#  Noncollapsed degeneration of Einstein $4$-manifolds II
Desingularization of Einstein metrics and obstructions

Tristan OZUCH  École Normale Supérieure, Université PSL

###### Abstract

In this second article, we prove that any desingularization in the Gromov-
Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation
procedure that we develop. This builds on our first paper where we proved that
a Gromov-Hausdorff convergence implied a much stronger convergence in suitable
weighted Hölder spaces, in which the analysis of the present paper takes
place.

The description of Einstein metrics as the result of a gluing-perturbation
procedure also sheds light on the local structure of the moduli space of
Einstein metrics near its boundary. More importantly here, we extend the
obstruction to the desingularization of Einstein orbifolds found by Biquard,
and prove that it holds for any desingularization by trees of quotients of
gravitational instantons only assuming a mere Gromov-Hausdorff convergence
instead of specific weighted Hölder spaces. This is conjecturally the general
case, but it can at least be ensured by topological assumptions such as a spin
structure on the degenerating manifolds. We also identify an obstruction to
desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE
spaces.

###### Contents

  1. 1 Orbifolds, ALE spaces and naïve desingularizations
    1. 1.1 Einstein orbifolds and ALE spaces
    2. 1.2 Naïve desingularizations
  2. 2 Weighted Hölder spaces and decoupling norms
    1. 2.1 Weighted Hölder spaces
      1. 2.1.1 Weighted Hölder spaces on orbifold and ALE spaces
      2. 2.1.2 Weighted Hölder spaces on trees of singularities
      3. 2.1.3 Weighted Schauder estimates
    2. 2.2 Decoupling norms
      1. 2.2.1 Estimates on annuli $A_{e}(\epsilon,\epsilon^{-1})$ of $(\mathbb{R}^{4},g_{e})$.
      2. 2.2.2 Approximate kernels
      3. 2.2.3 Estimates in the decoupling norms
      4. 2.2.4 Elliptic estimates for the decoupling norms
  3. 3 Reduced divergence-free gauge
    1. 3.1 Kernel of the linearization
    2. 3.2 Controls on the inverse of the linearization
    3. 3.3 Einstein metrics in gauge
  4. 4 Resolution of the Einstein equation modulo obstructions
    1. 4.1 Kernel and cokernel of the linearization on model spaces
    2. 4.2 Resolution modulo obstruction of the Einstein equation
  5. 5 Obstructions to the Gromov-Hausdorff desingularization of Einstein orbifolds
    1. 5.1 Better approximations and obstructions
    2. 5.2 Partial desingularizations
      1. 5.2.1 Weighted Hölder spaces on partial desingularizations
      2. 5.2.2 Resolution of the Einstein modulo obstructions equation for partial desingularization
    3. 5.3 An integrability issue
      1. 5.3.1 Weighted Hölder spaces and asymptotics of Ricci-flat ALE spaces
      2. 5.3.2 A new naïve desingularization and obstructions
    4. 5.4 Total desingularizations and approximations of Einstein modulo obstructions metric
    5. 5.5 Obstruction to the Gromov-Hausdorff desingularization of Einstein orbifolds
  6. 6 Obstructions for known singularity models
    1. 6.1 Kähler Ricci-flat ALE metrics and obstructions
    2. 6.2 Trees of Kähler Ricci-flat ALE orbifolds and obstructions
  7. 7 Obstructions under topological assumptions
    1. 7.1 Hitchin-Thorpe inequality and desingularization of Einstein orbifolds
    2. 7.2 Spin manifolds
    3. 7.3 Pinched Ricci curvature and the Einstein condition
  8. 8 A general obstruction for spherical and hyperbolic orbifolds
    1. 8.1 A general infinitesimal deformation for Ricci-flat ALE spaces
    2. 8.2 Obstructions to the desingularization of spherical and hyperbolic orbifolds

## Introduction

An Einstein metric, $g$ satisfies, for some real $\Lambda$, the equation

$\operatorname{\textup{Ric}}(g)=\Lambda g.$

In dimension $4$, they are considered optimal for the homogeneity of their
Ricci curvature, as critical points of the Einstein-Hilbert functional with
fixed volume, $g\mapsto\int_{M}\operatorname{\textup{R}}_{g}dvol_{g}$, and
more importantly as minimizers of the $L^{2}$-norm of Riemann curvature
tensor, $g\mapsto\int_{M}|\operatorname{\textup{Rm}}_{g}|^{2}dvol_{g}$.

From dimension $4$, even under natural assumptions of bounded diameter
(compactness) and lower bound on the volume (noncollapsing) Einstein metrics
can develop singularities. One major goal for $4$-dimensional geometry is
therefore to understand the set of Einstein metrics and to compactify it. The
possible limits and the singularity blow-ups in the Gromov-Hausdorff sense
have been understood for a long time in [And89, BKN89], they are respectively
_Einstein orbifolds_ and _Ricci-flat ALE orbifolds_.

An important question is to understand if the reverse of the degeneration, the
_desingularization_ , of Einstein orbifolds is possible. A natural way to
desingularize an orbifold is by a gluing-perturbation technique.

The goal of the present paper is to develop a gluing-perturbation procedure
which attains _any_ noncollapsed Einstein $4$-manifold which is sufficiently
close to an Einstein orbifold in the Gromov-Hausdorff sense. This in
particular elucidates the neighborhood of the boundary of the moduli space of
Einstein metrics, and we will use this description in future works. In this
paper, we will use it to prove that not all Einstein orbifolds can be
desingularized by Einstein metrics in the Gromov-Hausdorff sense.

### Desingularization of Einstein $4$-orbifolds and obstructions

A natural technique to desingularize orbifolds is the following procedure: we
glue Ricci-flat ALE manifolds to the singularities of the orbifold to obtain
an approximate Einstein metrics, and then try to perturbate it into an actual
Einstein metric. We will call such gluings, naïve desingularizations of the
orbifold and often denote them $g^{D}_{t}$ (see Definition 1.6), where $t$ is
the set of gluing parameters which are small positive real numbers. The main
result of [Ozu19a] is that these naïve desingularizations are good
approximations of the Einstein metrics close to a given orbifold in the sense
of a weighted Hölder norm denoted $C^{2,\alpha}_{\beta,*}$ which is bounded on
$2$-tensors decaying in the neck regions.

In the present paper, we will propose a partial converse by proving that any
naïve desingularization can be perturbated to a metric which is Einstein
modulo some _obstructions_ , which are elements of an approximate cokernel of
the linearization of the Einstein operator. We will call such a metric an
_Einstein modulo obstructions_ metric.

######  Theorem 0.1.

Let $g^{D}_{t}$ be a naïve desingularization of an Einstein orbifold
$(M_{o},g_{o})$ with small enough gluing parameters.

Then, there exists a small $C^{2,\alpha}_{\beta,*}$-neighborhood of
$g^{D}_{t}$ in which there exists a unique metric $\hat{g}_{t}$ which is
Einstein _modulo obstructions_ while satisfying some gauge conditions with
respect to $g^{D}_{t}$.

The proof relies on an inverse function theorem applied to the Einstein
operator in well chosen coordinates.

Together with the convergence of [Ozu19a], we have the following result as a
direct consequence.

######  Corollary 0.2.

Let $D_{0},v_{0}>0$, $l\in\mathbb{N}$, then, there exists
$\delta=\delta(D_{0},v_{0},l)>0$ such that if $(M,g^{\mathcal{E}})$ is an
Einstein manifold satisfying

  * •

the volume is bounded below by $v_{0}>0$,

  * •

the diameter is bounded by $D_{0}>0$,

  * •

the Ricci curvature is bounded $|\operatorname{\textup{Ric}}|\leq 3$.

and for which there exists an Einstein orbifold $(M_{o},g_{o})$ with

$d_{GH}\big{(}(M,g^{\mathcal{E}}),(M_{o},g_{o})\big{)}\leqslant\delta,$

then, $(M,g^{\mathcal{E}})$ is isometric to a result of the gluing-
perturbation procedure of Theorem 0.1.

### Obstructions to the Gromov-Hausdorff desingularization of Einstein
orbifolds

There is a well-known family of Ricci-flat ALE spaces which are the so-called
_gravitational instantons_. They have been classified in [Kro89] and their
Kähler quotients have been classified in [Suv11]. It is a famous conjecture
that all Ricci-flat ALE spaces are of these two types.

Our first goal here is to prove that an obstruction holds for any Gromov-
Hausdorff desingularization by trees of Kähler Ricci-flat ALE orbifolds, which
are conjecturally the only possibilities. The obstruction to satisfy is
$\det\mathbf{R}=0$ at a singular point of the orbifold metric, where
$\mathbf{R}$ is the Riemannian curvature seen as an endomorphism on the space
of $2$-forms.

######  Theorem 0.3.

Let $(M_{i},g_{i})$ be a sequence of Einstein manifolds converging in the
Gromov-Hausdorff sense to an Einstein orbifold $(M_{o},g_{o})$, and assume
that there exists a subsequence $(M_{i},g_{i})$ whose possible blow-up limits
are Kähler Ricci-flat ALE orbifolds.

Then, at every singular point $p$ of $(M_{o},g_{o})$, we have

$\det\mathbf{R}_{g_{o}}(p)=0.$

This answers positively a question from [Biq13]. It more precisely states that
the obstruction of [Biq13] holds for any known possible configuration of
singularity models, and that it holds even assuming the weakest possible
convergence instead of a convergence in particular weighted Hölder spaces.

Under topological assumptions, it is known that the singularity models
appearing are Kähler, in particular we have the following illustration of our
obstruction.

######  Example 0.4.

Consider $\mathbb{S}^{4}\subset\mathbb{R}^{5}$ and the quotient by
$\mathbb{Z}_{2}$ given by
$(x_{1},x_{2},x_{3},x_{4},x_{5})\sim(x_{1},-x_{2},-x_{3},-x_{4},-x_{5})$. We
will denote this space $\mathbb{S}^{4}/\mathbb{Z}_{2}$ which is an Einstein
orbifold with two $\mathbb{R}^{4}/\mathbb{Z}_{2}$ singularities. The minimal
resolutions of the two singularities
$\mathbb{R}^{4}/\mathbb{Z}_{2}\approx\mathbb{C}^{2}/\mathbb{Z}_{2}$ has the
topology
$M:=\mathbb{S}^{4}/\mathbb{Z}_{2}\\#T^{*}\mathbb{S}^{2}\\#T^{*}\mathbb{S}^{2}$,
where $\\#$ denotes the gluing of an ALE space to an orbifold along their
asymptotic cone. Then, for any $1\leqslant p<\infty$, there exists a sequence
of metrics $(M,g_{i})$ with both

$\|\operatorname{\textup{Ric}}(g_{i})-3g_{i}\|_{L^{p}(g_{i})}\to 0\text{ and
}\operatorname{\textup{Ric}}(g_{i})\geqslant 3g_{i}\text{ or
}\operatorname{\textup{Ric}}(g_{i})\leqslant 3g_{i}$

while

$(M,g_{i})\xrightarrow[]{GH}(\mathbb{S}^{4}/\mathbb{Z}_{2},g_{\mathbb{S}^{4}/\mathbb{Z}_{2}}),$

but there _does not_ exist any sequence of Einstein metrics satisfying

$\operatorname{\textup{Ric}}(g_{i})=3g_{i},$

and

$(M,g_{i})\xrightarrow[]{GH}(\mathbb{S}^{4}/\mathbb{Z}_{2},g_{\mathbb{S}^{4}/\mathbb{Z}_{2}}).$

In the same fashion, a conjecture of Anderson states that there is no sequence
of asymptotically hyperbolic Einstein metrics on $T^{*}\mathbb{S}^{2}$
desingularizing the hyperbolic orbifold $\mathbb{H}^{4}/\mathbb{Z}_{2}$
obtained by antipodal identification in a global geodesic chart. It was proven
in [Biq13] assuming among other things a convergence speed in weighted spaces
towards the orbifold depending on the maximum of the curvature. We can prove
it assuming a pointed Gromov-Hausdorff convergence together with a suitable
control in weighted spaces at infinity, this time independent on the maximum
of the curvature. It is again possible to desingularize
$\mathbb{H}^{4}/\mathbb{Z}_{2}$ with Ricci pinched in any $L^{p}$, space for
$1\leqslant p<\infty$ or with $\operatorname{\textup{Ric}}$ bounded above or
below by $-3$.

#### Hitchin-Thorpe inequality and degeneration of Einstein manifolds

The Hitchin-Thorpe inequality provides a topological obstruction to the
existence of Einstein metrics on a given $4$-dimensional differentiable
manifold $M$,

$2\chi(M)\geqslant 3|\tau(M)|,$

where $\chi$ is the Euler characteristic, and $\tau$ the signature. These
topological invariants have definitions adapted to orbifolds and ALE spaces
which we will denote $\tilde{\chi}$ and $\tilde{\tau}$, and any orbifold
$M_{o}$ admitting an Einstein metric satisfies

$2\tilde{\chi}(M_{o})\geqslant 3|\tilde{\tau}(M_{o})|.$

Any Gromov-Hausdorff desingularization deteriorates this inequality, and the
equality case implies the obstruction

######  Theorem 0.5.

Let $(M_{o},g_{o})$ be an Einstein orbifold, and assume that $(M,g_{i})$ is a
sequence of Einstein metrics converging to $(M_{o},g_{o})$ in the Gromov-
Hausdorff sense.

Then, we have the following inequality,

$2\chi(M)-3|\tau(M)|\geqslant 2\tilde{\chi}(M_{o})-3|\tilde{\tau}(M_{o})|.$

Moreover, there is equality if and only if $M$ is a desingularization of
$M_{o}$ by gluing of trees of Kähler Ricci-flat ALE orbifolds in the same
orientation (with the same sign for $\tilde{\tau}$). In this equality case, we
have the condition

$\det\mathbf{R}(g_{o})=0$

at every singular point.

#### Degeneration of Einstein metrics on spin manifolds

Another large class of manifolds on which we can prove our obstruction is the
class of $4$-manifolds admitting a spin structure.

######  Theorem 0.6.

Let $(M_{i},g_{i})$ be a sequence of _spin_ Einstein $4$-manifolds converging
to an Einstein orbifold $(M_{o},g_{o})$ in the Gromov-Hausdorff sense. Then,
$(M_{o},g_{o})$ is spin, and at any of its singular points whose group is in
$SU(2)$, we have the obstruction

$\det\mathbf{R}_{g_{o}}=0.$

##### General obstructions for spherical and hyperbolic orbifolds.

Our Theorem 0.2 holds for any singularity model which might be non-Kähler. We
will use it lastly to identify an obstruction to desingularizing any spherical
or hyperbolic orbifolds by _any_ Ricci-flat ALE manifold. This provides an
obstruction to any standard gluing-perturbation technique but will only imply
an actual obstruction to the Gromov-Hausdorff desingularization by Ricci-flat
ALE manifolds whose deformations are _integrable_ (this is the case of all
known examples).

