck{g},\widecheck{\pi})\in H^{2}_{-q-\delta}(\Sigma)\times
H^{1}_{-q-\delta-1}(\Sigma)$ for some $q\in\mathbb{N}^{*}$ and $0<\delta<1$.
For $\mathfrak{p}=(\breve{g},\breve{\pi})$ compactly supported we define

$(\hat{g},\hat{\pi})(\mathfrak{p})\vcentcolon=(\bar{g}+\widecheck{g},\bar{\pi}+\widecheck{\pi})+\mathfrak{p}.$
(5.1)

Following [11], we look for solutions of the constraint equations under the
form

$(g,\pi)=\left((1+\widecheck{u})^{4}\hat{g}(\mathfrak{p}),(1+\widecheck{u})^{2}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}\widecheck{X}\right)\right).$
(5.2)

######  Lemma 5.1.

If $(g,\pi)$ is defined by (5.2) then the constraint equations $\Phi(g,\pi)=0$
rewrite as

$\displaystyle P\left(\widecheck{\mathbb{U}}\right)$
$\displaystyle=D\Phi[\bar{g},\bar{\pi}]((\widecheck{g},\widecheck{\pi})+\mathfrak{p})+{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$
(5.3)

where the operator $P$ is defined in (4.1),
$\widecheck{\mathbb{U}}=\left(\widecheck{u},\widecheck{X}\right)$ and the
remainder is defined by

$\displaystyle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$
$\displaystyle\vcentcolon=\bigg{(}-Q_{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})$
$\displaystyle\hskip
28.45274pt-\widecheck{u}\left(D{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})+Q_{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})\right)$
$\displaystyle\hskip
28.45274pt-(1+\widecheck{u})Q_{\mathcal{H}}[\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})]\left(0,L_{\hat{g}(\mathfrak{p})}\widecheck{X}\right)-\widecheck{u}\left(-2\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{\widecheck{X}}\hat{g}(\mathfrak{p})_{ij}+\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}\widecheck{X}\right)$
$\displaystyle\hskip
28.45274pt+8\left(\Delta_{\hat{g}(\mathfrak{p})}-\Delta_{\bar{g}}\right)\widecheck{u}+2\left(\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{\widecheck{X}}\hat{g}(\mathfrak{p})_{ij}-\bar{\pi}^{ij}{\mathcal{L}}_{\widecheck{X}}\bar{g}_{ij}\right)-\left(\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}-\mathrm{tr}_{\bar{g}}\bar{\pi}\mathrm{div}_{\bar{g}}\right)\widecheck{X},$
$\displaystyle\hskip
28.45274pt2Q_{\mathcal{M}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})_{i}$
$\displaystyle\hskip
28.45274pt+4(1+\widecheck{u})^{-1}\hat{g}(\mathfrak{p})^{j\ell}\left(2L_{\hat{g}(\mathfrak{p})}\widecheck{X}_{i\ell}\partial_{j}\widecheck{u}-L_{\hat{g}(\mathfrak{p})}\widecheck{X}_{j\ell}\partial_{i}\widecheck{u}\right)$
$\displaystyle\hskip
28.45274pt+4((1+\widecheck{u})^{-1}-1)\hat{g}(\mathfrak{p})^{j\ell}\left(2\hat{\pi}(\mathfrak{p})_{i\ell}\partial_{j}\widecheck{u}-\hat{\pi}(\mathfrak{p})_{j\ell}\partial_{i}\widecheck{u}\right)$
$\displaystyle\hskip
28.45274pt+2\left(\mathrm{div}_{\hat{g}(\mathfrak{p})}L_{\hat{g}(\mathfrak{p})}-\mathrm{div}_{\bar{g}}L_{\bar{g}}\right)\widecheck{X}_{i}+8\left(\hat{g}(\mathfrak{p})^{j\ell}\hat{\pi}(\mathfrak{p})_{i\ell}-\bar{g}^{j\ell}\bar{\pi}_{i\ell}\right)\partial_{j}\widecheck{u}-4\left(\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})-\mathrm{tr}_{\bar{g}}\bar{\pi}\right)\partial_{i}\widecheck{u}\bigg{)}.$

###### Proof.

In the proof of Theorem 1 in [11] we find the following formulas:

$\displaystyle
u^{5}{\mathcal{H}}\left(u^{4}\hat{g}(\mathfrak{p}),u^{2}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)\right)$
$\displaystyle=-8\Delta_{\hat{g}(\mathfrak{p})}u+u{\mathcal{H}}\left(\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)$

and

$\displaystyle
u^{2}{\mathcal{M}}\left(u^{4}\hat{g}(\mathfrak{p}),u^{2}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)\right)_{i}$
$\displaystyle\qquad=\mathrm{div}_{\hat{g}(\mathfrak{p})}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)_{i}+2u^{-1}\hat{g}(\mathfrak{p})^{j\ell}\left(2\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)_{i\ell}\partial_{j}u-\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)_{j\ell}\partial_{i}u\right).$

On the one hand, thanks to (2.2) and (2.4), we can compute
$D{\mathcal{H}}[\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})](0,L_{\hat{g}}X)$
and rewrite the equation

$\displaystyle{\mathcal{H}}\left(u^{4}\hat{g}(\mathfrak{p}),u^{2}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)\right)=0$

as

$\displaystyle-8\Delta_{\hat{g}(\mathfrak{p})}u-2\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{X}\hat{g}(\mathfrak{p})_{ij}$
$\displaystyle+\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}X$
$\displaystyle=-{\mathcal{H}}(\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p}))$
$\displaystyle\quad-(u-1){\mathcal{H}}(\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p}))-uQ_{\mathcal{H}}[\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})](0,L_{\hat{g}(\mathfrak{p})}X)$
$\displaystyle\quad-(u-1)\left(-2\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{X}\hat{g}(\mathfrak{p})_{ij}+\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}X\right).$

Using now the definition (5.1), (2.2) and
${\mathcal{H}}(\bar{g},\bar{\pi})=0$, we rewrite this equation as

$\displaystyle-8\Delta_{\bar{g}}u-2\bar{\pi}^{ij}{\mathcal{L}}_{X}\bar{g}_{ij}+\mathrm{tr}_{\bar{g}}\bar{\pi}\mathrm{div}_{\bar{g}}X$
$\displaystyle=-D{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})$
$\displaystyle\quad-
Q_{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})$
$\displaystyle\quad-(u-1)\left(D{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})+Q_{\mathcal{H}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})\right)$
$\displaystyle\quad-
uQ_{\mathcal{H}}[\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})](0,L_{\hat{g}(\mathfrak{p})}X)$
$\displaystyle\quad-(u-1)\left(-2\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{X}\hat{g}(\mathfrak{p})_{ij}+\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}X\right)$
$\displaystyle\quad+8\Delta_{\hat{g}(\mathfrak{p})}u+2\hat{\pi}(\mathfrak{p})^{ij}{\mathcal{L}}_{X}\hat{g}(\mathfrak{p})_{ij}-\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\mathrm{div}_{\hat{g}(\mathfrak{p})}X$
$\displaystyle\quad-8\Delta_{\bar{g}}u-2\bar{\pi}^{ij}{\mathcal{L}}_{X}\bar{g}_{ij}+\mathrm{tr}_{\bar{g}}\bar{\pi}\mathrm{div}_{\bar{g}}X.$