######  Theorem 0.7.

Spherical and hyperbolic orbifolds cannot be desingularized in the Gromov-
Hausdorff sense by Ricci-flat ALE spaces which are integrable (see Definition
5.11).

### Outline of the paper

In Section 1, we give the principal definitions, and in Section 2, we
introduce and motivate the function spaces we will use throughout the paper,
and moreover restate the results of [Ozu19a] thanks to them.

In Section 3, we prove that we can always pull-back an Einstein metric which
is Gromov-Hausdorff close to an orbifold by a small diffeomorphism to ensure
that it satisfies some gauge condition with respect to a naïve
desingularization. The proof consists in a Lyapunov-Schmidt reduction in our
weighted norms where the relevant operators are proven to be Fredholm.

In Section 4, we prove that any naïve desingularization can be perturbated to
a metric which is Einstein modulo some obstruction, that is, an approximate
cokernel of the linearization of the gauged Einstein operator. The point is
that every possible Einstein metric is produced this way, and that whenever
the obstructions do not vanish, it is impossible to perturbate the naïve
desingularization to an Einstein metric. The proof again relies on a Lyapunov-
Schmidt reduction in our weighted Hölder spaces.

In Section 5, we estimate the obstructions to the above Einstein
desingularization modulo obstructions. To obtain such an obstruction at _all_
singular points, we need to develop an analysis on partial desingularizations
and produce better approximations of Einstein modulo obstructions metrics.

In Section 6, we test the above obstructions on degenerations of Einstein
manifolds forming trees of Kähler Ricci-flat ALE orbifolds. By developing our
analysis on trees of singularities, we prove that the obstruction of [Biq13]
for the Eguchi-Hanson metric extends to any tree of quotients of gravitational
instantons and holds under a mere Gromov-Hausdorff convergence.

In Section 7, we investigate topological conditions which ensure that a
sequence of Einstein manifold degenerating will only produce trees of Kähler
Ricci-flat ALE spaces. We mainly use the result of [Nak90] and consider the
behavior of the Hitchin-Thorpe inequality as well as the degeneration of
Einstein metrics on a spin manifold.

In Section 8, building on the notion of maximal volume for Ricci-flat ALE
spaces of [BH19], we prove that even without assuming that the trees of
singularities are Kähler, there is a non vanishing obstruction to the
desingularization of spherical and hyperbolic orbifolds. We can however only
prove that this is a Gromov-Hausdorff obstruction under the technical
assumption that the Ricci-flat ALE spaces have integrable deformations.

### Acknowledgements

I would like to thank my PhD advisor, Olivier Biquard, for his constant
support as well as his mathematical and writing advice. I would also like to
thank Aaron Naber for inviting me to Northwestern University in Spring 2017,
and for pointing out and discussing the issue addressed in Section 8.

##  1 Orbifolds, ALE spaces and naïve desingularizations

Let us start by definig the objects we will use throughout this article.

###  1.1 Einstein orbifolds and ALE spaces

######  Definition 1.1 (Einstein orbifold).

We will say that a metric space $(M_{o},g_{o})$ is an _Einstein orbifold_ if
there exists $\epsilon_{0}>0$ and a finite number of points $(p_{k})_{k}$ of
$M_{o}$ which we will call _singular_ such that we have the following:

  1. 1.

the space $(M_{o}\backslash\\{p_{k}\\}_{k},g_{o})$ is an Einstein manifold,

  2. 2.

for each singular point $p_{k}$ of $M_{o}$, there exists a neighborhood of
$p_{k}$, $U_{k}\subset M_{o}$, a finite subgroup acting freely on the sphere,
$\Gamma_{k}\subset SO(4)$, and a diffeomorphism
$\Phi_{k}:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{4}/\Gamma_{k}\to
U_{k}\subset M_{o}$ for which, $\Phi_{k}^{*}g_{o}$ is smooth on
$\mathbb{R}^{4}$ and for any $l\in\mathbb{N}$, there exists $C_{l}>0$ such
that

$r_{e}^{l}|\nabla^{l}(\Phi_{k}^{*}g_{o}-g_{e})|_{C^{2}(g_{e})}\leqslant
C_{l}r_{e}^{2}.$

######  Remark 1.2.

This definition implies that the metric obtained by pulling back $g_{o}$
restricted to $U_{k}$ on an open set of $\mathbb{R}^{4}/\Gamma_{k}$ and then
on an open set of $\mathbb{R}^{4}$ is actually an analytic metric.
Consequently, the analysis on an orbifold is exactly the same as the analysis
on a manifold up to using finite local coverings at the singular points.

######  Definition 1.3 (The function $r_{o}$ on an orbifold).

We define $r_{o}$, a smooth function on $M_{o}$ satisfying
$r_{o}:=\Phi_{k}^{*}r_{e}$ on each $U_{k}$, and such that on $M_{o}\backslash
U_{k}$, we have $\epsilon_{0}<r_{o}<1$ (the different choices will be
equivalent for our applications).

We will denote, for $0<\epsilon\leqslant\epsilon_{0}$,

$M_{o}(\epsilon):=\\{r_{o}>\epsilon\\}=M_{o}\backslash\Big{(}\bigcup_{k}\Phi_{k}(B_{e}(0,\epsilon))\Big{)}.$

Let us now turn to ALE Ricci-flat metrics.

######  Definition 1.4 (ALE Ricci-flat orbifold).

An ALE Ricci-flat orbifold $(N,g_{b})$ is a metric space for which there
exists $\epsilon_{0}>0$, singular points $(p_{k})_{k}$ and a compact $K\subset
N$ for which we have:

  1. 1.

$(N\backslash\\{p_{k}\\}_{k},g_{b})$ is a Ricci-flat Riemannian manifold,

  2. 2.

for each singular point $p_{k}$ of $N$, there exists $U_{k}$, a neighborhood
of $p_{k}$ in $N$, $\Gamma_{k}\subset SO(4)$, a finite subgroup acting freely
on the sphere $\mathbb{S}^{3}$, and a diffeomorphism
$\Psi_{k}:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{4}/\Gamma_{k}\to
U_{k}\subset M_{o}$ such that for all $l\in\mathbb{N}$, there exists $C_{l}>0$
for which

$r_{e}^{l}|\nabla^{l}(\Psi_{k}^{*}g_{b}-g_{e})|_{C^{2}(g_{e})}\leqslant
C_{l}r_{e}^{2},$

  3. 3.

there exists a diffeomorphism
$\Psi_{\infty}:(\mathbb{R}^{4}/\Gamma_{\infty})\backslash
B_{e}(0,\epsilon_{0}^{-1})\to N\backslash K$ such that we have

$r_{e}^{l}|\nabla^{l}(\Psi_{\infty}^{*}g_{b}-g_{e})|_{C^{2}(g_{e})}\leqslant
C_{l}r_{e}^{-4}.$

######  Definition 1.5 (The function $r_{b}$ on an ALE orbifold).

We define $r_{b}$ a smooth function on $N$ satisfying
$r_{b}:=\Psi_{k}^{*}r_{e}$ on each $U_{k}$, and
$r_{b}:=\Psi_{\infty}^{*}r_{e}$ on $U_{\infty}$, and such that
$\epsilon_{0}<r_{b}<\epsilon_{0}^{-1}$ on the rest of $N$ (the different
choices are equivalent for our applications).

For $0<\epsilon\leqslant\epsilon_{0}$, we will denote

$N(\epsilon):=\\{\epsilon<r_{b}<\epsilon^{-1}\\}=N\backslash\Big{(}\bigcup_{k}\Psi_{k}(B_{e}(0,\epsilon))\cup\Psi_{\infty}\big{(}(\mathbb{R}^{4}/\Gamma_{\infty})\backslash
B_{e}(0,\epsilon^{-1})\big{)}\Big{)}.$

###  1.2 Naïve desingularizations

Let us now recall the definition of a naïve desingularization of an orbifold
from [Ozu19a].

##### Gluing of ALE orbifolds to orbifold singularities

Let $0<2\epsilon<\epsilon_{0}$ be a fixed constant, $t>0$, and $(M_{o},g_{o})$
an orbifold and $\Phi:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{4}/\Gamma\to U$
a local chart around a singular point $p\in M_{o}$ such that the metric
$\Phi^{*}g_{o}$ is analytic. Let also $(N,g_{b})$ be an ALE orbifold
asymptotic to $\mathbb{R}^{4}/\Gamma$, and
$\Psi_{\infty}:(\mathbb{R}^{4}/\Gamma_{\infty})\backslash
B_{e}(0,\epsilon_{0}^{-1})\to N\backslash K$ a chart at infinity in which
$\Psi_{\infty}^{*}g_{b}$ equals a power series in $r_{e}^{-1}$.

######  Definition 1.6 (Naïve gluing of an ALE space to an orbifold).

We define a _naïve gluing_ of $(N,g_{b})$ at scale $t$ to $(M_{o},g_{o})$ at
the singular point $p$, which we will denote $(M_{o}\\#N,g_{o}\\#_{p,t}g_{b})$
by putting $g_{o}\\#_{p,t}g_{b}=g_{o}$ on $M\backslash U$,
$g_{o}\\#_{p,t}g_{b}=tg_{b}$ on $K$, and

$g_{o}\\#_{p,t}g_{b}=\chi(t^{-\frac{1}{4}}r_{e})\Phi^{*}g_{o}+\big{(}1-\chi(t^{-\frac{1}{4}}r_{e})\big{)}\Psi_{\infty}^{*}g_{b}$

on $A(t,\epsilon):=A_{e}(\epsilon^{-1}\sqrt{t},2\epsilon)$.

More generally, it is possible to desingularize iteratively by trees of Ricci-
flat ALE orbifolds. Consider $(M_{o},g_{o})$ an Einstein orbifold, and $S_{o}$
a subset of its singular points and $(N_{j},g_{b_{j}})_{j}$ a family of Ricci-
flat ALE spaces asymptotic at infinity to $\mathbb{R}^{4}/\Gamma_{j}$ and
$(S_{b_{j}})_{j}$ a subset of their singular points. Let us finally assume
that there is a one to one map $p:j\mapsto p_{j}\in
S_{o}\cup\bigcup_{k}S_{b_{k}}$, where the singularity at $p_{j}$ is
$\mathbb{R}^{4}/\Gamma_{j}$. We will call
$D:=\big{(}(M_{o},g_{o},S_{o}),(N_{j},g_{b_{j}},S_{b_{j}})_{j},p\big{)}$ a
_desingularization pattern_.

######  Definition 1.7 (Naïve desingularization by a tree of singularities).

Let $0<2\epsilon<\epsilon_{0}$, $D$ be a desingularization pattern for
$(M_{o},g_{o})$, and let $0<t_{j}<\epsilon^{4}$ be relative gluing scales. The
metric $g^{D}_{t}$ is then the result of the following finite iteration: start
with a deepest bubble $(N_{j},g_{b_{j}})$, that is, $j$ such that
$S_{j}=\emptyset$. If $p_{j}\in N_{k}$ and we can replace
$(N_{k},g_{b_{k}},S_{j})$ and $(N_{j},g_{b_{j}},\emptyset)$ by
$(N_{k}\\#N_{j},g_{b_{k}}\\#_{p_{j},t_{j}},S_{k}\backslash\\{p_{j}\\})$ and
restrict $p$ as $l\to p_{l}$ for $l\neq j$ in $D$ and consider another deepest
bubble, the same works if $p_{j}\in M_{o}$.

We moreover define, for $t=(t_{j})_{j}$, if $N_{j}$ is glued to $p_{j}\in
N_{j_{1}}$, and $N_{j_{1}}$ is glued to $p_{j_{1}}\in N_{j_{2}}$, …,
$N_{j_{k-1}}$ is glued to $N_{j_{k}}$, which is glued to $M_{o}$, we define
$T_{j}:=t_{j_{1}}t_{j_{2}}...t_{j_{k}}$. This way, on each $N_{j}(b\epsilon)$,
the metric is $T_{j}g_{b_{j}}$.

For $\Gamma$ a finite subgroup of $SO(4)$ acting freely on $\mathbb{S}^{3}$,
let us denote $(\mathbb{R}^{4}/\Gamma,g_{e})$ the flat orbifold obtained by
the quotient by the action of $\Gamma$, and $r_{e}:=d_{e}(.,0)$. Let
$(M_{o},g_{o})$ be an Einstein orbifold, and $(M,g^{D})$ a naïve
desingularization of $(M_{o},g_{o})$ by a tree of ALE Ricci-flat orbifolds
$(N_{j},g_{b_{j}})$ glued at scales $T_{j}>0$.

In [Ozu19a], for $0<\epsilon\leqslant\epsilon_{0}$ we have used a covering

$M=M_{o}(\epsilon)\cup\bigcup_{j}N_{j}(\epsilon)\cup\bigcup_{k}\mathcal{A}_{k}(\epsilon),$

where $\mathcal{A}_{k}(\epsilon)$ is the annulus between $N_{k}(32\epsilon)$
and $M_{o}(32\epsilon)$ if $N_{k}$ is glued to $M_{o}$, or the annulus between
$N_{k}(32\epsilon)$ and $N_{j}(32\epsilon)$ if $N_{k}$ is glued to $N_{j}$.

Here, the manifold $M$ is also covered as
$M=M_{o}^{t}\cup\bigcup_{j}N_{j}^{t}$, where

$M_{o}^{t}:=M_{o}\backslash\big{(}\bigcup_{k}\Phi_{k}(B_{e}(0,t_{k}^{\frac{1}{4}}))\big{)},$

where $t_{k}>0$ the relative gluing scale of $N_{k}$ at the singular point
$p_{k}\in M_{o}$, and where

$N_{j}^{t}:=\big{(}N_{j}\backslash\Psi_{\infty}\big{(}(\mathbb{R}^{4}/\Gamma_{\infty})\backslash
B_{e}(0,2t_{j}^{-\frac{1}{4}})\big{)}\big{)}\backslash\big{(}\bigcup_{k}\Psi_{k}(B_{e}(0,t_{k}^{\frac{1}{4}})\big{)}.$

On $M_{o}^{16t}\subset M_{o}^{t}$, we have $g^{D}=g_{o}$ and on each
$N_{j}^{16t}\subset N_{j}^{t}$, we have $g^{D}=T_{j}g_{b_{j}}$. We also define
$t_{\max}:=\max_{j}t_{j}$. By definition of the gluings, on the intersection
$N_{j}^{t}\cap M_{o}^{t}$ we then have $\sqrt{T_{j}}r_{b_{j}}=r_{o}$, and on
the intersection $N_{j}^{t}\cap N_{k}^{t}$, we have
$\sqrt{T_{j}}r_{b_{j}}=\sqrt{T_{k}}r_{b_{k}}$.

######  Definition 1.8 (Function $r_{D}$ on a naïve desingularization).

On a naïve desingularization $(M,g^{D})$, we define a function $r_{D}$ in the
following way:

  1. 1.

$r_{D}=r_{o}$ on $M_{o}^{t}$,

  2. 2.

$r_{D}=\sqrt{T_{j}}r_{b_{j}}$ on each $N_{j}^{t}$.

The function $r_{D}$ is smooth on $M$.

##  2 Weighted Hölder spaces and decoupling norms

###  2.1 Weighted Hölder spaces

Let us construct weighted Hölder spaces adapted to our situation. Let
$(M,g^{D}_{t})$ be a naïve desingularization of an orbifold $(M_{o},g_{o})$ by
Ricci-flat ALE orbifolds $(N_{j},g_{b_{j}},p_{j})$ at scales $T_{j}>0$.

####  2.1.1 Weighted Hölder spaces on orbifold and ALE spaces

Let us first define weighted spaces on manifolds asymptotic to cones or with
conical singularities. For a tensor $s$, a point $x$, $\alpha>0$ and a metric
$g$, the Hölder seminorm of $s$ on $M$ is defined as

$[s]_{C^{\alpha}(g)}(x):=\sup_{\\{y\in
T_{x}M,|y|<\textup{inj}_{g}(x)\\}}\Big{|}\frac{s(x)-s(\exp^{g}_{x}(y))}{|y|^{\alpha}}\Big{|}_{g}.$

For orbifolds, we will consider a norm which is bounded for tensors decaying
at the singular points.

######  Definition 2.1 (Weighted Hölder normes on an orbifold).

Let $\beta\in\mathbb{R}$, $k\in\mathbb{N}$, $0<\alpha<1$ and $(M_{o},g_{o})$
an orbifold. Then, for all tensor $s$ on $M_{o}$, we define

$\displaystyle\|s\|_{C^{k,\alpha}_{\beta}(g_{o})}$
$\displaystyle:=\sup_{M_{o}}r_{o}^{-\beta}\Big{(}\sum_{i=0}^{k}r_{o}^{i}|\nabla_{g_{o}}^{i}s|_{g_{o}}+r_{o}^{k+\alpha}[\nabla_{g_{o}}^{k}s]_{C^{\alpha}(g_{o})}\Big{)}.$

######  Remark 2.2.

The injectivity radius at a point $x\in M_{o}$ is equivalent to $r_{o}$.

For ALE orbifolds, we will consider a norm which is bounded for tensors
decaying at infinity and at the singular points.

######  Definition 2.3 (Weighted norm on ALE orbifolds).

For $\beta\in\mathbb{R}$, $k\in\mathbb{N}$ and $0<\alpha<1$ on an orbifold ALE
$(N,g_{b})$, we define

$\displaystyle\|s\|_{C^{k,\alpha}_{\beta}(g_{b})}:=\sup_{N}\Big{\\{}\max(r_{b}^{\beta},r_{b}^{-\beta})\Big{(}\sum_{i=0}^{k}r_{b}^{i}|\nabla_{g_{b}}^{i}s|_{g_{b}}+r_{b}^{k+\alpha}[\nabla_{g_{b}}^{k}s]_{C^{\alpha}({g_{b}})}\Big{)}\Big{\\}}.$

######  Remark 2.4.

The injectivity radius at a point $x\in N$ is equivalent to $r_{b_{j}}$.

####  2.1.2 Weighted Hölder spaces on trees of singularities

Let us assume that $(M,g^{D}_{t})$ is a naïve desingularization of
$(M_{o},g_{o})$ by a tree of singularities $(N_{j},g_{b_{j}})$. For
$t_{\max}<\epsilon_{0}^{4}$, for $\epsilon_{0}>0$ the constant of the
subsection 1.2 only depending on $g_{o}$ and the $g_{b_{j}}$ and
$t_{\max}:=\max_{j}t_{j}$, we define the global weighted norm in the following
way.

On $M$, we consider a partition of unity,

$1=\chi_{M_{o}^{t}}+\sum_{j}\chi_{N_{j}^{t}},$

where $\chi_{M_{o}^{t}}$ is supported in $M_{o}^{t}$ and equal to $1$ on
$M_{o}^{16t}$ and $\chi_{N_{j}^{t}}$ supported in $N_{j}^{t}$ and equal to $1$
on $N_{j}^{16t}$, are cut-off functions satisfying for all $k\in\mathbb{N}$,
there exists $C_{k}>0$ such that
$|\nabla^{k}_{g^{D}}\chi(x)|_{g^{D}}\leqslant\frac{C_{k}}{r_{D}(x)^{k}}$.

######  Definition 2.5 (Weighted Hölder norm on a naïve desingularization).

Let $\beta\in\mathbb{R}$ and $k\in\mathbb{N}$, $0<\alpha<1$. We define for
$s\in TM^{\otimes l_{+}}\otimes T^{*}M^{\otimes l_{-}}$ a tensor
$(l_{+},l_{-})\in\mathbb{N}^{2}$, with $l:=l_{+}-l_{-}$ the associated
conformal weight.

$\|s\|_{C^{k,\alpha}_{\beta}(g^{D})}:=\|\chi_{M_{o}^{t}}s\|_{C^{k,\alpha}_{\beta}(g_{o})}+\sum_{j}T_{j}^{\frac{l}{2}}\|\chi_{N_{j}^{t}}s\|_{C^{k,\alpha}_{\beta}(g_{b_{j}})}.$

######  Remark 2.6.

The factor $T_{j}^{\frac{l}{2}}$ comes from the fact that on $N_{j}^{t}$, the
metric $g^{D}$ is close to $T_{j}g_{b_{j}}$. For a tensor $s$ of conformal
weight $l$, we have

$|s|_{T_{j}g_{b_{j}}}=T_{j}^{\frac{l}{2}}|s|_{g_{b_{j}}}.$

Thanks to this norm, we can for example rewrite and extend the statement of
[Ozu19a, Theorem 6.4].

######  Corollary 2.7.

Let $D_{0},v_{0}>0$, $l\in\mathbb{N}$, and $\beta=\beta(v_{0},D_{0})>0$
obtained in [Ozu19a, Theorem 6.4]. Then, for all $\epsilon>0$, there exists
$\delta=\delta(\epsilon,D_{0},v_{0},l)>0$ such that if $(M,g^{\mathcal{E}})$
is an Einstein manifold satisfying

  * •

the volume is bounded below by $v_{0}>0$,

  * •

the diameter is bounded above by $D_{0}$,

  * •

the Ricci curvature is bounded $|\operatorname{\textup{Ric}}|\leq 3$.

and for an Einstein orbifold $(M_{o},g_{o})$,

$d_{GH}\big{(}(M,g^{\mathcal{E}}),(M_{o},g_{o})\big{)}\leqslant\delta,$

then, there exists a naïve desingularization $(M,g^{D})$ of $(M_{o},g_{o})$ by
a tree of singularities and a diffeomorphism $\phi:M\to M$ such that

$\big{\|}\phi^{*}g^{\mathcal{E}}-g^{D}\big{\|}_{C^{l}_{\beta}(g^{D})}\leqslant\epsilon.$

###### Proof.

Let $l\in\mathbb{N}$. Let us give a proof by contradiction and consider a
sequence of counter examples, that is a sequence of Einstein manifolds
$(M_{i},g_{i})$ such that $\operatorname{\textup{Vol}}(g_{i})\geqslant
v_{0}>0$, $diam(g_{i})<D_{0}$ and
$|\operatorname{\textup{Ric}}(g_{i})|_{g_{i}}\leqslant 3$ converging in the
Gromov-Hausdorff sense to an Einstein orbifold $(M_{o},g_{o})$, but such that
there exists $\epsilon>0$ for which, for all $i\in\mathbb{N}$ and all naïve
desingularization $(M_{i},g^{D}_{i})$ of $(M_{o},g_{o})$, and all
diffeomorphism $\Phi_{i}:M_{i}\to M_{i}$, we have
$\|\Phi_{i}^{*}g_{i}-g^{D}_{i}\|_{C^{l}_{\beta}(g^{D})}>\epsilon$.

According to Corollary [Ozu19a, Theorem 6.4], this implies that there exists
subsequence $(M,g_{i})$ with fixed topology, and a sequence $(M,g^{D}_{i})$
contradicting the assumption for $i$ large enough by definition of the
weighted norm. ∎

The weighted norm is equivalent to the sum of the ones of $M_{o}$ and the
$N_{j}$ for tensors respectively supported in $M_{o}^{t}$ and the $N_{j}^{t}$.
On the annuli of low curvature $\mathcal{A}_{k}(\epsilon_{0})$ pulled back on
flat annuli $A_{e}(\rho_{1},\rho_{2})\subset\mathbb{R}^{4}/\Gamma$, it is
equivalent to a particular norm which allows us to control _independently of
the radii_ the sum of tensors decaying at the center of the annulus and of
tensors decaying at infinity.

######  Definition 2.8 (Weighted norm adapted to an annulus).

Let $0<\rho_{1}<\rho_{2}$, $\beta\in\mathbb{R}$, $k\in\mathbb{N}$,
$0<\alpha<1$ and a tensor $s$ on $(A_{e}(\rho_{1},\rho_{2}),g_{e})$ and the
function $r_{e}:=d_{e}(0,.)$. We define
$\eta(r_{e}):=\max\Big{(}\big{(}\frac{\rho_{1}}{r_{e}}\big{)}^{\beta},\big{(}\frac{r_{e}}{\rho_{2}}\big{)}^{\beta}\Big{)}\leqslant
1$, and the norm,

$\displaystyle\|s\|_{C^{k,\alpha}_{\beta}(A_{e}(\rho_{1},\rho_{2}))}:$
$\displaystyle=\sup_{A_{e}(\rho_{1},\rho_{2})}\Big{[}\eta^{-1}(r_{e})\Big{(}\sum_{i=0}^{k}r_{e}^{i}|\nabla_{g_{e}}^{i}s|_{g_{e}}+r_{e}^{k+\alpha}[\nabla_{g_{e}}^{k}s]_{C^{\alpha}(g_{e})}\Big{)}\Big{]}.$

In the rest of this article, we will often use spaces denoted
$fC^{k,\alpha}_{\beta}$ for a positive function $f$. They will always be
equipped with the following norm

$\|s\|_{fC^{k,\alpha}_{\beta}}:=\Big{\|}\frac{s}{f}\Big{\|}_{C^{k,\alpha}_{\beta}}.$

######  Remark 2.9.

By definition of $r_{D}$, for all $m$, there exists a constant $C>0$ only
depending on the cut-off functions such that

$\frac{1}{C}\|s\|_{r^{m}_{D}C^{k,\alpha}_{\beta}(g^{D})}\leqslant\|\chi_{M_{o}^{t}}s\|_{r^{m}_{o}C^{k,\alpha}_{\beta}(g_{o})}+\sum_{j}T_{j}^{\frac{l-m}{2}}\|\chi_{N_{j}^{t}}s\|_{r^{m}_{b_{j}}C^{k,\alpha}_{\beta}(g_{b_{j}})}\leqslant
C\|s\|_{r^{m}_{D}C^{k,\alpha}_{\beta}(g^{D})}.$ (1)

######  Remark 2.10.

Let $\beta\leqslant\beta^{\prime}$, $k+\alpha\leqslant
k^{\prime}+\alpha^{\prime}$, and $m,m^{\prime}\in\mathbb{Z}$.

For all the previously mentioned weighted Hölder spaces generically denoted
$r^{m}C^{k,\alpha}_{\beta}$, we have the following properties: for any tensors
$s$ and $s^{\prime}$

  * •

$\|s\|_{r^{m}C^{k,\alpha}_{\beta}}\leqslant\|s\|_{r^{m}C^{k^{\prime},\alpha^{\prime}}_{\beta^{\prime}}}$,

  * •

$\|\nabla^{k}s\|_{r^{m}C^{k^{\prime}-k,\alpha}_{\beta}}\leqslant\|s\|_{r^{m+k}C^{k^{\prime},\alpha}_{\beta}}$

  * •

if $*$ is a composition, a product of a contraction of tensors, there exists
$C=C(*,k,\alpha)>0$ such that

$\|s*s^{\prime}\|_{r^{m+m^{\prime}}C^{k,\alpha}_{\beta+\beta^{\prime}}}\leqslant
C\|s\|_{r^{m}C^{k,\alpha}_{\beta}}\|s^{\prime}\|_{r^{m^{\prime}}C^{k,\alpha}_{\beta^{\prime}}}.$

####  2.1.3 Weighted Schauder estimates

Weighted Schauder estimates hold in these norms for the operator
$P:=\frac{1}{2}\nabla^{*}\nabla-\mathring{\operatorname{\textup{R}}}$.

######  Proposition 2.11.

For all $\beta>0$ and $0<\alpha<1$, there exists $C>0$ and $\epsilon>0$ such
that if $h$ is a symmetric $2$-tensor on $(M,g^{D})$, and $g$ a metric on $M$
satisfying

$\|g-g^{D}\|_{C^{2,\alpha}_{\beta}(g^{D})}\leqslant\epsilon,$

then, we have

$\|h\|_{C^{2,\alpha}_{\beta}(g^{D})}\leqslant
C\big{(}\|P_{g}h\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}+\|h\|_{C^{0}_{\beta}(g^{D})}\big{)}.$

###### Proof.

Let $g$ be a metric on $M$ satisfying
$\|g-g^{D}\|_{C^{2,\alpha}_{\beta}(g^{D})}\leqslant\epsilon$, for $\epsilon>0$
which we will choose small enough along the proof.