On the other hand, thanks to (2.3) and (2.5), as well as (5.1) and
$\mathrm{div}_{\bar{g}}\bar{\pi}=0$, we rewrite the equation

$\displaystyle{\mathcal{M}}\left(u^{4}\hat{g}(\mathfrak{p}),u^{2}\left(\hat{\pi}(\mathfrak{p})+L_{\hat{g}(\mathfrak{p})}X\right)\right)_{i}=0$

as

$\displaystyle-2\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}-8\bar{g}^{j\ell}\bar{\pi}_{i\ell}\partial_{j}u+4\mathrm{tr}_{\bar{g}}\bar{\pi}\partial_{i}u$
$\displaystyle=2D{\mathcal{M}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})_{i}+2Q_{\mathcal{M}}[\bar{g},\bar{\pi}](\widecheck{g}+\breve{g},\widecheck{\pi}+\breve{\pi})_{i}$
$\displaystyle\quad+8u^{-1}\hat{g}(\mathfrak{p})^{j\ell}L_{\hat{g}(\mathfrak{p})}X_{i\ell}\partial_{j}u-4u^{-1}\mathrm{tr}_{\hat{g}(\mathfrak{p})}L_{\hat{g}(\mathfrak{p})}X\partial_{i}u$
$\displaystyle\quad+8(u^{-1}-1)\hat{g}(\mathfrak{p})^{j\ell}\hat{\pi}(\mathfrak{p})_{i\ell}\partial_{j}u-4(u^{-1}-1)\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\partial_{i}u$
$\displaystyle\quad+2\mathrm{div}_{\hat{g}(\mathfrak{p})}L_{\hat{g}(\mathfrak{p})}X_{i}+8\hat{g}(\mathfrak{p})^{j\ell}\hat{\pi}(\mathfrak{p})_{i\ell}\partial_{j}u-4\mathrm{tr}_{\hat{g}(\mathfrak{p})}\hat{\pi}(\mathfrak{p})\partial_{i}u$
$\displaystyle\quad-2\mathrm{div}_{\bar{g}}L_{\bar{g}}X_{i}-8\bar{g}^{j\ell}\bar{\pi}_{i\ell}\partial_{j}u+4\mathrm{tr}_{\bar{g}}\bar{\pi}\partial_{i}u.$

Applying the above formula to $u=1+\widecheck{u}$ and $X=\widecheck{X}$
concludes the proof of the lemma. ∎

The proof of Theorem 3.1 reduces to whether we can construct $\mathfrak{p}$
such that (5.3) admits a solution $\widecheck{\mathbb{U}}\in
H^{2}_{-q-\delta}(\Sigma)$. Thanks to Proposition 4.12, we need $\mathfrak{p}$
to ensure orthogonality conditions with the $\mathbb{W}_{j,\ell,{\alpha}}$ for
$1\leq j\leq q$. Because of the remainder term in (5.3), the definition of
$\mathfrak{p}$ is non-linear and will follow from a fixed point argument
performed in Section 5.3. More precisely for each $\widecheck{\mathbb{U}}\in
H^{2}_{-q-\delta}(\Sigma)$ small enough we will construct and control
$\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)$ so that the right-hand side
of (5.3) is orthogonal to $\mathbb{W}_{j,\ell,{\alpha}}$ for $1\leq j\leq q$.
We then solve the elliptic system in Section 5.4 and thus conclude the proof
of Theorem 3.1.

###  5.2. Estimating the remainder

We start by estimating the remainder term in (5.3), i.e.
${\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$. For the sake
of clarity, we introduce the schematic notations

$\displaystyle\bar{\Gamma}=\\{\nabla\bar{g},\bar{\pi}\\},\quad\widecheck{\Gamma}=\\{\nabla\widecheck{g},\widecheck{\pi}\\},\quad\breve{\Gamma}=\\{\nabla\breve{g},\breve{\pi}\\},\quad\hat{\Gamma}(\mathfrak{p})=\\{\nabla\hat{g}(\mathfrak{p}),\hat{\pi}(\mathfrak{p})\\}.$

Since $(\widecheck{g},\widecheck{\pi})$, $\mathfrak{p}$ and
$\widecheck{\mathbb{U}}$ will be small in appropriate spaces, we only write
down quadratic terms when manipulating
${\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$ since higher
order terms are smaller and enjoy more decay and thus are easier to treat.
Thanks to its expression given in Lemma 5.1 and the various schematic
expressions of Lemma 2.4 we find

$\displaystyle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$
$\displaystyle={\mathcal{R}}_{1}\left(\mathfrak{p}\right)+{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)+{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)+\text{higher-
order terms}$

where we have schematically

$\displaystyle{\mathcal{R}}_{1}\left(\mathfrak{p}\right)$
$\displaystyle=(\widecheck{g}+\breve{g})\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+\bar{\Gamma}(\widecheck{g}+\breve{g})\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+\left(\widecheck{\Gamma}+\breve{\Gamma}\right)^{2}+\bar{\Gamma}^{2}(\widecheck{g}+\breve{g})^{2},$
$\displaystyle{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$
$\displaystyle=\hat{g}(\mathfrak{p})\left(\nabla\widecheck{\mathbb{U}}\right)^{2}+\hat{\Gamma}(\mathfrak{p})\widecheck{\mathbb{U}}\left(\nabla\widecheck{\mathbb{U}}+\hat{\Gamma}(\mathfrak{p})\widecheck{\mathbb{U}}\right),$
$\displaystyle{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$
$\displaystyle=(\widecheck{g}+\breve{g})\nabla^{2}\widecheck{\mathbb{U}}+\left((\widecheck{g}+\breve{g})\bar{\Gamma}+\widecheck{\Gamma}+\breve{\Gamma}\right)\nabla\widecheck{\mathbb{U}}$
$\displaystyle\quad+\left(\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+(\widecheck{g}+\breve{g})\nabla\bar{\Gamma}+\bar{\Gamma}\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+\bar{\Gamma}^{2}(\widecheck{g}+\breve{g})\right)\widecheck{\mathbb{U}}.$

Note that we have omitted undifferentiated components of $\bar{g}$ since they
are $O(1)$.