On the compacts $M_{o}(\epsilon_{0})$ of the orbifold and
$N_{j}(\epsilon_{0})$ of the ALE orbifolds minus their singular points, we
have an elliptic estimate for the operators $P_{g_{o}}$ and $P_{g_{b_{j}}}$:
there exists $C_{1}>0$ such that for all $2$-tensors $h_{o}$ supported in
$M_{o}(\epsilon_{0})$ and $h_{j}$ supported in $N_{j}(\epsilon_{0})$, we have

$\|h_{o}\|_{C^{2,\alpha}(g_{o})}\leqslant
C_{1}\big{(}\|P_{g_{o}}h_{o}\|_{C^{\alpha}(g_{o})}+\|h_{o}\|_{C^{0}(g_{o})}\big{)},$

and

$\|h_{j}\|_{C^{2,\alpha}(g_{b_{j}})}\leqslant
C_{1}\big{(}\|P_{g_{b_{j}}}h_{j}\|_{C^{\alpha}(g_{b_{j}})}+\|h_{j}\|_{C^{0}(g_{b_{j}})}\big{)}.$

By assumption, there exists $C>0$ only depending on $g_{o}$ and $g_{b_{j}}$
such that $\|g-g_{o}\|_{C^{2,\alpha}(g_{o})}\leqslant C\epsilon$ on
$M_{o}(\epsilon_{0})$ and
$\big{\|}\frac{g}{T_{j}}-g_{b_{j}}\big{\|}_{C^{2,\alpha}(g_{b_{j}})}\leqslant
C\epsilon$ on $N_{j}(\epsilon_{0})$. We conclude that for $\epsilon$ small
enough, the operators $P_{g}$ and $P_{\frac{g}{T_{j}}}$, which are close to
the operators $P_{g_{o}}$ and $P_{g_{b_{j}}}$, satisfy for all $h_{o}$
supported in $M_{o}(\epsilon_{0})$ and $h_{j}$ supported in
$N_{j}(\epsilon_{0})$,

$\|h_{o}\|_{C^{2,\alpha}(g_{o})}\leqslant
2C_{1}\big{(}\|P_{g}h_{o}\|_{C^{\alpha}(g_{o})}+\|h_{o}\|_{C^{0}(g_{o})}\big{)},$

and

$\|h_{j}\|_{C^{2,\alpha}(g_{b_{j}})}\leqslant
2C_{1}\big{(}\big{\|}P_{\frac{g}{T_{j}}}h_{j}\big{\|}_{C^{\alpha}(g_{b_{j}})}+\|h_{j}\|_{C^{0}(g_{b_{j}})}\big{)}.$

On each almost flat annulus $\mathcal{A}_{k}(\epsilon_{0})$, on the annulus
$\mathcal{A}_{\rho}:=\\{\rho\leqslant r_{D}\leqslant 2\rho\\}$, there exists a
diffeomorphism $\phi_{\rho}:A_{e}(1,2)\to\mathcal{A}_{\rho}$ such that

$\Big{\|}\frac{\phi_{\rho}^{*}g}{\rho^{2}}-g_{e}\Big{\|}_{C^{1,\alpha}}\leqslant
C\eta(\rho)\epsilon.$

then, by ellipticity, for $\epsilon$ small enough, there exists $C_{2}>0$,
such that for all $2$-tensor $h$ on $A_{e}(1,2)$ we have,

$\|h\|_{C^{2,\alpha}(A_{e}(1,2))}\leqslant
2C_{2}\Big{(}\|h\|_{C^{0}(A_{e}(1,2))}+\big{\|}P_{\frac{\phi_{\rho}^{*}g}{\rho^{2}}}h\big{\|}_{C^{\alpha}(A_{e}(1,2))}\Big{)}.$

Coming back to $(M,g)$, this implies that for $\epsilon$ small enough, we have

$\|h\|_{C^{2,\alpha}(\mathcal{A}_{\rho},\frac{g}{\rho^{2}})}\leqslant
4C_{2}\left(\|h\|_{C^{0}(\mathcal{A}_{\rho},\frac{g}{\rho^{2}})}+\big{\|}P_{\frac{g}{\rho^{2}}}h\big{\|}_{C^{\alpha}(\mathcal{A}_{\rho},\frac{g}{\rho^{2}})}\right).$

The norm of a $2$-tensor $s$ behaves in the following way by rescaling, for
$t>0$

$|s|_{\frac{g}{t}}=t|s|_{g},$

and the operator $P$ behaves in the following way by rescaling, for $t>0$:
$P_{\frac{g}{t}}=tP_{g}$. Multiplying both sides of the equality by
$r_{D}^{-2}$, we get

$\|h\|_{C^{2,\alpha}(\mathcal{A}_{\rho},g)}\leqslant
4C_{2}\left(\|h\|_{C^{0}(\mathcal{A}_{\rho},g)}+r_{D}^{2}\|P_{g}h\|_{C^{\alpha}(\mathcal{A}_{\rho},g)}\right).$

Given the controls on the derivatives of $r_{D}$, we deduce the stated result
by definition of the weighted norms by multiplying both sides of the
inequality by the weight of the norm. ∎

Analogous estimates also hold for the elliptic operator $\delta\delta^{*}$
with the same proof.

######  Proposition 2.12.

For all $\beta>0$ and $0<\alpha<1$ there exists $C>0$ and $\epsilon>0$ such
that if $X$ is a vector field on $(M,g^{D})$, and $g$ a metric on $M$
satisfying

$\|g-g^{D}\|_{C^{2,\alpha}_{\beta}(g^{D})}\leqslant\epsilon,$

then, we have

$\|X\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}\leqslant
C\big{(}\|\delta_{g}\delta^{*}_{g}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}+\|X\|_{r_{D}C^{0}_{\beta}(g^{D})}\big{)}.$

###  2.2 Decoupling norms

We will see here that to expect good controls for the operators $P$ and
$\delta\delta^{*}$ in the annular regions of our manifold, we need to consider
separately the influence of traceless constant $2$-tensors for $P$ and linear
vector fields of the kernel of $\delta_{e}\delta^{*}_{e}$ for
$\delta\delta^{*}$.

####  2.2.1 Estimates on annuli $A_{e}(\epsilon,\epsilon^{-1})$ of
$(\mathbb{R}^{4},g_{e})$.

Let us start by studying the situation on flat annuli to motivate our new
norms.

######  Proposition 2.13.

Let $0<\beta<1$, $0<\alpha<1$, and
$P=\frac{1}{2}\nabla^{*}\nabla-\mathring{\operatorname{\textup{R}}}$. There
exists $C_{e}>0$, and $\epsilon_{e}>0$ such that for any $2$-tensor $h$ on an
annulus of radii $0<\epsilon<\epsilon_{e}$ and $\frac{1}{\epsilon}$, there
exists a constant $2$-tensor $H_{0}$ and a $2$-tensor $H_{*}$ satisfying

$\nabla^{*}_{e}\nabla_{e}H_{*}=0,$
$\|H_{*}\|_{C^{2,\alpha}_{1}(A_{e}(2\epsilon,(1/2)\epsilon^{-1}))}\leqslant
C_{e}\|h-H_{0}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))},$ (2)

(notice the norm $C^{2,\alpha}_{\mathbf{1}}$ for the left hand side) and,

$\|h-H_{0}-H_{*}\|_{C^{2,\alpha}_{\beta}(A_{e}(2\epsilon,(1/2)\epsilon^{-1}))}\leqslant
C_{e}\|P_{g_{e}}h\|_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}.$
(3)

This implies in particular the following control: for all $x\in A_{e}(1/2,2)$,

$\displaystyle|h-H_{0}(x)|_{g_{e}}+|$ $\displaystyle\nabla
h(x)|_{g_{e}}+|\nabla^{2}h(x)|_{g_{e}}+[\nabla^{2}h]_{C^{\alpha}(g)}(x)$
$\displaystyle\leqslant$
$\displaystyle\;C_{e}\big{(}(2\epsilon)^{\beta}\|P_{g_{e}}h\|_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}+2\epsilon\|h-H_{0}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\big{)}.$
(4)

######  Remark 2.14.

This is a strictly better estimate than the elliptic estimates of Proposition
2.12 which would only have given

$\displaystyle|h-H_{0}(x)|_{g_{e}}+|$ $\displaystyle\nabla
h(x)|_{g_{e}}+|\nabla^{2}h(x)|_{g_{e}}+[\nabla^{2}h]_{C^{\alpha}(g)}(x)$
$\displaystyle\leqslant$
$\displaystyle\;C_{e}\big{(}(2\epsilon)^{\beta}\|P_{g_{e}}h\|_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}+(2\epsilon)^{\beta}\|h-H_{0}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\big{)}.$
(5)

###### Proof.

Let us start by noting that (4) is a consequence of (2) and (3). Indeed,
$h-H_{0}=(h-H_{0}-H_{*})+H_{*}$, and we have therefore, denoting

$\|s\|_{C^{2,\alpha}(1/2,2)}:=\sup_{x\in A_{e}(1/2,2)}|s(x)|_{g_{e}}+|\nabla
s(x)|_{g_{e}}+|\nabla^{2}s(x)|_{g_{e}}+[\nabla^{2}s]_{C^{\alpha}(g)}(x),$
$\displaystyle\|h-H_{0}\|_{C^{2,\alpha}(1/2,2)}\leqslant$
$\displaystyle\;\|h-H_{0}-H_{*}\|_{C^{2,\alpha}(1/2,2)}+\|H_{*}\|_{C^{2,\alpha}(1/2,2)}$
$\displaystyle\leqslant$
$\displaystyle\;(2\epsilon)^{\beta}\|h-H_{0}-H_{*}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}+2\epsilon\|H_{*}\|_{C^{2,\alpha}_{1}(A_{e}(\epsilon,\epsilon^{-1}))}$
$\displaystyle\leqslant$
$\displaystyle\;C_{e}\big{(}(2\epsilon)^{\beta}\|P_{g_{e}}h\|_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}+2\epsilon\|h-H_{0}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\big{)},$

by definition of the weighted norms and assuming, for $C_{e}>0$, the
inequalities (3) and (2).

On $\mathbb{R}^{4}\backslash\\{0\\}$, the harmonic $2$-tensors are sum of
homogeneous harmonic $2$-tensors whose coefficients in the canonical basis of
$\mathbb{R}^{4}$ are proportional to $r^{j}$ for
$j\in\mathbb{Z}\backslash\\{-1\\}$. These harmonic $2$-tensors are more
precisely of the form $r_{e}^{k}H_{k}$ or $r^{-2-k}_{e}H_{k}$ for
$k\in\mathbb{N}$, where $H_{k}$ is a radial $2$-tensor whose coefficients,
once restricted to the unit sphere are eingenfunctions of the Laplacian with
eingenvalue $-k(k+2)$. In particular, we see that any non constant harmonic
$2$-tensor is the sum of a constant $2$-tensor and a tensor of order
$r_{e}^{-2}$ in $0$ or of order $r_{e}$ at infinity.

For any $2$-tensor $h$ on $A_{e}(\epsilon,\epsilon^{-1})$, let us define
$\tilde{H}$ the solution of the following Dirichlet problem, denoting for
$r>0$, $S_{e}(r):=\\{r_{e}=r\\}$,

$\left\\{\begin{aligned} &\nabla^{*}_{e}\nabla_{e}\tilde{H}=0,\\\
&\tilde{H}=h\text{ on }S_{e}(\epsilon)\cup
S_{e}(\epsilon^{-1}).\end{aligned}\right.$

More precisely, $\tilde{H}=\sum_{k\geqslant 0}(\epsilon
r_{e})^{k}\tilde{H}_{k}^{+}+(\epsilon^{-1}r_{e})^{-2-k}\tilde{H}_{k}^{-}$
where, if we decompose in spherical harmonics
$h_{|S_{e}(\epsilon)}=\sum_{k}H_{k}(\epsilon)$ and
$h_{|S_{e}(\epsilon^{-1})}=\sum_{k}H_{k}(\epsilon^{-1})$, we have the system

$\left\\{\begin{aligned}
&H_{k}(\epsilon^{-1})=\tilde{H}_{k}^{+}+\epsilon^{4+2k}\tilde{H}_{k}^{-},\\\
&H_{k}(\epsilon)=\epsilon^{2k}\tilde{H}_{k}^{+}+\tilde{H}_{k}^{-},\end{aligned}\right.$

and therefore,

$\left\\{\begin{aligned}
&\tilde{H}_{k}^{+}=\frac{1}{1-\epsilon^{4+4k}}\big{(}H_{k}(\epsilon^{-1})-\epsilon^{4+2k}H_{k}(\epsilon)\big{)},\\\
&\tilde{H}_{k}^{-}=\frac{1}{1-\epsilon^{4+4k}}\big{(}H_{k}(\epsilon)-\epsilon^{2k}H_{k}(\epsilon^{-1})\big{)},\end{aligned}\right.$
(6)

Denote $\tilde{H}_{*}:=\tilde{H}-\tilde{H}_{0}^{+}$. Since
$\nabla^{*}_{e}\nabla_{e}\tilde{H}_{*}=0$, by elliptic regularity on the
annulus $A(\rho/2,4\rho)\subset A_{e}(\epsilon,\epsilon^{-1})$, there exists a
constant $C>0$ independent of $h$ such that we have,

$\|\tilde{H}_{*}\|_{C^{0}(A(\rho,2\rho))}\leqslant\frac{C}{\rho^{2}}\|\tilde{H}_{*}\|_{L^{2}(A_{e}(\rho/2,4\rho))},$
(7)

so to control the norm $C^{0}_{1}(A_{e}(\epsilon,\epsilon^{-1}))$ of
$\tilde{H}_{*}$, we just have to control the $L^{2}$-norm of $\tilde{H}_{*}$
on the different annuli $A_{e}(\rho,2\rho)\subset A(\epsilon,\epsilon^{-1})$.
The equalities (6) and the fact that the decompositions in spherical harmonics
are orthogonal give the following estimate for $\epsilon$ small enough and a
constant $C>0$,

$\displaystyle\|\tilde{H}_{*}\|_{L^{2}(A_{e}(\rho,2\rho))}^{2}\leqslant$
$\displaystyle\;2\int_{A_{e}(\rho,2\rho)}\sum_{k\geqslant
0}\big{|}(r_{e}\epsilon)^{k+1}\big{(}H_{k+1}(\epsilon^{-1})-\epsilon^{4+2(k+1)}H_{k+1}(\epsilon)\big{)}$
$\displaystyle+(r_{e}\epsilon^{-1})^{-2-k}\big{(}H_{k}(\epsilon)-\epsilon^{2k}H_{k}(\epsilon^{-1})\big{)}\big{|}^{2}$
$\displaystyle\leqslant$
$\displaystyle\;C\int_{A_{e}(\rho,2\rho)}(r_{e}\epsilon)^{2}\big{(}|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon)}|^{2}+\epsilon^{8}|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon^{-1})}|^{2}\big{)}$
$\displaystyle+(r_{e}\epsilon^{-1})^{-4}\big{(}|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon)}|^{2}+|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon^{-1})}|^{2}\big{)}$
$\displaystyle\leqslant$
$\displaystyle\;C^{2}\sum_{k}\rho^{4}\epsilon^{2}\big{(}\rho^{1}+\rho^{-1}\big{)}^{2}\|h-\tilde{H}_{0}^{+}\|_{C^{0}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}^{2}$
(8)

because
$|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon)}|_{g_{e}}\leqslant\|h-\tilde{H}_{0}^{+}\|_{C^{0}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}$
and
$|(h-\tilde{H}_{0}^{+})_{|S_{e}(\epsilon^{-1})}|_{g_{e}}\leqslant\|h-\tilde{H}_{0}^{+}\|_{C^{0}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}$
by definition of the norm. Combining (7) and (8), we get

$\|\tilde{H}_{*}\|_{C^{0}_{1}(A_{e}(\epsilon,\epsilon^{-1}))}\leqslant
C\|h-\tilde{H}_{0}^{+}\|_{C^{0}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}$ (9)

Let us fix $x_{0}\in S_{e}(1)$ and modify our $2$-tensor $\tilde{H}$ to get a
$2$-tensor $H$ such that $h-H$ vanishes at $x_{0}$ and on $S_{e}(\epsilon)$
while being constant on $S_{e}(\epsilon^{-1})$. The only possible choice with
harmonic $2$-tensors is

$H:=\Big{(}\tilde{H}_{0}^{+}-\frac{c_{0}}{1-\epsilon^{2}}\Big{)}+\Big{(}\tilde{H}_{*}+\frac{\epsilon^{2}}{1-\epsilon^{2}}\frac{c_{0}}{r_{e}^{2}}\Big{)},$

where $c_{0}=(h-\tilde{H})(x_{0})$. We will show that $h-H$ satisfies the
estimate (3), but let us start by proving the control (2) stated. Denote
$H_{0}:=\tilde{H}_{0}^{+}-\frac{c_{0}}{1-\epsilon^{2}}$ the constant part of
$H$, and
$H_{*}:=\tilde{H}_{*}+\frac{1}{1-\epsilon^{2}}\frac{c_{0}}{(\epsilon^{-1}r_{e})^{2}}$
its varying part.

According to (8), the part $\tilde{H}_{*}=\sum_{k\geqslant 1}(\epsilon
r_{e})^{k}\tilde{H}_{k}^{+}+(\epsilon^{-1}r_{e})^{-2-k}\tilde{H}_{k}^{-}$ is
well controlled in $L^{2}$-norm by the varying parts of $h-H_{0}$ on
$S_{e}(\epsilon)$ and $S_{e}(\epsilon)$ (which are the same as the varying
parts of $h-\tilde{H}_{0}$). There remains to control the part in
$r_{e}^{-2}$, that is

$\frac{1}{1-\epsilon^{2}}\frac{(1-\epsilon^{2})\tilde{H}_{0}^{-}+c_{0}}{\epsilon^{-2}r_{e}^{2}}.$

according to (6), $|\tilde{H}_{0}^{-}|$ is controlled by the difference
between the mean value of $h-H_{0}$ on the sphere $S_{e}(\epsilon)$ and on the
sphere $S_{e}(\epsilon^{-1})$ (which is the same as for $h-\tilde{H}_{0}$),
and we therefore have $|\tilde{H}_{0}^{-}|\leqslant
3\|h-H_{0}\|_{C^{0}_{\beta}}$ for $\epsilon$ small enough. Finally, to control
$|c_{0}|$ by a function of $\|h-H_{0}\|_{C^{0}_{\beta}}$, we just notice that
the mean value of $h-H_{0}$ on $S_{e}(\epsilon^{-1})$ is
$\epsilon^{4}\tilde{H}_{0}^{-}+c_{0}$, and that we therefore have
$|c_{0}|_{g_{e}}\leqslant(1+3\epsilon^{4})\|h-H_{0}\|_{C^{0}_{\beta}}$ because
$|\tilde{H}_{0}^{-}|_{g_{e}}\leqslant 3\|h-H_{0}\|_{C^{0}_{\beta}}$.

Hence we finally have the existence of a constant $C>0$ such that we have,
going from $L^{2}$-controls to $C^{0}$-controls thanks to (7) applied to
$H_{*}$,

$\|H_{*}\|_{C^{0}_{1}}\leqslant C\|h-H_{0}\|_{C^{0}_{\beta}},$

and therefore the stated inequality (2).

Let us prove the estimate (3), and assume towards a contradiction that there
exists a sequence of positive numbers $\epsilon_{i}\to 0$, and a sequence of
$2$-tensors $h_{i}$ on annuli $A_{e}(\epsilon_{i},\epsilon_{i}^{-1})$
satisfying,
$\|h_{i}-H_{i}\|_{C^{0}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}=1$, and
$\|P_{e}h_{i}\|_{r^{-2}_{e}C^{0}_{\beta}(A_{e}(\epsilon_{i},\epsilon_{i}^{-1}))}\leqslant\frac{1}{i}$.

Let then $(x_{i})_{i}$ be a sequence of points of
$A_{e}(\epsilon_{i},\epsilon_{i}^{-1})$ where the
$C^{0}_{\beta}(A_{e}(\epsilon_{i},\epsilon_{i}^{-1}))$-norm of
$\bar{h}_{i}:=h_{i}-H_{i}$ is reached. We can extract a subsequence with one
of the following behaviors:

  1. 1.

$r_{e}(x_{i})\to+\infty$, and $\epsilon_{i}r_{e}(x_{i})\to 0$,

  2. 2.

$r_{e}(x_{i})\to+\infty$, and $\epsilon_{i}r_{e}(x_{i})\to c>0$,

  3. 3.

$r_{e}(x_{i})\to 0$ and $\epsilon_{i}^{-1}r_{e}(x_{i})\to+\infty$,

  4. 4.

$r_{e}(x_{i})\to 0$ and $\epsilon_{i}^{-1}r_{e}(x_{i})\to c>0$,

  5. 5.

$r_{e}(x_{i})\to c>0$.

In all cases, we rescale to fix $r_{e}(x_{i})=1$ by defining, for all $x$,

$h_{i}^{\prime}(x):=\frac{\bar{h}_{i}\big{(}r_{e}(x_{i})x\big{)}}{\epsilon_{i}^{\beta}\big{(}r_{e}(x_{i})^{\beta}+r_{e}(x_{i})^{-\beta}\big{)}},$

which satisfies

$\big{(}\nabla_{e}^{*}\nabla_{e}h_{i}^{\prime}\big{)}(x)=r_{e}(x_{i})^{2}\big{(}\nabla_{e}^{*}\nabla_{e}\bar{h}_{i}\big{)}(r_{e}(x_{i})x).$

Since we had by assumption the controls

$\bar{h}_{i}(x)\leqslant\epsilon_{i}^{\beta}\big{(}r_{e}(x)^{\beta}+r_{e}(x)^{-\beta}\big{)},$

and

$|\nabla_{e}^{*}\nabla_{e}\bar{h}_{i}|(x)\leqslant\frac{1}{i}r_{e}(x)^{-2}\epsilon_{i}^{\beta}\big{(}r_{e}(x)^{\beta}+r_{e}(x)^{-\beta}\big{)},$

our new $2$-tensor $h^{\prime}_{i}$ vanishes at $\frac{x_{0}}{r_{e}(x_{i})}$
and on $S_{e}(\epsilon_{i}r_{e}(x_{i})^{-1})$, and is constant on
$S_{e}(\epsilon_{i}^{-1}r_{e}(x_{i})^{-1})$. It moreover satisfies

$|h_{i}^{\prime}|(x)\leqslant\frac{\big{(}(r_{e}(x_{i})r_{e}(x))^{\beta}+(r_{e}(x_{i})r_{e}(x))^{-\beta}\big{)}}{r_{e}(x_{i})^{\beta}+r_{e}(x_{i})^{-\beta}}$

and

$|\nabla_{e}^{*}\nabla_{e}h_{i}^{\prime}|(x)\leqslant\frac{1}{i}r_{e}(x)^{-2}\frac{\big{(}(r_{e}(x_{i})r_{e}(x))^{\beta}+(r_{e}(x_{i})r_{e}(x))^{-\beta}\big{)}}{r_{e}(x_{i})^{\beta}+r_{e}(x_{i})^{-\beta}}.$

In the different situations, up to extracting a subsequence, we finally get
one of the following limits

  1. 1.

on $\mathbb{R}^{4}\backslash\\{0\\}$, a solution $h^{\prime}_{\infty}$ of
$P_{e}h^{\prime}_{\infty}=\frac{1}{2}\nabla_{e}^{*}\nabla_{e}h^{\prime}_{\infty}=0$,
and $\sup r^{-\beta}h^{\prime}_{\infty}=1$, but there does not exist such a
solution because the harmonic $2$-tensors decay at least as $\mathcal{O}(r)$
at $0$ if they vanish at $0$ and must therefore grow at this rate at infinity.
This is a contradiction.

  2. 2.

on $B_{e}(1/c)\backslash\\{0\\}$, a solution $h^{\prime}_{\infty}$ of
$P_{e}h^{\prime}_{\infty}=0$, and $\sup r^{-\beta}h^{\prime}_{\infty}=1$, and
such that $(h_{\infty}^{\prime})_{|S_{e}(1/c)}$ is constant. The unique
solution to the Dirichlet problem with the zero condition at $0$ and a
constant condition on $S_{e}(1/c)$ is $h_{\infty}^{\prime}=0$. This is a
contradiction.

  3. 3.

on $\mathbb{R}^{4}\backslash\\{0\\}$, a solution $h^{\prime}_{\infty}$ of
$P_{e}h^{\prime}_{\infty}=0$, and $\sup r^{\beta}h^{\prime}_{\infty}=1$, but
there does not exist such a solution because the harmonic $2$-tensors decaying
at infinity decay at least like $\mathcal{O}(r^{-2})$, and therefore blow up
at least at this rate at $0$, and finally, $h_{\infty}^{\prime}=0$. This is a
contradiction.

  4. 4.

on $\mathbb{R}^{4}\backslash B_{e}(1/c)$, a solution $h^{\prime}_{\infty}$ of
$P_{e}h^{\prime}_{\infty}=0$, and $\sup r^{\beta}h^{\prime}_{\infty}=1$ and
$(h^{\prime}_{\infty})_{|S_{e}(1/c)}=0$. The unique solution to the Dirichlet
problem on $\mathbb{R}^{4}\backslash B_{e}(1/c)$, decaying at infinity and
vanishing on $S_{e}(1/c)$ being zero, we have $h_{\infty}=0$. This is a
contradiction.

  5. 5.

on $\mathbb{R}^{4}\backslash\\{0\\}$, a solution $h^{\prime}_{\infty}$ of
$P_{e}h^{\prime}_{\infty}=0$, and
$\sup(r^{\beta}+r^{-\beta})h^{\prime}_{\infty}=1$ satisfying
$h^{\prime}_{\infty}\big{(}\frac{x_{0}}{c}\big{)}=0$. The conditions
$P_{e}h^{\prime}_{\infty}=0$, and
$\sup(r^{\beta}+r^{-\beta})h^{\prime}_{\infty}=1$ imply that
$h^{\prime}_{\infty}$ is constant, since $h^{\prime}_{\infty}$ vanishes at
$\frac{x_{0}}{c}$, we have $h^{\prime}_{\infty}=0$. This is a contradiction.

We therefore deduce that there exists $\epsilon_{e}>0$ and $C_{e}>0$ such that
for all $0<\epsilon<\epsilon_{e}$ and all $2$-tensor $h$ on the annulus
$A_{e}(\epsilon,\epsilon^{-1})$, we have

$\|h-H_{0}\|_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\leqslant
C_{e}\|P_{g}h\|_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}.$

To go from $C^{0}_{\beta}$-controls to $C^{2,\alpha}_{\beta}$-controls we use
elliptic estimates which are satisfied on the flat annuli according to the end
of the proof of Proposition 2.11. ∎

With a completely analogous proof using the harmonic decomposition of
$1$-forms on a cone of [CT94, (2.16)-(2.19)] (see also Section 3.1 for
particular case of flat cones), we have the same result for vector fields, but
this time, we treat the linear kernel of $\delta\delta^{*}$ on
$\mathbb{R}^{4}/\Gamma$ separately. On $\mathbb{R}^{4}$, the elliptic operator

$\delta\delta^{*}=\nabla^{*}\nabla-\frac{1}{2}d^{*}d=dd^{*}+\frac{1}{2}d^{*}d,$

has its kernel equal to the linear vector fields of the kernel of
$\delta_{e}\delta^{*}_{e}$ among the vector fields of order
$\mathcal{O}(r^{1-\beta}+r^{1+\beta})$ for $0<\beta<1$, see Lemma 3.2 for a
proof of this and Section 3.1 for a description of the kernel.

######  Proposition 2.15.

Let $0<\beta<1$. There exists $C_{e}>0$, and $\epsilon_{e}>0$ such that for
any vector field $X$ on an annulus of radii $0<\epsilon<\epsilon_{e}$ and
$\frac{1}{\epsilon}$, there exists $Y_{0}$, a linear vector field of the
kernel of $\delta_{e}\delta^{*}_{e}$, and an element $Y_{*}$ of the kernel of
$\delta_{e}\delta_{e}^{*}$ satisfying

$\|Y_{*}\|_{r_{e}C^{3,\alpha}_{1}(A_{e}(\epsilon,\epsilon^{-1}))}\leqslant
C_{e}\|X-Y_{0}\|_{r_{e}C^{3,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))},$
$\|X-Y_{0}-Y_{*}\|_{r_{e}C^{3,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\leqslant
C_{e}\|\delta_{g_{e}}\delta^{*}_{g_{e}}X\|_{r^{-1}_{e}C^{1,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}.$

In particular, this implies the following control, for all $x\in
A_{e}(1/2,2)$,

$\displaystyle|(X-Y_{0})(x)|_{g_{e}}+|$
$\displaystyle\nabla(X-Y_{0})(x)|_{g_{e}}+|\nabla^{2}(X-Y_{0})(x)|_{g_{e}}+[\nabla^{2}(X-Y_{0})]_{C^{\alpha}(g)}(x)$
$\displaystyle\leqslant$
$\displaystyle\;C_{e}\big{(}(2\epsilon)^{\beta}\|\delta_{g}\delta_{g}^{*}\|_{r^{-1}_{e}C^{1,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}+2\epsilon\|X-Y_{0}\|_{r_{e}C^{3,\alpha}_{\beta}(A_{e}(\epsilon,\epsilon^{-1}))}\big{)}.$
(10)

####  2.2.2 Approximate kernels

Let $(M,g^{D})$ be a naïve desingularization of an Einstein orbifold. For each
annulus $\mathcal{A}_{k}(\epsilon)$ between $N_{k}$ and $N_{j}$ or $N_{k}$ and
$M_{o}$, there exists a diffeomorphism

$\Phi_{k}:A_{e}\big{(}\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}\big{)}\subset\mathbb{R}^{4}/\Gamma_{k}\to\mathcal{A}_{k}(\epsilon)\subset
M,$

such that there exists $C>0$ for which, for all $0<\beta<1$,

$\big{\|}\Phi^{*}_{k}g^{D}-g_{e}\big{\|}_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}))}\leqslant
C\epsilon^{2-\beta}.$ (11)

Because of the above constant $2$-tensors and the linear vector fields, we
cannot expect estimates independent of the gluing scales in the definition of
$(M,g^{D})$ of the type $\|h\|_{C^{2,\alpha}_{\beta}(g^{D})}\leqslant
C\|P_{g^{D}}h\|_{r_{D}^{-2}C^{\alpha}_{\beta}(g^{D})}$ which are needed to
apply an inverse function theorem. Indeed, we have the following estimates
according to Proposition 2.23 proven in the following.

######  Note 2.16.

In most of the rest of the paper, we will often abusively forget the
diffeomorphism $\Phi_{k}$ to simplify the notations. For instance, a
$2$-tensor $\Phi_{k}^{*}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}H\big{)}$ will
be denoted $\chi_{\mathcal{A}_{k}(\epsilon)}H$ on $M$.