######  Lemma 5.2.

Assume

$\displaystyle\left\|\widecheck{\mathbb{U}}\right\|_{H^{2}_{-q-\delta}(\Sigma)}$
$\displaystyle\leq C_{0}\varepsilon,$ (5.4)
$\displaystyle\left\|\breve{g}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}\right\|_{W^{1,\infty}(\Sigma)}$
$\displaystyle\leq D_{0}\varepsilon,$ (5.5)

for some $C_{0},D_{0}>0$. If $C_{0}$ and $D_{0}$ are large enough and
$\varepsilon$ is small enough compared to $C_{0}^{-1}$ and $D_{0}^{-1}$ then

$\displaystyle\left\|{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim C(C_{0},D_{0})\varepsilon^{2},$ (5.6)
$\displaystyle\left|\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|$
$\displaystyle\lesssim C(C_{0},D_{0})\varepsilon^{2}.$ (5.7)

###### Proof.

Recall that $\mathfrak{p}=(\breve{g},\breve{\pi})$ is compactly supported and
that its support will be fixed in the whole proof so that when estimating
terms containing either $\breve{g}$ or $\breve{\pi}$ we can drop the weights
at infinity. Moreover, it suffices to estimate
${\mathcal{R}}_{1}\left(\mathfrak{p}\right)$,
${\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$ and
${\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$ since the
higher order terms in
${\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$ enjoy better
decay and are smaller. We start with
${\mathcal{R}}_{1}\left(\mathfrak{p}\right)$:

$\displaystyle\left\|{\mathcal{R}}_{1}\left(\mathfrak{p}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim\left\|(\widecheck{g}+\breve{g})\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|(\widecheck{g}+\breve{g})\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\right\|_{L^{2}_{-q-\delta}(\Sigma)}$
$\displaystyle\quad+\left\|\left(\widecheck{\Gamma}+\breve{\Gamma}\right)^{2}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|(\widecheck{g}+\breve{g})^{2}\right\|_{L^{2}_{-q-\delta+2}(\Sigma)},$

where we used asymptotic flatness of $(\bar{g},\bar{\pi})$ from Definition
2.1, which implies $|\bar{\Gamma}|\lesssim(1+r)^{-2}$. Using now the
continuous bilinear embeddings from Lemma 2.3 we obtain

$\displaystyle\left\|{\mathcal{R}}_{1}\left(\mathfrak{p}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim\left\|\widecheck{g}+\breve{g}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\left(\left\|\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|\widecheck{\Gamma}+\breve{\Gamma}\right\|_{H^{1}_{-q-\delta-1}(\Sigma)}\right)$
$\displaystyle\quad+\left\|\widecheck{\Gamma}+\breve{\Gamma}\right\|_{H^{1}_{-q-\delta-1}(\Sigma)}^{2}+\left\|\widecheck{g}+\breve{g}\right\|_{H^{2}_{-q-\delta}(\Sigma)}^{2}.$

Using now (3.1) and (5.5) this gives

$\displaystyle\left\|{\mathcal{R}}_{1}\left(\mathfrak{p}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim C(D_{0})\varepsilon^{2}.$ (5.8)

We now estimate
${\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$:

$\displaystyle\left\|{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim(1+C(D_{0})\varepsilon)\left(\left\|\left(\nabla\widecheck{\mathbb{U}}\right)^{2}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|\widecheck{\mathbb{U}}\nabla\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta}(\Sigma)}+\left\|\widecheck{\mathbb{U}}^{2}\right\|_{L^{2}_{-q-\delta+2}(\Sigma)}\right),$

where we used (5.1), asymptotic flatness, (3.1) and (5.5) which imply
$\hat{g}(\mathfrak{p})=1+O\left(r^{-1}\right)+O\left(\varepsilon
C(D_{0})r^{-q-\delta}\right)$ and
$\hat{\Gamma}(\mathfrak{p})=O\left(r^{-2}\right)+O\left(\varepsilon
C(D_{0})r^{-q-\delta-1}\right)$. The assumption (5.4) and the continuous
bilinear embeddings from Lemma 2.3 then imply

$\displaystyle\left\|{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim C(C_{0},D_{0})\varepsilon^{2}.$ (5.9)

We finally estimate
${\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)$:

$\displaystyle\left\|{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim\left\|(\widecheck{g}+\breve{g})\nabla^{2}\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\nabla\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|\left(\widecheck{g}+\breve{g}\right)\nabla\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta}(\Sigma)}$
$\displaystyle\quad+\left\|\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}+\left\|(\widecheck{g}+\breve{g})\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta+1}(\Sigma)}$
$\displaystyle\quad+\left\|\left(\widecheck{\Gamma}+\breve{\Gamma}\right)\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta}(\Sigma)}+\left\|(\widecheck{g}+\breve{g})\widecheck{\mathbb{U}}\right\|_{L^{2}_{-q-\delta+2}(\Sigma)},$

where we used asymptotic flatness to write $\bar{\Gamma}=O\left(r^{-2}\right)$
and $\nabla\bar{\Gamma}=O\left(r^{-3}\right)$. As above the continuous
bilinear embeddings from Lemma 2.3 together with (3.1), (5.4) and (5.5) imply

$\displaystyle\left\|{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim C(C_{0},D_{0})\varepsilon^{2}.$ (5.10)

Putting (5.8), (5.9) and (5.10) together concludes the proof of (5.6).
Finally, since $j\leq q$ and $\delta>0$ imply
$\left|\left\langle\mathbb{V},\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|\lesssim\left\|\mathbb{V}\right\|_{L^{2}_{-q-\delta-2}}$
(where we also used $|\mathbb{W}_{j,\ell,{\alpha}}|\lesssim r^{j-1}$), (5.6)
implies (5.7). ∎

######  Lemma 5.3.

Let $\widecheck{\mathbb{U}},\widecheck{\mathbb{U}}^{\prime}$ satisfying (5.4)
and $\mathfrak{p},\mathfrak{p}^{\prime}$ satisfying (5.5). If $C_{0}$ and
$D_{0}$ are large enough and $\varepsilon$ is small enough compared to
$C_{0}^{-1}$ and $D_{0}^{-1}$ then

$\displaystyle\left\|{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
(5.11) $\displaystyle\hskip 56.9055pt\lesssim
C(C_{0},D_{0})\varepsilon\bigg{(}\left\|\breve{g}-\breve{g}^{\prime}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}-\breve{\pi}^{\prime}\right\|_{W^{1,\infty}(\Sigma)}+\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\bigg{)},$
$\displaystyle\left|\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|$
(5.12) $\displaystyle\hskip 56.9055pt\lesssim
C(C_{0},D_{0})\varepsilon\bigg{(}\left\|\breve{g}-\breve{g}^{\prime}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}-\breve{\pi}^{\prime}\right\|_{W^{1,\infty}(\Sigma)}+\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\bigg{)}.$

###### Proof.

We have

$\displaystyle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle={\mathcal{R}}_{1}\left(\mathfrak{p}\right)-{\mathcal{R}}_{1}\left(\mathfrak{p}^{\prime}\right)+{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{2}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle\quad+{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{3}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)+\text{difference
of higher order terms}.$

We have

$\displaystyle{\mathcal{R}}_{1}\left(\mathfrak{p}\right)-{\mathcal{R}}_{1}\left(\mathfrak{p}^{\prime}\right)$
$\displaystyle=(\breve{g}-\breve{g}^{\prime})\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+(\widecheck{g}+\breve{g}^{\prime})\nabla\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)$
$\displaystyle\quad+\bar{\Gamma}\left((\breve{g}-\breve{g}^{\prime})\left(\widecheck{\Gamma}+\breve{\Gamma}\right)+(\widecheck{g}+\breve{g}^{\prime})\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)\right)$
$\displaystyle\quad+\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)\left(\widecheck{\Gamma}+\breve{\Gamma}+\breve{\Gamma}^{\prime}\right)+\bar{\Gamma}^{2}\left(\breve{g}-\breve{g}^{\prime}\right)\left(\widecheck{g}+\breve{g}+\breve{g}^{\prime}\right).$

By following the same lines as the proof of (5.8) we obtain

$\displaystyle\left\|{\mathcal{R}}_{1}\left(\mathfrak{p}\right)-{\mathcal{R}}_{1}\left(\mathfrak{p}^{\prime}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim
C(C_{0})\varepsilon\left(\left\|\breve{g}-\breve{g}^{\prime}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}-\breve{\pi}^{\prime}\right\|_{W^{1,\infty}(\Sigma)}\right).$
(5.13)

Next, we have

$\displaystyle{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{2}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle=(\breve{g}-\breve{g}^{\prime})\left(\nabla\widecheck{\mathbb{U}}\right)^{2}+\hat{g}(\mathfrak{p}^{\prime})\nabla\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)\nabla\left(\widecheck{\mathbb{U}}+\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle\quad+\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)\widecheck{\mathbb{U}}\nabla\widecheck{\mathbb{U}}+\hat{\Gamma}(\mathfrak{p}^{\prime})\left(\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)\nabla\widecheck{\mathbb{U}}+\widecheck{\mathbb{U}}^{\prime}\nabla\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)\right)$
$\displaystyle\quad+\left(\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)\widecheck{\mathbb{U}}+\hat{\Gamma}(\mathfrak{p}^{\prime})\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)\right)\left(\hat{\Gamma}(\mathfrak{p})\widecheck{\mathbb{U}}+\hat{\Gamma}(\mathfrak{p}^{\prime})\widecheck{\mathbb{U}}^{\prime}\right).$

By following the same lines as the proof of (5.9) we obtain

$\displaystyle\left\|{\mathcal{R}}_{2}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{2}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}\lesssim{}$
$\displaystyle
C(C_{0},D_{0})\varepsilon\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}$
(5.14)
$\displaystyle+C(C_{0},D_{0})\varepsilon^{2}\left(\left\|\breve{g}-\breve{g}^{\prime}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}-\breve{\pi}^{\prime}\right\|_{W^{1,\infty}(\Sigma)}\right).$

Finally, we have

$\displaystyle{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{3}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)={}$
$\displaystyle(\breve{g}-\breve{g}^{\prime})\nabla^{2}\widecheck{\mathbb{U}}+(\widecheck{g}+\breve{g}^{\prime})\nabla^{2}\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle+\left(\breve{\Gamma}-\breve{\Gamma}+(\breve{g}-\breve{g}^{\prime})\bar{\Gamma}\right)\nabla\widecheck{\mathbb{U}}+\left(\widecheck{\Gamma}+\breve{\Gamma}^{\prime}+(\widecheck{g}+\breve{g}^{\prime})\bar{\Gamma}\right)\nabla\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right)$
$\displaystyle+\left(\nabla\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)+(\breve{g}-\breve{g}^{\prime})\nabla\bar{\Gamma}+\bar{\Gamma}\left(\breve{\Gamma}-\breve{\Gamma}^{\prime}\right)+\bar{\Gamma}^{2}(\breve{g}-\breve{g}^{\prime})\right)\widecheck{\mathbb{U}}$
$\displaystyle+\left(\nabla\left(\widecheck{\Gamma}+\breve{\Gamma}^{\prime}\right)+(\widecheck{g}+\breve{g}^{\prime})\nabla\bar{\Gamma}+\bar{\Gamma}\left(\widecheck{\Gamma}+\breve{\Gamma}^{\prime}\right)+\bar{\Gamma}^{2}(\widecheck{g}+\breve{g}^{\prime})\right)\left(\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right).$

By following the same lines as the proof of (5.10) we obtain

$\begin{split}&\left\|{\mathcal{R}}_{3}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}_{3}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}\\\
\lesssim{}&C(C_{0},D_{0})\varepsilon\bigg{(}\left\|\breve{g}-\breve{g}^{\prime}\right\|_{W^{2,\infty}(\Sigma)}+\left\|\breve{\pi}-\breve{\pi}^{\prime}\right\|_{W^{1,\infty}(\Sigma)}+\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\bigg{)}.\end{split}$
(5.15)

By putting (5.13), (5.14) and (5.15) together we obtain (5.11). We conclude
the proof of the lemma by noting that (5.11) implies (5.12), as (5.6) implies
(5.7). ∎

###  5.3. Solving for the corrector

In this section, we construct the corrector $\mathfrak{p}$, starting by
deducing from the analyticity assumptions on
$\left(\Omega,\bar{g}_{|_{\Omega}},\bar{\pi}_{|_{\Omega}}\right)$ (see Theorem
3.1) that KIDS, i.e. elements of $\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$, enjoy
similar regularity.

######  Lemma 5.4.

Any element of $\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$ is analytic on
$\\{r_{0}\leq r<r_{1}\\}$.

###### Proof.

If $(f,X)\in\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$, then (recall (2.10) and
(2.11)) it satisfies a system of the form

$\displaystyle\mathrm{Hess}_{\bar{g}}(f)^{ij}$
$\displaystyle=\bar{A}^{ij}_{k\ell}D^{k}X^{\ell}+\bar{A}^{ij}_{0}f+\bar{A}^{ij}_{k}X^{k},$
$\displaystyle{\mathcal{L}}_{X}\bar{g}^{ij}$ $\displaystyle=\bar{B}^{ij}f,$

where the coefficients $\bar{A}^{ij}_{k\ell}$, $\bar{A}^{ij}_{0}$,
$\bar{A}^{ij}_{k}$ and $\bar{B}^{ij}$ depends only on $\bar{g}$ and
$\bar{\pi}$ and are analytic on $\\{r_{0}\leq r<r_{1}\\}$. To simplify the
exposition, in what follows we don’t write the analytic coefficients of the
equations satisfied by $f$ and $X$. In particular the above equations rewrite
schematically as

$\displaystyle\mathrm{Hess}_{\bar{g}}(f)$ $\displaystyle=DX+f+X,$ (5.16)
$\displaystyle{\mathcal{L}}_{X}\bar{g}$ $\displaystyle=f.$ (5.17)

Taking the trace of (5.17) implies $\mathrm{div}_{\bar{g}}X=f$. Taking also
the divergence of (5.17) together with the identity

$\displaystyle\mathrm{div}_{\bar{g}}{\mathcal{L}}_{X}\bar{g}_{i}$
$\displaystyle=\Delta_{\bar{g}}^{(1)}X_{i}+\partial_{i}\mathrm{div}_{\bar{g}}X+Ric(\bar{g})(\partial_{j},X)$

implies that $f$ and $X$ solve a system of the form

$\displaystyle\Delta_{\bar{g}}f$ $\displaystyle=DX+f+X,$ (5.18)
$\displaystyle\Delta_{\bar{g}}^{(1)}X$ $\displaystyle=Df+f+X,$ (5.19)

where we also took the trace of (5.16) to get the first equation. Therefore,
$f$ and $X$ solve an elliptic system with analytic coefficients and thus are
analytic on $\\{r_{0}<r<r_{1}\\}$ thanks to standard elliptic regularity. It
remains to prove that there are analytic on the boundary $\partial\Sigma$, and
for that we will derive from (5.16)-(5.17) an elliptic system on
$\partial\Sigma$ with analytic coefficients.