######  Proposition 2.17.

On a naïve desingularization $(M,g^{D})$, for all $0<\beta<1$, there exists
$C>0$ such that for $H_{k}$ a constant $2$-tensor,

$\|P_{g^{D}}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}\big{)}\|_{r_{D}^{-2}C^{\alpha}_{\beta}(g^{D})}\leqslant
C|H_{k}|_{g_{e}},$

but

$\|\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}\|_{C^{0}_{\beta}(g^{D})}\geqslant\frac{1}{2}t_{\max}^{-\frac{\beta}{4}}|H_{k}|_{g_{e}}.$

Linear vector fields in the kernel of $\delta\delta^{*}$ also rule out the
existence of estimates independent of $t$ for the operator $\delta\delta^{*}$
according to Proposition 2.23.

######  Proposition 2.18.

On a naïve desingularization $(M,g^{D})$, for all $0<\beta<1$, there exists
$C>0$ such that for $X_{k}$ a linear vector field in the kernel of
$\delta_{e}\delta^{*}_{e}$,

$\|\delta_{g^{D}}\delta_{g^{D}}^{*}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}\big{)}\|_{r_{D}^{-1}C^{1,\alpha}_{\beta}(g^{D})}\leqslant
C\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})},$

but

$\|\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}\|_{C^{0}_{\beta}(g^{D})}\geqslant\frac{1}{2}t_{\max}^{-\frac{\beta}{4}}\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}.$

##### Weighted decoupling norms.

Propositions 2.13 and 2.15 actually show that we can control the inverses of
our operators once we solve our equations modulo constant $2$-tensors and the
linear vector fields of the kernel of $\delta_{e}\delta^{*}_{e}$ on
$\mathbb{R}^{4}$ and Propositions 2.17 and 2.18 show that we cannot expect
better. We therefore introduce new norms to reflect this. They are similar to
the norms introduced in [Bam12] for similar reasons.

######  Definition 2.19 (Norm $\|.\|_{C^{k,\alpha}_{\beta,*}}$ on
$2$-tensors).

Let $h$ be a $2$-tensor on $(M,g^{D})$, (respectively $(M_{o},g_{o})$ or
$(N,g_{b})$). We define its $C^{k,\alpha}_{\beta,*}$-norm by

$\|h\|_{C^{k,\alpha}_{\beta,*}}:=\inf_{h_{*},H_{k}}\|h_{*}\|_{C^{k,\alpha}_{\beta}}+\sum_{k}|H_{k}|_{g_{e}},$

where the infimum is taken on the $(h_{*},H_{k})$ satisfying
$h=h_{*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}$ (respectively
$h=h_{*}+\sum_{k}\chi_{B_{o}(\epsilon)}H_{k}$ or
$h=h_{*}+\sum_{k}\chi_{B_{b}(\epsilon)}H_{k}$), for $H_{k}$ a constant
traceless $2$-tensor on $\mathbb{R}^{4}/\Gamma_{k}$.

######  Definition 2.20 (Norm $\|.\|_{rC^{k,\alpha}_{\beta,*}}$ on vector
fields).

Let $X$ a vector field on $(M,g^{D})$ (respectively $(M_{o},g_{o})$ or
$(N,g_{b})$). We define its $rC^{k,\alpha}_{\beta,*}$-norm, where $r$ is the
function $r_{D}$, $r_{o}$ or $r_{b}$ corresponding by

$\|X\|_{rC^{k,\alpha}_{\beta,*}}:=\inf_{X_{*},X_{k}}\|X_{*}\|_{rC^{k,\alpha}_{\beta}}+\sum_{k}\|X_{k}\|_{rC^{0}_{0}(g_{e})},$

where the infimum is taken on the $(X_{*},X_{k})$ satisfying
$X=X_{*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}$ (respectively
$X=X_{*}+\sum_{k}\chi_{B_{o}(\epsilon)}X_{k}$ or
$X=X_{*}+\chi_{B_{b}(\epsilon)}X_{k}$).

######  Remark 2.21.

By definition of the weighted norms, on an orbifold or orbifold ALE, the
decompositions $h=h_{*}+\sum_{k}\chi_{B(\epsilon)}H_{k}$ and
$X=X_{*}+\sum_{k}\chi_{B(\epsilon)}B_{k}$ are unique and determined
respectively by the limits of $h$ and of $\frac{X}{r}$ when $r\to 0$ (where
$r=r_{o}$ or $r=r_{b}$). Indeed, in other cases, the expression we minimize is
infinite.

######  Remark 2.22.

By definition, we have

$\|.\|_{C^{k,\alpha}_{\beta,*}}\leqslant\|.\|_{C^{k,\alpha}_{\beta}}\text{,
and }\;\|.\|_{rC^{k,\alpha}_{\beta,*}}\leqslant\|.\|_{rC^{k,\alpha}_{\beta}},$

and the spaces $(C^{k,\alpha}_{\beta,*},\|.\|_{C^{k,\alpha}_{\beta,*}})$ and
$(rC^{k,\alpha}_{\beta,*},\|.\|_{rC^{k,\alpha}_{\beta,*}})$ are clearly Banach
spaces.

####  2.2.3 Estimates in the decoupling norms

Let us show that it is possible to control thanks to the
$r^{m}C^{k,\alpha}_{\beta}$-norm the images by the operators $P$ and
$\delta\delta^{*}$ of elements of $r^{m+2}C^{k+2,\alpha}_{\beta,*}$.

######  Proposition 2.23.

Let $0<\beta<1$, and $(M_{\bullet},g_{\bullet})$ one of the spaces
$(M_{o},g_{o})$, $(N_{j},g_{b_{j}})$ or $(M,g^{D})$, $g$ a metric, $h$ a
symmetric $2$-tensor, and $X$ a vector field on $M_{\bullet}$. We then have,
the following controls:

$\|P_{g_{\bullet}}h\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}\leqslant
C\|h\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})},$
$\|\delta_{g_{\bullet}}\delta^{*}_{g_{\bullet}}X\|_{r^{-1}_{\bullet}C^{1,\alpha}_{\beta}(g_{\bullet})}\leqslant
C\|X\|_{r_{\bullet}C^{3,\alpha}_{\beta,*}(g_{\bullet})},$
$\|P_{g}(h)-P_{g_{\bullet}}(h)\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}\leqslant
C\|g-g_{\bullet}\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\|h\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})},$

and

$\|\delta_{g}\delta^{*}_{g}(X)-\delta_{g_{\bullet}}\delta^{*}_{g_{\bullet}}(X)\|_{r^{-1}_{\bullet}C^{1,\alpha}_{\beta}(g_{\bullet})}\leqslant
C\|g-g_{\bullet}\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\|X\|_{r_{\bullet}C^{3,\alpha}_{\beta,*}(g_{\bullet})}.$

###### Proof.

Let us show the result for $g^{D}$, the proof for other spaces is very
similar. For the two first inequalities, consider $h$ a $2$-tensor and $X$ a
vector field on $M$, and some decompositions
$h=h_{*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}$ and
$X=X_{*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}$. Remark 2.10 implies
that we have the following controls for $h_{*}$ and $X_{*}$,

$\|P_{g^{D}}h_{*}\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}\leqslant
C\|h_{*}\|_{C^{2,\alpha}_{\beta}(g^{D})},$

and

$\|\delta_{g^{D}}\delta^{*}_{g^{D}}X_{*}\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\leqslant
C\|X_{*}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}.$

On $\mathbb{R}^{4}/\Gamma$, we have $P_{e}H_{k}=0$ and
$\delta_{e}\delta_{e}^{*}X_{k}=0$, hence, since for all $l\in\mathbb{N}$, we
have

$|\nabla^{l}\chi_{k}|_{g^{D}}\leqslant C_{l}r_{D}^{-l},$ (12)

and thanks to the control (11), we have

$\|P_{g^{D}}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}\big{)}\|_{C^{\alpha}_{\beta}(g^{D})}\leqslant
C|H_{k}|_{g_{e}},$

and

$\|\delta_{g^{D}}\delta_{g^{D}}^{*}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}\big{)}\|_{r_{D}^{-1}C^{1,\alpha}_{\beta}(g^{D})}\leqslant
C\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})},$

where we pulled-back thanks to the diffeomorphism

$\Phi_{k}:A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}})\subset\mathbb{R}^{4}/\Gamma_{k}\to\mathcal{A}_{k}(\epsilon)\subset
M,$

which gives the two first inequalities.

Let us now focus on the two last inequalities, which are more difficult. The
control we want is local, let us therefore write down the expressions of our
operators in local coordinates in an orthonormal basis $(e_{i})$. For a
$2$-tensor $h$, and a vector field $X$, denoting $h_{ij}=h(e_{i},e_{j})$ and
$R_{ijkl}$ the Riemannian curvature in coordinates, we have

$P_{g}(h)_{ij}=\frac{1}{2}\big{(}\nabla_{g}^{*}\nabla_{g}h\big{)}_{ij}-g^{kp}g^{lq}R_{ikjl}h_{pq},$
(13)

and, for $\omega=g(X,.)$, we have

$(\delta_{g}\delta_{g}^{*}X)_{i}=-\frac{1}{2}g^{ki}\nabla_{k}\big{(}\nabla_{i}\omega_{j}+\nabla_{j}\omega_{i}\big{)},$
(14)

where $\nabla_{i}$ is the covariant derivative for $g$ in the direction
$e_{i}$. We directly see thanks to the estimates of Remark 2.10 that we have
the controls:

$\|P_{g}(h)-P_{g^{D}}(h)\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}\leqslant
C\|g-g^{D}\|_{C^{2,\alpha}_{\beta}(g^{D})}\|h\|_{C^{2,\alpha}_{\beta}(g^{D})},$

and

$\|\delta_{g}\delta^{*}_{g}(X)-\delta_{g^{D}}\delta^{*}_{g^{D}}(X)\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\leqslant
C\|g-g^{D}\|_{C^{2,\alpha}_{\beta}(g^{D})}\|X\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}.$

Let us now consider $\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}$,
$\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}^{\prime}$, and
$\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}$. These tensors being all
supported in the annuli $\mathcal{A}_{k}(\epsilon)$, we just need to restrict
our attention to them. The crucial remark is that in (13), every term involves
at least a derivative of $h$ or of $g-g^{D}$, and that in (14), every term
involves at least two derivatives of $X$ or one of $g$ and one of $X$. Hence,
we have a more precise control on $h$ a $2$-tensor and $X$ a vector field
supported in $\mathcal{A}_{k}(\epsilon_{0})$

$\displaystyle\big{\|}P_{g}(h)-P_{g^{D}}(h)\big{\|}_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}\leqslant$
$\displaystyle
C\Big{(}\|g-g^{D}\|_{C^{2,\alpha}_{0}(g^{D})}\|\nabla^{2}h\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}$
$\displaystyle+\|\nabla(g-g^{D})\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\|\nabla
h\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$
$\displaystyle+\|\nabla^{2}(g-g^{D})\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}\|h\|_{C^{2,\alpha}_{0}(g^{D})}\Big{)},$

(notice the norms $C^{2,\alpha}_{0}(g^{D})$ in which we have
$\|\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}\|_{C^{2,\alpha}_{0}(g^{D})}\leqslant
C|H_{k}|_{g_{e}}$ and
$\|\chi_{\mathcal{A}_{k}(\epsilon)}H^{\prime}_{k}\|_{C^{2,\alpha}_{0}(g^{D})}\leqslant
C|H_{k}^{\prime}|_{g_{e}}$), and similarly

$\displaystyle\big{\|}\delta_{g}\delta^{*}_{g}(X)-\delta_{g^{D}}\delta^{*}_{g^{D}}(X)\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\leqslant$
$\displaystyle
C\Big{(}\|g-g^{D}\|_{C^{2,\alpha}_{0}(g^{D})}\|\nabla^{2}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$
$\displaystyle+\|\nabla(g-g^{D})\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\|\nabla
X\|_{C^{2,\alpha}_{0}(g^{D})}\Big{)}.$

and with the control
$\|\nabla\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}\big{)}\|_{C^{2,\alpha}_{0}(g^{D})}\leqslant
C\|X_{k}\|_{r_{D}C^{0}_{0}(g_{e})}$.

There remains to control the derivatives of the tensors
$\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}$,
$\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}^{\prime}$ and
$\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}$. Since the $H_{k}$ and
$H^{\prime}_{k}$ are constant on $\mathbb{R}^{4}$, and $X_{k}$ is linear, we
have $\nabla_{e}H_{k}=0$, $\nabla_{e}H^{\prime}_{k}=0$ and
$\nabla_{e}^{2}X_{k}=0$. Since the cut off functions are bounded in
$C^{2}_{0}(g^{D})$ by (12), for $i\in\\{1,2\\}$, we have

$\|\nabla^{i}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}\big{)}\|_{r^{-i}_{D}C^{2-i,\alpha}_{\beta}(g^{D})}\leqslant
C|H_{k}|_{g_{e}},$
$\|\nabla^{i}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}H_{k}^{\prime}\big{)}\|_{r^{-i}_{D}C^{2-i,\alpha}_{\beta}(g^{D})}\leqslant
C|H_{k}^{\prime}|_{g_{e}},$

and

$\|\nabla^{i}\big{(}\chi_{\mathcal{A}_{k}(\epsilon)}X_{k}\big{)}\|_{r^{1-i}_{D}C^{3-i,\alpha}_{\beta}(g^{D})}\leqslant
C\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})},$

which are enough to conclude. ∎

####  2.2.4 Elliptic estimates for the decoupling norms

Some elliptic estimates are still satisfied in these norms.

######  Proposition 2.24.

Let $0<\beta<1$, $g$ a metric, $h$ a symmetric $2$-tensor and $X$ a vector
field on $M_{o}$ (respectively $N_{j}$, or $M$). Then, there exists
$\epsilon_{*}=\epsilon_{*}(g_{o},g_{b_{j}},g^{D},\beta)>0$ and $C>0$ such that
if we have
$\|g-g_{\bullet}\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\leqslant\epsilon_{*}$,
where $g_{\bullet}$ is one of the norms $g_{o}$, $g_{b_{j}}$ or $g^{D}$, then,

$\|h\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\leqslant
C\big{(}\|P_{g}h\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}+\|h\|_{C^{0}_{\beta,*}(g_{\bullet})}\big{)},$

and

$\|X\|_{r_{\bullet}C^{3,\alpha}_{\beta,*}(g_{\bullet})}\leqslant
C\big{(}\|\delta_{g}\delta^{*}_{g}X\|_{r^{-1}\bullet
C^{1,\alpha}_{\beta}(g_{\bullet})}+\|X\|_{r_{\bullet}C^{0}_{\beta,*}(g_{\bullet})}\big{)}.$

###### Proof.

Let $g_{\bullet}$ be one of the metrics $g_{o}$, $g_{b_{j}}$ or $g^{D}$, and
for all $k$, $H_{k}$ a traceless constant $2$-tensor on
$\mathbb{R}^{4}/\Gamma_{k}$, and $X_{k}$ a Killing vector field on
$\mathbb{R}^{4}/\Gamma_{k}$. Let moreover $h_{*}$ be a $2$-tensor of
$C^{2,\alpha}_{\beta}(g_{\bullet})$ and $X_{*}$ be a vector field of
$r_{\bullet}C^{3,\alpha}_{\beta}(g_{\bullet})$, and define
$h=h_{*}+\sum_{k}\chi_{\bullet}X_{k}$ and
$X=X_{*}+\sum_{k}\chi_{\bullet}X_{k}$, where $\chi_{\bullet}$ is
$\chi_{\mathcal{A}_{k}(\epsilon)}$ or $\chi_{B_{g_{\bullet}}(\epsilon)}$
depending on the metric.

We then have the following controls:

$\|P_{g_{\bullet}}\chi_{\bullet}H_{k}\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}\leqslant
C\|H_{k}\|_{C^{0}_{0}(g_{e})},$

and

$\|\delta_{g_{\bullet}}\delta_{g_{\bullet}}^{*}\chi_{\bullet}X_{k}\|_{r^{-1}_{\bullet}C^{1,\alpha}_{\beta}(g_{\bullet})}\leqslant
C|X_{k}|_{g_{e}}.$

Hence, for $h_{*}$, we have

$\|P_{g_{\bullet}}h_{*}\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}}\leqslant
C\big{(}\|P_{g_{\bullet}}h\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}+\sum_{k}|H_{k}|_{g_{e}}\big{)},$

and the expected estimate for $g=g_{\bullet}$ is then a consequence of the
elliptic estimates in the weighted spaces of Lemma 2.11 which give

$\|h_{*}\|_{C^{2,\alpha}_{\beta}(g_{\bullet})}\leqslant
C\big{(}\|P_{g_{\bullet}}h_{*}\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}+\|h_{*}\|_{C^{0}_{\beta}(g_{\bullet})}\big{)},$

and imply therefore that

$\|h\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\leqslant
2C^{2}\big{(}\|P_{g}h\|_{r^{-2}_{\bullet}C^{\alpha}_{\beta}(g_{\bullet})}+\|h\|_{C^{0}_{\beta,*}(g_{\bullet})}\big{)}.$

The same argument works for the operator $\delta\delta^{*}$ on the vector
fields thanks to the elliptic estimates of Lemma 2.12.

Proposition 2.23 finally lets us go from the metric $g_{\bullet}$ to a metric
$g$ satisfying
$\|g-g_{\bullet}\|_{C^{2,\alpha}_{\beta,*}(g_{\bullet})}\leqslant\epsilon_{*}$.
∎

##  3 Reduced divergence-free gauge

When the Einstein orbifold which we approximate has nonpositive scalar
curvature, we can always put our Einstein metrics in Bianchi gauge with
respect to a naïve desingularization (see [Biq13, Lemme 8.2] adapted to our
norms). When the Ricci curvature of our Einstein manifolds is positive, this
is not necessarily true, but we can still use the divergence-free gauge. This
is the goal of this section whose main result is Proposition 3.8. To show
this, we will use a Banach fixed point theorem approach which necessitate the
study of the linearized equation:

$\delta\delta^{*}X=-\delta h,$

where $X$ is a vector field, and $h$ a $2$-tensor.

In our degenerating situation, we want to obtain estimates in our weighted
norms which are independent of the gluing scales. A difficulty is that our
limit orbifold might have more symmetries than the Ricci-flat ALE spaces (for
example, $\mathbb{S}^{4}/\mathbb{Z}_{2}$ desingularized by Eguchi-Hanson
metrics). The associated Killing vector fields would give an approximate
kernel for $\delta\delta^{*}$ which would not be an actual kernel or cokernel.
We will need to define a reduced divergence-free gauge to obtain uniform
estimates as the gluing scales go to zero.

######  Note 3.1.

All along this section, if nothing is precised, an Einstein orbifold
$(M_{o},g_{o})$ will be either compact or ALE.

###  3.1 Kernel of the linearization

Let us focus on the operator $\delta\delta^{*}$ on a flat cone
$(\mathbb{R}^{4}/\Gamma,g_{e})$, on an orbifold $(M_{o},g_{o})$, and on Ricci-
flat ALE orbifolds $(N_{j},g_{b_{j}})$.

##### On a flat cone.

On the flat cone
$(\mathbb{R}^{4}/\Gamma,g_{e})=(\mathbb{R}^{+}\times\mathbb{S}^{3}/\Gamma,dr^{2}+r^{2}g_{S})$,
according to [CT94, Section 2], any $1$-form on $\mathbb{R}^{4}/\Gamma$ is a
countable sum of $1$-forms of one of the following types which are preserved
by $\delta\delta^{*}$:

  1. 1.

$p(r)\psi$, where $\delta_{S}\psi=0$, and $d^{*}_{S}d_{S}\psi=\mu\psi$,

  2. 2.

$r^{-1}l(r)\phi dr+u(r)rd_{S}\phi$, and $d^{*}_{S}d_{S}\phi=\nu\phi$,

where $p,l,u:\mathbb{R}^{+}\to\mathbb{R}$ and
$\phi:\mathbb{S}^{3}/\Gamma\to\mathbb{R}$ are functions, and where $\psi$ is a
$1$-form on $\mathbb{S}^{3}/\Gamma$.

According to [AV12, Section 4.1], thanks to the computation of the
eingenvalues of the Laplacian and of the Hodge Laplacian on the $1$-forms of
the sphere [Fol89, Theorem C], the solutions to
$\delta_{e}\delta_{e}^{*}\omega=0$ are countable sums of $1$-forms of the
following types

  1. 1.

$r^{a^{\pm}_{j}}\psi$ with $a^{\pm}_{j}:=\pm(1+j)$, $j\in\mathbb{N}^{*}$,

  2. 2.

$r^{b^{\pm}_{j}}d_{S}\phi\;+\;b^{\pm}_{j}r^{b^{\pm}_{j}-1}\phi dr$, or
$2r^{b^{\pm}_{j}+2}d_{S}\phi\;+\;b^{\mp}_{j}r^{b^{\pm}_{j}+1}\phi dr$, with
$b_{j}^{\pm}=-1\pm(1+j)$, $j\in\mathbb{N}$.

Since we are interested in solving an equation

$\delta\delta^{*}X=-\delta h,$

where $X$ is a vector field, and $h$ a $2$-tensor is in
$C^{2,\alpha}_{\beta,*}$, we are naturally looking for $X$ in
$r_{D}C^{3,\alpha}_{\beta,*}$. The _exceptional values_ of
$\delta_{e}\delta_{e}^{*}$ are the values $\gamma\in\mathbb{R}$ such that
there exists a homogeneous $1$-form whose norm is proportional to
$r_{e}^{\gamma}$ in the kernel of $\delta_{e}\delta_{e}^{*}$. We are therefore
interested in the exceptional values around the exceptional value $1$
associated to the linear vector fields of the kernel of
$\delta_{e}\delta^{*}_{e}$.

######  Lemma 3.2.

On $(\mathbb{R}^{4}/\Gamma)\backslash\\{0\\}$ for $\Gamma\neq\\{e\\}$, $1$ is
the only exceptional value between $-3$ and $2$.

###### Proof.

According to the above discussion, the exceptional values are a priori of the
form $a^{\pm}_{j}-1=-1\pm(1+j)$ for $j\in\mathbb{N}^{*}$,
$b^{\pm}_{j}-1=-2\pm(1+j)$ with $j\in\mathbb{N}$, or $b^{\pm}_{j}+1=\pm(1+j)$
with $j\in\mathbb{N}$. Let us first note that $a^{\pm}_{j}-1\in(-3,2)$ for
$j\in\mathbb{N}^{*}$ implies that $a^{\pm}_{j}-1=1$, and therefore that no
other exceptional value between $-3$ and $2$ come from the first type of
$1$-form.

For $b^{\pm}_{j}+1$, the values $0$ and $-1$ are a priori possible, and for
$b^{\pm}_{j}-1$, $-1$ and $-2$ are a priori possible. However, these values
cannot appear on a flat cone $\mathbb{R}^{4}/\Gamma$ for $\Gamma\neq\\{e\\}$.
Indeed, the values $b^{\pm}_{j}-1=0$ and $b^{\pm}+1=-2$ only appear if $-3$ is
an eingenvalue of the Laplacian on the link of the cone, but this is not the
case for $\mathbb{S}^{3}/\Gamma$ because there does not exist any non zero
$\Gamma$-invariant linear function on $\mathbb{R}^{4}$.

For the values $b^{\pm}_{j}-1=-1$ and $b^{\pm}_{j}+1=-1$, we use the form of
the solutions. In the first case, $b^{\pm}_{j}=0$ gives

$r^{b^{\pm}_{j}}d_{S}\phi\;+\;b^{\pm}_{j}r^{b^{\pm}_{j}-1}\phi dr=d_{S}\phi,$

for $\Delta_{S}\phi=0$, therefore $\phi$ is constant and finally
$d_{S}\phi=0$. In the second case, the equality $b^{\pm}_{j}=-2$, that is
$b^{\mp}_{j}=0$, gives

$2r^{b^{\pm}_{j}+2}d_{S}\phi\;+\;b^{\mp}_{j}r^{b^{\pm}_{j}+1}\phi
dr=d_{S}\phi,$

for $\Delta_{S}\phi=0$, therefore $\phi$ is constant and finally
$d_{S}\phi=0$. ∎

The $1$-forms associated to the exceptional value $1$ are sum of $1$-forms of
the three following types:

  1. 1.

$r^{2}\psi$, where $\psi$ is the dual of a Killing vector field of
$\mathbb{S}^{3}$,

  2. 2.

$rdr$, (associated to the eingenvalue $\nu=0$),

  3. 3.

$2r\phi dr+r^{2}d_{S}\phi$, (associated to the eingenvalue $\nu=8$).

##### On an orbifold or an ALE space.

Since there is no exceptional value other than $1$ in $(-3,2)$, we have the
following result on an orbifold ALE.

######  Proposition 3.3.

Let $(N_{j},g_{b_{j}})$ be a Ricci-flat ALE orbifold. For $0<\beta<1$, the
operator

$\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}:r_{b_{j}}C^{3,\alpha}_{\beta,*}\to
r_{b_{j}}^{-1}C^{1,\alpha}_{\beta}$

is bijective.

Let $(M_{o},g_{o})$ be a compact Einstein orbifold. For $0<\beta<1$, the
operator

$\delta_{g_{o}}\delta_{g_{o}}^{*}:r_{o}C^{3,\alpha}_{\beta,*}\to
r_{o}^{-1}C^{1,\alpha}_{\beta}$

is Fredholm and both its kernel and its cokernel are equal to
$\mathbf{K}_{o}$, the set of Killing vector fields of $(M_{o},g_{o})$.

As a consequence, there exist $C_{o}>0$ and $\epsilon_{o}>0$ depending on
$g_{o}$ such that if
$\|g-g_{o}\|_{C^{2,\alpha}_{\beta,*}(M_{o})}\leqslant\epsilon_{o}$, then we
have for any vector field $X\in\mathbf{K}_{o}^{\perp}$ on $M_{o}$

$\|X\|_{r_{o}C^{3,\alpha}_{\beta,*}(g_{o})}\leqslant
C_{o}\|\delta_{g}\delta_{g}^{*}X\|_{r_{o}^{-1}C^{1,\alpha}_{\beta}(g_{o})}.$

There also exists $C_{j}>0$ and $\epsilon_{j}>0$ depending on $g_{b_{j}}$ such
that if $\|g-g_{b_{j}}\|_{C^{2,\alpha}_{\beta,*}(N_{j})}\leqslant\epsilon_{j}$
then we have for any vector field $X$ on $N_{j}$,

$\|X\|_{r_{b_{j}}C^{3,\alpha}_{\beta,*}(g_{b_{j}})}\leqslant
C_{j}\|\delta_{g}\delta_{g}^{*}X\|_{r_{b_{j}}^{-1}C^{1,\alpha}_{\beta}(g_{b_{j}})}.$

###### Proof.

For orbifold singularities, we will first authorize our tensors to behave like
$r^{1-\beta}$ for $0<\beta<1$ at the singularities, instead of being in
$rC^{3,\alpha}_{\beta,*}$ to use the theory of elliptic operators in weighted
Hölder spaces. Let us start by considering an Einstein orbifold
$(M_{o},g_{o})$ and the operator
$\delta_{g_{o}}\delta_{g_{o}}^{*}:r_{o}C^{3,\alpha}_{-\beta}\to
r_{o}^{-1}C^{1,\alpha}_{-\beta}$ (notice the $-\beta$). Its kernel is composed
of Killing vector fields of $g_{o}$ which we will denote $\mathbf{K}_{o}$
because if for $X\in r_{o}C^{3,\alpha}_{-\beta}$ we have
$\delta_{g_{o}}\delta_{g_{o}}^{*}X$, integrating by parts yields,

$\displaystyle 0=$
$\displaystyle\;\int_{M_{o}}\langle\delta_{g_{o}}\delta_{g_{o}}^{*}X,X\rangle
dv_{o}$
$\displaystyle=\int_{M_{o}}|\delta_{g_{o}}^{*}X|^{2}_{g_{o}}dv_{o}+\lim_{r\to
0}\int_{\\{r_{o}=r\\}}\delta^{*}_{g_{o}}X(n,X)$
$\displaystyle=\int_{M_{o}}|\delta_{g_{o}}^{*}X|^{2}_{g_{o}}dv_{o},$

where $n=\frac{\nabla r_{o}}{|\nabla r_{o}|}$, because the boundary term which
is schematically $\lim_{r\to 0}(\mathcal{O}(r^{-\beta+1-\beta+3}))$ vanishes.
Similarly, its cokernel is equal to the kernel of
$\delta_{g_{o}}\delta_{g_{o}}^{*}$ on $r_{o}^{-3}C^{1,\alpha}_{\beta}$ which
is also reduced to $\mathbf{K}_{o}$ because there is no exceptional value
between $-3$ and $1$.