For this, we introduce the standard notations from [8], i.e.
${D}\mkern-13.0mu/\,$, ${\mathrm{Hess}}\mkern-13.0mu/\,$ and
${\Delta}\mkern-13.0mu/\,$ denote covariant derivative, Hessian and Laplace-
Beltrami operator associated to the induced metric ${\bar{g}}\mkern-9.0mu/\,$
on $\partial\Sigma$ by $\bar{g}$. We also decompose the vector field
$X=N\nu+{X}\mkern-13.0mu/\,$ where ${X}\mkern-13.0mu/\,\in T\partial\Sigma$
and $N$ is a scalar function. By assumption, every quantity related to
$\bar{g}$ is analytic and for simplicity we don’t write the analytic
coefficients appearing in the following formulas such as the second
fundamental form of $\partial\Sigma$ as a submanifold of $\Sigma$. By plugging
(5.18) into (5.16) we obtain a simpler form of the system for $(f,X)$:

$\displaystyle\mathrm{Hess}_{\bar{g}}(f)$ $\displaystyle=DX+f+X,$ (5.20)
$\displaystyle{\mathcal{L}}_{X}\bar{g}$ $\displaystyle=f.$ (5.21)

We first contract (5.21) twice with $\nu$ to obtain

$\displaystyle\partial_{\nu}N=f+{X}\mkern-13.0mu/\,.$ (5.22)

Then, we contract (5.21) with one tangential vector field and $\nu$ to get

$\displaystyle{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,N=f+N+{X}\mkern-13.0mu/\,,$
(5.23)

which after taking the tangential divergence implies

$\displaystyle{\Delta}\mkern-13.0mu/\,N$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq
1}({D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,+f+N+{X}\mkern-13.0mu/\,).$
(5.24)

Finally, we contract (5.21) with tangential vector fields, take the trace to
obtain ${\mathrm{div}}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,=f$ and then take the
tangential divergence of (5.21) to obtain the following elliptic equation on
${X}\mkern-13.0mu/\,$

$\displaystyle{\Delta}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq 1}(f+N+{X}\mkern-13.0mu/\,).$ (5.25)

Using that $[{\Delta}\mkern-13.0mu/\,,\partial_{\nu}]$ involves
${D}\mkern-13.0mu/\,\partial_{\nu}$ and ${D}\mkern-13.0mu/\,^{2}$ we obtain
from (5.25) the following

$\displaystyle{\Delta}\mkern-13.0mu/\,{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,+\partial_{\nu}f+{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,),$
(5.26)

where we have used (5.22) to replace $\partial_{\nu}N$. We also commute (5.25)
with ${D}\mkern-13.0mu/\,$ to obtain

$\displaystyle{\Delta}\mkern-13.0mu/\,{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,)+{D}\mkern-13.0mu/\,^{2}(f+N+{X}\mkern-13.0mu/\,).$

Using (5.23) we replace ${D}\mkern-13.0mu/\,^{2}N$ by
${D}\mkern-13.0mu/\,{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,$ and
${D}\mkern-13.0mu/\,N$ to obtain

$\displaystyle{\Delta}\mkern-13.0mu/\,{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,)+{D}\mkern-13.0mu/\,^{2}f.$
(5.27)

We now look at the equation for $f$, i.e. (5.20). By contracting it in
tangential directions we obtain

$\displaystyle{\mathrm{Hess}}\mkern-13.0mu/\,f$
$\displaystyle=\partial_{\nu}f+{D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,),$ (5.28)

where we have used (5.22) and (5.23) to replace $\partial_{\nu}N$ and
${D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,$. Taking the tangential trace of
(5.28) we obtain the equation for $f$:

$\displaystyle{\Delta}\mkern-13.0mu/\,f$
$\displaystyle=\partial_{\nu}f+{D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,).$ (5.29)

Moreover, by using (5.28) we can also control ${D}\mkern-13.0mu/\,^{2}f$ in
(5.27) which becomes

$\displaystyle{\Delta}\mkern-13.0mu/\,{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,$
$\displaystyle=\partial_{\nu}f+{D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,+{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,).$
(5.30)

Also, we commute (5.29) with $\partial_{\nu}$ to obtain an equation on
$\partial_{\nu}f$, using again the fact that
$[{\Delta}\mkern-13.0mu/\,,\partial_{\nu}]$ involves
${D}\mkern-13.0mu/\,\partial_{\nu}$ and ${D}\mkern-13.0mu/\,^{2}$. We get:

$\displaystyle{\Delta}\mkern-13.0mu/\,\partial_{\nu}f$
$\displaystyle=\partial_{\nu}^{2}f+{D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,+\partial_{\nu}f+{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,)$
(5.31)

where we have used (5.22) and (5.28) to replace $\partial_{\nu}N$ and
${D}\mkern-13.0mu/\,^{2}f$. Finally, using the double contraction by $\nu$ of
(5.20) we can replace $\partial_{\nu}^{2}$ in (5.31) to finally obtain

$\displaystyle{\Delta}\mkern-13.0mu/\,\partial_{\nu}f$
$\displaystyle={D}\mkern-13.0mu/\,^{\leq
1}(f+N+{X}\mkern-13.0mu/\,+\partial_{\nu}f+{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,).$
(5.32)

By using the schematic notation

$\displaystyle
V=\\{f,N,{X}\mkern-13.0mu/\,,\partial_{\nu}f,{D}\mkern-13.0mu/\,_{\nu}{X}\mkern-13.0mu/\,,{D}\mkern-13.0mu/\,{X}\mkern-13.0mu/\,\\},$

we remark that combining equations (5.24), (5.25), (5.26), (5.29), (5.30) and
(5.32) gives an elliptic system on $\partial\Sigma$ of the form

$\displaystyle{\Delta}\mkern-13.0mu/\,V={D}\mkern-13.0mu/\,^{\leq 1}V.$

In particular, the coefficients of this system are analytic by assumption and
thus standard elliptic regularity implies that $V$ is analytic on
$\partial\Sigma$. This concludes the proof of Lemma 5.4. ∎

We now define the space $\mathcal{Z}_{q}$ of correctors.

######  Definition 5.5.

We fix $I\subset(r_{0},r_{1})$ an open interval in ${\mathbb{R}}$.

  1. (i)

We define $\chi_{corr}:{\mathbb{R}}\longrightarrow[0,1]$ to be a smooth
function such that $\chi_{corr|_{{\mathbb{R}}\setminus I}}=0$ and
$\chi_{corr|_{I}}>0$.

  2. (ii)

For $j\geq 1$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$ we define

$\displaystyle Z_{j,\ell,{\alpha}}$
$\displaystyle\vcentcolon=\chi_{corr}(r)D\Phi[\bar{g},\bar{\pi}]^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right).$

  3. (iii)

For $q\in\mathbb{N}^{*}$ we define the $4q^{2}$ dimensional space of
correctors

$\displaystyle\mathcal{Z}_{q}$
$\displaystyle\vcentcolon=\mathrm{span}\big{(}Z_{j,\ell,{\alpha}},\text{$1\leq
j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$}\big{)},$

endowed with the norm $\left\|\cdot\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}$.

The following lemma will allow us to solve the equation for the corrector.

######  Lemma 5.6.

Let $q\in\mathbb{N}^{*}$.

  1. (i)

The matrix

$\displaystyle\left(\left\langle
Z_{j,\ell,{\alpha}},D\Phi[\bar{g},\bar{\pi}]^{*}\left(\mathbb{W}_{j^{\prime},\ell^{\prime},{\alpha}^{\prime}}\right)\right\rangle_{L^{2}(\Sigma)}\right)_{1\leq
j,j^{\prime}\leq q,-(j-1)\leq\ell\leq j-1,-(j^{\prime}-1)\leq\ell^{\prime}\leq
j^{\prime}-1,0\leq{\alpha},{\alpha}^{\prime}\leq 3}$

is invertible.

  2. (ii)

Let $\mu_{j,\ell,{\alpha}}$ be real numbers for $1\leq j\leq q$,
$-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$. There exists a unique
$\mathfrak{p}\in\mathcal{Z}_{q}$ such that

$\displaystyle\left\langle\mathfrak{p},D\Phi[\bar{g},\bar{\pi}]^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right)\right\rangle_{L^{2}(\Sigma)}=\mu_{j,\ell,{\alpha}},$

and moreover $\mathfrak{p}$ satisfies

$\displaystyle\left\|\mathfrak{p}\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}\lesssim\sup_{j,\ell,{\alpha}}|\mu_{j,\ell,{\alpha}}|.$

###### Proof.

We start with the first point of the lemma. Let $\lambda^{j,\ell,{\alpha}}$ be
real numbers defined such that

$\displaystyle\left\langle\lambda^{j,\ell,{\alpha}}\chi_{corr}(r)D\Phi[\bar{g},\bar{\pi}]^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right),D\Phi[\bar{g},\bar{\pi}]^{*}\left(\mathbb{W}_{j^{\prime},\ell^{\prime},{\alpha}^{\prime}}\right)\right\rangle_{L^{2}(\Sigma)}=0,$

for all $1\leq j^{\prime}\leq q$, $-(j^{\prime}-1)\leq\ell^{\prime}\leq
j^{\prime}-1$ and ${\alpha}^{\prime}=0,1,2,3$. We multiply each equality by
$\lambda^{j^{\prime},\ell^{\prime},{\alpha}^{\prime}}$ and sum over the
indices $j^{\prime}$, $\ell^{\prime}$ and ${\alpha}^{\prime}$ to get

$\displaystyle\left\langle\chi_{corr}D\Phi[\bar{g},\bar{\pi}]^{*}\left(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}\right),D\Phi[\bar{g},\bar{\pi}]^{*}\left(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}\right)\right\rangle_{L^{2}(\Sigma)}=0,$

Since $\chi_{corr|_{{\mathbb{R}}\setminus I}}=0$ and $\chi_{corr|_{I}}>0$ this
implies

$\displaystyle
D\Phi[\bar{g},\bar{\pi}]^{*}\left(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}\right)=0,\quad\text{on
$\\{r\in I\\}$.}$ (5.33)

Recall from Lemma 4.10 that $P^{*}\left(\mathbb{W}_{j,\ell,{\alpha}}\right)=0$
on $\Sigma$ and by assumption (see Theorem 3.1) the coefficients of $P^{*}$
are analytic in the region $\\{r_{0}<r<r_{1}\\}$. Standard elliptic regularity
thus implies that $\mathbb{W}_{j,\ell,{\alpha}}$ is analytic in the region
$\\{r_{0}<r<r_{1}\\}$. Since the assumptions also imply that the coefficients
of $D\Phi[\bar{g},\bar{\pi}]^{*}$ are analytic in the region
$\\{r_{0}<r<r_{1}\\}$, we can extend (5.33) to this region, and by continuity
to $\\{r_{0}\leq r<r_{1}\\}$. We have thus proved

$\displaystyle
D\Phi[\bar{g},\bar{\pi}]^{*}\left(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}\right)=0,\quad\text{on
$\\{r_{0}\leq r<r_{1}\\}$.}$ (5.34)

Now, Lemma 5.4 implies that
$\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}$ is analytic on the
boundary $\partial\Sigma$. Recall from Proposition 4.12 that
$(B_{\nu}^{*}+F_{\delta}^{*})(\mathbb{W}_{j,\ell,{\alpha}})=0$ on
$\partial\Sigma$. Restricting this to the open subset $\mathcal{U}$ of
$\partial\Sigma$ where $F_{\delta}=0$ (which implies that $F_{\delta}^{*}=0$
on the same open subset) we obtain
$B_{\nu}^{*}(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}})_{|_{\mathcal{U}}}=0$.
Since $\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}$ is analytic on
the boundary, we extend this equality to
$B_{\nu}^{*}(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}})=0$ on
$\partial\Sigma$. Returning to the boundary conditions, this implies that
$F_{\delta}^{*}(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}})=0$ on
$\partial\Sigma$. Restricting this to the open subset $\mathcal{V}$ of
$\partial\Sigma$ where $F_{\delta}=\mathrm{Id}$ (which implies that
$F_{\delta}^{*}=\mathrm{Id}$ on the same open subset), we obtain
$\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}|_{\mathcal{V}}}=0$ which
again extends to $\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}=0$ on
$\partial\Sigma$ thanks to analyticity. Together with Lemma 4.10 this implies
that $P^{*}(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}})=0$ on
$\Sigma$,
$B_{\nu}^{*}(\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}})=\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}=0$
on $\partial\Sigma$. Lemma 4.4 thus implies that

$\displaystyle\lambda^{j,\ell,{\alpha}}\mathbb{W}_{j,\ell,{\alpha}}=0,\quad\text{on
$\Sigma$.}$ (5.35)

Since the family formed by the $\mathbb{W}_{j,\ell,{\alpha}}$ is linearly
independent on $\Sigma$, this implies that $\lambda^{j,\ell,{\alpha}}=0$ for
all indices and concludes the proof of Lemma 5.6, since the second point
follows directly from the first one. ∎

######  Remark 5.7.

The proof of Lemma 5.6 shows how the KIDS are obstructions to the construction
of the corrector. In order to go from (5.34) to (5.35), we need to use a
property distinguishing elements of $\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$ and
elements of $\ker(P^{*}_{F_{\delta},q-1+\delta})$. The key is to note that
$P^{*}(\mathbb{U})=0$ for all
$\mathbb{U}\in\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$, which follows from the
following alternative definition of $P$

$\displaystyle P=D\Phi[\bar{g},\bar{\pi}]\circ\Pi,$

with $\Pi(u,X)=\left(4u\bar{g},2u\bar{\pi}+L_{\bar{g}}X\right)$. This implies
that the only way to distinguish $\ker(D\Phi[\bar{g},\bar{\pi}]^{*})$ from
$\ker(P^{*}_{F_{\delta},q-1+\delta})$ is the boundary condition, and in
particular the regularity at the boundary.

We now construct the corrector map. If $\widecheck{\mathbb{U}}$ is such that

$\displaystyle\left\|\widecheck{\mathbb{U}}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\leq
C_{0}\varepsilon,$ (5.36)

we want the corrector $\mathfrak{p}\in\mathcal{Z}_{q}$ to be such that the
following orthogonality conditions hold

$\displaystyle\left\langle
D\Phi[\bar{g},\bar{\pi}]((\widecheck{g},\widecheck{\pi})+\mathfrak{p})+{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}$
$\displaystyle=0,$ (5.37)

for all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$.
According to Proposition 4.12 this will ensure the solvability of (5.3). Since
each $\mathfrak{p}\in\mathcal{Z}_{q}$ is compactly supported in $\Sigma$ we
can perform the following integration by part without picking up boundary
terms

$\displaystyle\left\langle
D\Phi[\bar{g},\bar{\pi}](\mathfrak{p}),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}=\left\langle\mathfrak{p},D\Phi[\bar{g},\bar{\pi}]^{*}(\mathbb{W}_{j,\ell,{\alpha}})\right\rangle_{L^{2}(\Sigma)},$

so that (5.37) rewrites

$\displaystyle\left\langle\mathfrak{p},D\Phi[\bar{g},\bar{\pi}]^{*}(\mathbb{W}_{j,\ell,{\alpha}})\right\rangle_{L^{2}(\Sigma)}$
$\displaystyle=-\left\langle
D\Phi[\bar{g},\bar{\pi}](\widecheck{g},\widecheck{\pi}),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}-\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}.$
(5.38)

######  Definition 5.8.

Let $D_{0}>0$. The _corrector map_ $\Psi^{(c)}$ is defined by

$\displaystyle\Psi^{(c)}:B_{\mathcal{Z}_{q}}(0,D_{0}\varepsilon)$
$\displaystyle\longrightarrow B_{\mathcal{Z}_{q}}(0,D_{0}\varepsilon),$
$\displaystyle\mathfrak{p}$
$\displaystyle\longmapsto\Psi^{(c)}(\mathfrak{p}),$

such that $\Psi^{(c)}(\mathfrak{p})$ solves

$\displaystyle\left\langle\Psi^{(c)}(\mathfrak{p}),D\Phi[\bar{g},\bar{\pi}]^{*}(\mathbb{W}_{j,\ell,{\alpha}})\right\rangle_{L^{2}(\Sigma)}$
$\displaystyle=-\left\langle
D\Phi[\bar{g},\bar{\pi}](\widecheck{g},\widecheck{\pi}),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}-\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}$
(5.39)

for all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$.

######  Lemma 5.9.

Let $\widecheck{\mathbb{U}}$ satisfying (5.36). If $D_{0}$ is large enough and
if $\varepsilon$ is small enough compared to $D_{0}^{-1}$ and $C_{0}^{-1}$,
then the corrector map $\Psi^{(c)}$ is a well-defined contraction.

###### Proof.

Let $\mathfrak{p}\in B_{\mathcal{Z}_{q}}(0,D_{0}\varepsilon)$. Thanks to Lemma
5.6 there exists a unique $\Psi^{(c)}(\mathfrak{p})\in\mathcal{Z}_{q}$
satisfying (5.39) for all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and
${\alpha}=0,1,2,3$ satisfying moreover

$\displaystyle\left\|\Psi^{(c)}(\mathfrak{p})\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}\lesssim\sup_{j,\ell,{\alpha}}\left|\left\langle
D\Phi[\bar{g},\bar{\pi}](\widecheck{g},\widecheck{\pi}),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}+\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|.$

We estimate both terms on the right-hand side. First since $j\leq q$ and
$\delta>0$ we have

$\displaystyle\left|\left\langle
D\Phi[\bar{g},\bar{\pi}](\widecheck{g},\widecheck{\pi}),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle\right|$
$\displaystyle\lesssim\left\|D\Phi[\bar{g},\bar{\pi}](\widecheck{g},\widecheck{\pi})\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim\left\|\widecheck{g}\right\|_{H^{2}_{-q-\delta}(\Sigma)}+\left\|\widecheck{\pi}\right\|_{H^{1}_{-q-\delta-1}(\Sigma)}$
$\displaystyle\lesssim\varepsilon$

where we have used (2.4), (2.5), asymptotic flatness and (3.1). For the second
term in the estimate for $\Psi^{(c)}(\mathfrak{p})$ we use (5.7). We obtain

$\displaystyle\left\|\Psi^{(c)}(\mathfrak{p})\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}\lesssim\varepsilon\left(1+C(C_{0},D_{0})\varepsilon\right).$

Taking $D_{0}$ large enough and $\varepsilon$ small enough compared to
$D_{0}^{-1}$ and $C_{0}^{-1}$ shows that $\Psi^{(c)}(\mathfrak{p})\in
B_{\mathcal{Z}_{q}}(0,D_{0}\varepsilon)$ and thus that the corrector map
$\Psi^{(c)}$ is well-defined.

Next let $\mathfrak{p},\mathfrak{p}^{\prime}\in
B_{\mathcal{Z}_{q}}(0,D_{0}\varepsilon)$. Taking the difference of the
relations defining $\Psi^{(c)}(\mathfrak{p})$ and
$\Psi^{(c)}(\mathfrak{p}^{\prime})$ and using Lemma 5.6 we obtain

$\displaystyle\left\|\Psi^{(c)}(\mathfrak{p})-\Psi^{(c)}(\mathfrak{p}^{\prime})\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}$
$\displaystyle\lesssim\sup_{j,\ell,{\alpha}}\left|\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|$
$\displaystyle\lesssim
C(C_{0},D_{0})\varepsilon\left\|\mathfrak{p}-\mathfrak{p}^{\prime}\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)},$

where we have used (5.12). Taking $\varepsilon$ small enough compared to
$D_{0}^{-1}$ and $C_{0}^{-1}$ shows that $\Psi^{(c)}$ is a contraction. ∎

######  Corollary 5.10.

Let $\widecheck{\mathbb{U}}$ satisfying (5.36). If $\varepsilon$ is small
enough compared to $C_{0}^{-1}$, there exists a unique
$\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\in\mathcal{Z}_{q}$ such that
(5.37) holds. Moreover we have

$\displaystyle\left\|\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}\lesssim\varepsilon$ (5.40)

and

$\displaystyle\left\|\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)-\mathfrak{p}\left(\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}\lesssim
C(C_{0})\varepsilon\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)},$
(5.41)

if $\widecheck{\mathbb{U}}$ and $\widecheck{\mathbb{U}}^{\prime}$ both satisfy
(5.36).

###### Proof.

The proof of the first part of the corollary follows from Lemma 5.9 and the
Banach fixed point theorem. For the proof of (5.41) we take the difference of
the relations defining
$\mathfrak{p}\vcentcolon=\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)$ and
$\mathfrak{p}^{\prime}\vcentcolon=\mathfrak{p}\left(\widecheck{\mathbb{U}}^{\prime}\right)$
together with Lemma 5.6 and (5.12) to obtain

$\displaystyle\left\|\mathfrak{p}-\mathfrak{p}^{\prime}\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}$
$\displaystyle\lesssim\sup_{j,\ell,{\alpha}}\left|\left\langle{\mathcal{R}}\left(\mathfrak{p},\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}^{\prime},\widecheck{\mathbb{U}}^{\prime}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}\right|$
$\displaystyle\lesssim
C(C_{0})\varepsilon\bigg{(}\left\|\mathfrak{p}-\mathfrak{p}^{\prime}\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}+\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\bigg{)}.$

By taking $\varepsilon$ small enough compared to $C_{0}^{-1}$ we can absorb
$C(C_{0})\varepsilon\left\|\mathfrak{p}-\mathfrak{p}^{\prime}\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}$ by the LHS and obtain (5.41). ∎

###  5.4. Solving the elliptic system

In this section we conclude the proof of Theorem 3.1 by solving the elliptic
system for $\widecheck{\mathbb{U}}$ with the corrector
$\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)$ constructed in Section 5.3.
We want to solve

$\left\\{\begin{aligned}
P\left(\widecheck{\mathbb{U}}\right)&=D\Phi[\bar{g},\bar{\pi}]\left((\widecheck{g},\widecheck{\pi})+\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\right)+{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right),\widecheck{\mathbb{U}}\right),\quad\text{on
$\Sigma$,}\\\
(B_{\nu}+F_{\delta})\left(\widecheck{\mathbb{U}}\right)&=0,\quad\text{on
$\partial\Sigma$.}\end{aligned}\right.$ (5.42)

######  Definition 5.11.

Let $C_{0}>0$. The _solution map_ $\Psi^{(s)}$ is defined by

$\displaystyle\Psi^{(s)}:B_{H^{2}_{-q-\delta}(\Sigma)}(0,C_{0}\varepsilon)$
$\displaystyle\longrightarrow
B_{H^{2}_{-q-\delta}(\Sigma)}(0,C_{0}\varepsilon)$
$\displaystyle\longmapsto\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)$

such that $\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)$ solves

$\left\\{\begin{aligned}
P\left(\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)\right)&=D\Phi[\bar{g},\bar{\pi}]\left((\widecheck{g},\widecheck{\pi})+\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\right)+{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right),\widecheck{\mathbb{U}}\right),\quad\text{on
$\Sigma$,}\\\
(B_{\nu}+F_{\delta})\left(\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)\right)&=0,\quad\text{on
$\partial\Sigma$.}\end{aligned}\right.$ (5.43)

######  Lemma 5.12.

If $C_{0}$ is large enough and $\varepsilon$ is small enough compared to
$C_{0}^{-1}$, the solution map $\Psi^{(s)}$ is a well-defined contraction.

###### Proof.

Let $\widecheck{\mathbb{U}}\in
B_{H^{2}_{-q-\delta}(\Sigma)}(0,C_{0}\varepsilon)$. First note that if
$\varepsilon$ is small enough compared to $C_{0}^{-1}$ then Corollary 5.10
applies and the corrector $\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)$
exists and satisfy the bounds of Corollary 5.10. By construction, this
corrector is such that

$\displaystyle\left\langle
D\Phi[\bar{g},\bar{\pi}]\left((\widecheck{g},\widecheck{\pi})+\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\right)+{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right),\widecheck{\mathbb{U}}\right),\mathbb{W}_{j,\ell,{\alpha}}\right\rangle_{L^{2}(\Sigma)}$
$\displaystyle=0,$

for all $1\leq j\leq q$, $-(j-1)\leq\ell\leq j-1$ and ${\alpha}=0,1,2,3$.
Moreover, using (2.4), (2.5), asymptotic flatness, (3.1), (5.6) and (5.40) we
obtain

$\displaystyle\left\|D\Phi[\bar{g},\bar{\pi}]\left((\widecheck{g},\widecheck{\pi})+\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)\right)+{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right),\widecheck{\mathbb{U}}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim\varepsilon(1+C(C_{0})\varepsilon).$

Therefore, the third point in Proposition 4.12 implies that there exists a
unique $\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)$ solving (5.43) with the
bound

$\displaystyle\left\|\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)\right\|_{H^{2}_{-q-\delta}(\Sigma)}\lesssim\varepsilon(1+C(C_{0})\varepsilon).$

Choosing first $C_{0}$ large enough and then $\varepsilon$ small enough
compared to $C_{0}^{-1}$ implies that
$\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)\in
B_{H^{2}_{-q-\delta}(\Sigma)}(0,C_{0}\varepsilon)$ and thus that the solution
map $\Psi^{(s)}$ is well-defined.

Next let $\widecheck{\mathbb{U}},\widecheck{\mathbb{U}}^{\prime}\in
B_{H^{2}_{-q-\delta}(\Sigma)}(0,C_{0}\varepsilon)$. We substract the two
systems defining $\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)$ and
$\Psi^{(s)}\left(\widecheck{\mathbb{U}}^{\prime}\right)$ together with
Proposition 4.12 and (5.11) to obtain

$\displaystyle\left\|\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)-\Psi^{(s)}\left(\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{H^{2}_{-q-\delta}(\Sigma)}$
$\displaystyle\lesssim\left\|D\Phi[\bar{g},\bar{\pi}]\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)-\mathfrak{p}\left(\widecheck{\mathbb{U}}^{\prime}\right)\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\quad+\left\|{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}\right),\widecheck{\mathbb{U}}\right)-{\mathcal{R}}\left(\mathfrak{p}\left(\widecheck{\mathbb{U}}^{\prime}\right),\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{L^{2}_{-q-\delta-2}(\Sigma)}$
$\displaystyle\lesssim(1+C(C_{0})\varepsilon)\left\|\mathfrak{p}\left(\widecheck{\mathbb{U}}\right)-\mathfrak{p}\left(\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{W^{2,\infty}(\Sigma)\times
W^{1,\infty}(\Sigma)}$
$\displaystyle\quad+C(C_{0})\varepsilon\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}.$

We now use (5.41) to obtain

$\displaystyle\left\|\Psi^{(s)}\left(\widecheck{\mathbb{U}}\right)-\Psi^{(s)}\left(\widecheck{\mathbb{U}}^{\prime}\right)\right\|_{H^{2}_{-q-\delta}(\Sigma)}$
$\displaystyle\lesssim
C(C_{0})\varepsilon\left\|\widecheck{\mathbb{U}}-\widecheck{\mathbb{U}}^{\prime}\right\|_{H^{2}_{-q-\delta}(\Sigma)}.$

Taking $\varepsilon$ small enough compared to $C_{0}^{-1}$ shows that
$\Psi^{(s)}$ is a contraction. ∎

In view of Lemma 5.12, the Banach fixed point theorem implies that if
$\varepsilon$ is small enough then there exists a unique solution
$\widecheck{\mathbb{U}}$ to the system (5.3) with the bound

$\displaystyle\left\|\widecheck{\mathbb{U}}\right\|_{H^{2}_{-q-\delta}(\Sigma)}\lesssim\varepsilon.$

This concludes the proof of Theorem 3.1.

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