On an ALE orbifold $(N,g_{b})$, let us assume that a vector field $X\in
r_{b}^{1-\beta}C^{3,\alpha}_{0}$ satisfies
$\delta_{g_{b}}\delta_{g_{b}}^{*}X=0$. Since there is no exceptional value
between $-3$ and $1$, we actually have $X=\mathcal{O}(r_{b})$ when $r_{b}\to
0$ at the singular points of $(N,g_{b})$, and $X=\mathcal{O}(r_{b}^{-3})$ at
infinity. Let us then consider the following integration by parts,

$\displaystyle 0=$
$\displaystyle\;\int_{N}\langle\delta_{g_{b}}\delta_{g_{b}}^{*}X,X\rangle
dv_{b}$
$\displaystyle=\int_{N}|\delta_{g_{b}}^{*}X|^{2}_{g_{b}}dv_{b}-\lim_{\rho\to\infty}\int_{\\{r_{b}=\rho\\}}\delta^{*}_{g_{b}}X(n,X)+\lim_{r\to
0}\int_{\\{r_{b}=r\\}}\delta^{*}_{g_{b}}X(n,X)$
$\displaystyle=\int_{N}|\delta_{g_{b}}^{*}X|^{2}_{g_{b}}dv_{b},$

where the boundary term vanishes because it is the sum of the limit for
$r_{b}\to\infty$ of a $\mathcal{O}(|X|_{g_{b}}|\nabla
X|_{g_{b}}r_{b}^{3})=\mathcal{O}(r_{b}^{-4})$ and of the limit when $r_{b}\to
0$ of a $\mathcal{O}(|X|_{g_{b}}|\nabla
X|_{g_{b}}r_{b}^{3})=\mathcal{O}(r_{b}^{4})$. Hence, we have
$\delta_{g_{b}}^{*}X=0$, and since $g_{b}$ is Ricci-flat,
$\big{(}\delta_{b}+\frac{1}{2}d\textup{tr}_{b}\big{)}\delta_{g_{b}}^{*}X=\nabla_{b}^{*}\nabla_{b}X=0$,
which implies that $\nabla_{b}X=0$ by integration by parts against $X$, and
finally, that $X$ is parallel on $N$. Since $X$ tends to $0$ at infinity, we
have $X=0$. The operator
$\delta_{g_{b}}\delta_{g_{b}}^{*}:r_{b}^{1-\beta}C^{3,\alpha}_{0}\to
r_{b}^{-1-\beta}C^{1,\alpha}_{0}$ is therefore injective.

The cokernel of the self adjoint operator
$\delta_{g_{b}}\delta_{g_{b}}^{*}:r_{b}^{1-\beta}C^{3,\alpha}_{0}\to
r_{b}^{-1-\beta}C^{1,\alpha}_{0}$ is equal to the kernel of
$\delta_{g_{b}}\delta_{g_{b}}^{*}$ on $r_{b}^{-3+\beta}C^{1,\alpha}_{0}$ which
is also reduced to $\\{0\\}$ because there is no exceptional values between
$-3+\beta$ and $1-\beta$. The operator
$\delta_{g_{b}}\delta_{g_{b}}^{*}:r_{b}^{1-\beta}C^{3,\alpha}_{0}\to
r_{b}^{-1-\beta}C^{1,\alpha}_{0}$ is therefore bijective.

Let us finally work in the norms we are interested in and study the operators
$\delta_{g_{o}}\delta_{g_{o}}^{*}:r_{o}C^{3,\alpha}_{\beta,*}\to
r_{o}^{-1}C^{1,\alpha}_{\beta}$ and
$\delta_{g_{b}}\delta_{g_{b}}^{*}:r_{b}C^{3,\alpha}_{\beta,*}\to
r_{b}^{-1}C^{1,\alpha}_{\beta}$. Since the spaces
$r_{b}C^{3,\alpha}_{\beta,*}$ and $r_{o}C^{3,\alpha}_{\beta,*}$ are
respectively only the direct sum of $r_{b}C^{3,\alpha}_{\beta}$ and
$r_{o}C^{3,\alpha}_{\beta}$ with a space of finite dimension composed of cut-
off of linear vector fields, the image remains closed and of finite
codimension. We can be more precise by noticing that

$\delta_{g_{o}}\delta_{g_{o}}^{*}\big{(}r_{o}C^{3,\alpha}_{\beta,*}\big{)}=\delta_{g_{o}}\delta_{g_{o}}^{*}\big{(}r_{o}C^{3,\alpha}_{-\beta}\big{)}\cap
r_{o}^{-1}C^{1,\alpha}_{\beta}.$

Indeed, we have
$\delta_{g_{o}}\delta_{g_{o}}^{*}\big{(}r_{o}C^{3,\alpha}_{\beta,*}\big{)}\subset\delta_{g_{o}}\delta_{g_{o}}^{*}\big{(}r_{o}C^{3,\alpha}_{-\beta}\big{)}\cap
r_{o}^{-1}C^{1,\alpha}_{\beta}$ because $r_{o}C^{3,\alpha}_{\beta,*}\subset
r_{o}C^{3,\alpha}_{-\beta}$ and thanks to Proposition 2.23. Conversely, if for
$X\in r_{o}C^{3,\alpha}_{-\beta}$ we have
$\delta_{g_{o}}\delta_{g_{o}}^{*}X\in r_{o}^{-1}C^{1,\alpha}_{\beta}$, then,
since the only exceptional value between $1-\beta$ and $1+\beta$ is $1$ and
corresponds to the linear kernel of $\delta_{e}\delta_{e}^{*}$, we have $X\in
r_{o}C^{3,\alpha}_{\beta,*}$. Similarly, we conclude that

$\delta_{g_{b}}\delta_{g_{b}}^{*}\big{(}r_{b}C^{3,\alpha}_{\beta,*}\big{)}=\delta_{g_{b}}\delta_{g_{b}}^{*}\big{(}r_{b}^{1-\beta}C^{3,\alpha}_{0}\big{)}\cap
r_{b}^{-1}C^{1,\alpha}_{\beta},$

and finally, $\delta_{g_{o}}\delta_{g_{o}}^{*}:r_{o}C^{3,\alpha}_{\beta,*}\to
r_{o}^{-1}C^{1,\alpha}_{\beta}$ is Fredholm with $\mathbf{K}_{o}$ as kernel
and cokernel, and
$\delta_{g_{b}}\delta_{g_{b}}^{*}:r_{b}C^{3,\alpha}_{\beta,*}\to
r_{b}^{-1}C^{1,\alpha}_{\beta}$ is bijective. We can then conclude by the open
mapping theorem between Banach spaces which is stable by small perturbation of
the operator. ∎

###  3.2 Controls on the inverse of the linearization

These controls will help us treat the case of trees of singularities with
small enough gluing parameters. For this, we approximate the kernel
$\mathbf{K}_{o}$ on our naïve desingularization $(M,g^{D})$ in the following
way. For all $\mathbf{X}_{o}\in\mathbf{K}_{o}$, according to Remark 2.21, on
an orbifold, there exists a unique decomposition

$\mathbf{X}_{o}=\mathbf{X}_{o,*}+\sum_{k}\chi_{B_{o}(p_{k},\epsilon_{0})}\mathbf{X}_{o,k},$

such that
$\|\mathbf{X}_{o}\|_{r_{o}C^{3,\alpha}_{\beta,*}}=\|\mathbf{X}_{o,*}\|_{r_{o}C^{3,\alpha}_{\beta}}+\sum_{k}\|\mathbf{X}_{o,k}\|_{r_{e}C^{0}_{0}}$
(other decompositions make the value infinite). We then define
$\tilde{\mathbf{K}}_{o}$ as the space of the following vector fields on $M$

$\tilde{\mathbf{X}}_{o,t}:=\chi_{M_{o}^{16t}}\mathbf{X}_{o,*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\mathbf{X}_{o,k},$

for $\mathbf{X}_{o}\in\mathbf{K}_{o}$. Note that we therefore have
$\tilde{\mathbf{X}}_{o,t}=\mathbf{X}_{o}$ on $M_{o}^{16t}$.

######  Definition 3.4 (Reduced divergence-free gauge).

We define the _reduced divergence_ operator,
$\tilde{\delta}_{g}:=\pi_{\tilde{\mathbf{K}}_{o}^{\perp}}\delta_{g}$, where
$\pi_{\tilde{\mathbf{K}}_{o}^{\perp}}$ is the $L^{2}(g^{D})$-orthogonal
projection on $\tilde{\mathbf{K}}_{o}^{\perp}$. We will say that a metric
$g_{1}$ is in _reduced divergence-free_ gauge with respect to a metric $g_{2}$
if $\tilde{\delta}_{g_{2}}g_{1}=0$.

Let us start by noticing that the operator $\tilde{\delta}_{g^{D}}$ is
actually very close to $\delta_{g^{D}}$ for a naïve desingularization $g^{D}$
with small enough gluing parameters.

######  Lemma 3.5.

There exists $C>0$ such that for all $2$-tensor $h\in
C^{2,\alpha}_{\beta,*}(g^{D})$, we have,

$\big{\|}(\tilde{\delta}_{g^{D}}-\delta_{g^{D}})h\big{\|}_{r_{D}^{-1}C^{1,\alpha}_{\beta}(g^{D})}\leqslant
Ct_{\max}\|h\|_{C^{2,\alpha}_{\beta,*}(g^{D})}.$ (15)

###### Proof.

It is enough to show that the projection on $\tilde{\mathbf{K}}_{o}$ of
$\delta_{g^{D}}h$ is small to show the result. We naturally proceed by
integration by parts. Let $\tilde{\mathbf{X}}_{o,t}\in\tilde{\mathbf{K}}_{o}$
for $\mathbf{X}_{o}\in\mathbf{K}_{o}$ be an approximate Killing vector field.
We have,

$\displaystyle\Big{|}\int_{M}(\delta_{g^{D}}h)\big{(}\tilde{\mathbf{X}}_{o,t}\big{)}dv_{g^{D}}\Big{|}=$
$\displaystyle\;\Big{|}\int_{M}\langle
h,\delta_{g^{D}}^{*}(\tilde{\mathbf{X}}_{o,t})\rangle_{g^{D}}dv_{g^{D}}\Big{|},$

but, we have $\delta_{g^{D}}=\delta_{g_{o}}$ on $M_{o}^{16t}\subset
M_{o}^{t}$, and therefore,

$\displaystyle\delta_{g^{D}}^{*}(\tilde{\mathbf{X}}_{o,t})=$
$\displaystyle\;\delta^{*}_{g_{o}}\mathbf{X}_{o}+\delta^{*}_{g_{o}}((\chi_{M_{o}^{t}}-1)\mathbf{X}_{o,*})+\delta^{*}_{g_{o}}\Big{(}(\chi_{\mathcal{A}_{k}(\epsilon_{0})}-\chi_{B_{o}(p_{k},\epsilon_{0})})\sum_{k}\mathbf{X}_{o,k}\Big{)}$
$\displaystyle+\big{(}\delta_{g_{o}}^{*}-\delta_{g^{D}}^{*}\big{)}\Big{(}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\sum_{k}\mathbf{X}_{o,k}\Big{)},$

where by definition $\delta^{*}_{g_{o}}\mathbf{X}_{o}=0$. But, on
$M_{o}^{16t}$, $\chi_{M^{16t}_{o}}\mathbf{X}_{o,*}$ is equal to
$\mathbf{X}_{o,*}$ except on the annuli of radii $2t_{k}^{\frac{1}{4}}$ and
$4t_{k}^{\frac{1}{4}}$, and in the same way
$\chi_{M_{o}^{\epsilon_{0}^{-4}t^{2}}}-1$ is supported in
$\epsilon_{0}^{-1}\sqrt{t}_{k}<r_{D}<2\epsilon_{0}^{-1}\sqrt{t}_{k}$, the cut
off functions satisfy for all $\epsilon_{o}^{-1}\sqrt{t_{k}}<r<\epsilon_{o}$
and for all $l\in\mathbb{N}^{*}$, there exists $C_{l}>0$ such that, if we
denote $\mathbb{1}_{A}$ the indicator function of $A$,

$r_{D}^{l}|\nabla^{l}\chi_{M_{o}^{r^{4}}}|_{g_{o}}\leqslant
C_{l}\mathbb{1}_{\\{r<r_{D}<2r\\}},$

for the vector fields, we have

$r_{D}^{l}\Big{(}|\nabla^{l}\mathbf{X}_{o,*}|_{g_{o}}+\Big{|}\nabla^{l}\sum_{k}\mathbf{X}_{o,k}\Big{|}_{g_{o}}\Big{)}\leqslant
C_{l}\|X_{o}\|_{C^{l}_{\beta,*}}r_{D},$

and finally for the metric, we have

$r_{D}^{l}\big{|}\nabla^{l}\big{(}g^{D}-g_{o})\big{|}_{g_{o}}\leqslant
C_{l}\mathbb{1}_{\\{r_{D}<t_{k}^{\frac{1}{4}}\\}}\big{(}r_{D}^{2}+t_{k}^{2}r_{D}^{-4}\big{)},$

As a consequence, because of the properties of the norms detailed in Remark
2.10, we have

$|\delta^{*}_{g_{o}}((\chi_{M_{o}^{t}}-1)\mathbf{X}_{o,*})|_{g_{o}}\leqslant\;C\|\mathbf{X}_{o}\|_{r_{o}C^{0}_{0}(g_{o})},$

and on the annulus of radii, we have
$\epsilon_{0}^{-1}\sqrt{t}_{k}<r_{D}<2\epsilon_{0}^{-1}\sqrt{t}_{k}$,

$\Big{|}\delta^{*}_{g_{o}}\Big{(}\big{(}\chi_{\mathcal{A}_{k}(\epsilon_{0})}-\chi_{B_{o}(p_{k},\epsilon_{0})})\sum_{k}\mathbf{X}_{o,k}\Big{)}\Big{|}\leqslant
C\mathbb{1}_{\\{\epsilon_{0}^{-1}\sqrt{t}_{k}<r_{D}<2\epsilon_{0}^{-1}\sqrt{t}_{k}\\}}\|X_{o}\|_{C^{l}_{\beta,*}},$

and

$\Big{|}\Big{(}\delta_{g_{o}}^{*}-\delta_{g^{D}}^{*}\big{)}\Big{(}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\sum_{k}\mathbf{X}_{o,k}\Big{)}\Big{|}\leqslant
C\mathbb{1}_{\\{\epsilon_{0}^{-1}\sqrt{t}_{k}<r_{D}<t_{k}^{\frac{1}{4}}\\}}t_{k}^{2}r_{D}^{-4}\|X_{o}\|_{C^{l}_{\beta,*}}.$

Finally, since we have
$\operatorname{\textup{Vol}}\big{(}A_{e}\big{(}t_{k}^{\frac{1}{4}},2t_{k}^{\frac{1}{4}}\big{)}\big{)}\approx
t_{k},$
$\operatorname{\textup{Vol}}\big{(}A_{e}\big{(}\epsilon_{0}^{-1}\sqrt{t_{k}},2\epsilon_{0}^{-1}\sqrt{t_{k}}\big{)}\big{)}\approx
t_{k}^{2}$, and also
$\int_{\\{\epsilon_{0}^{-1}\sqrt{t_{k}}<r_{D}<t_{k}^{\frac{1}{4}}\\}}t_{k}^{2}r_{D}^{-4}.r_{D}^{3}\approx
t_{k}^{2}|\log t_{k}|$, we have

$\Big{|}\int_{M}\langle\delta_{g^{D}}h,\tilde{\mathbf{X}}_{o,t}\rangle_{g^{D}}dv_{g^{D}}\Big{|}\leqslant
Ct_{\max}\|h\|_{C^{0}_{0}(g^{D})}\|\mathbf{X}_{o}\|_{r_{o}C^{0}_{0}(g_{o})}.$

We finally conclude that
$\tilde{\delta}_{g^{D}}h=\pi_{\tilde{\mathbf{K}}_{o}}=\delta_{g^{D}}h+\tilde{\mathbf{X}}_{o,t}$
with

$\|\tilde{\mathbf{X}}_{o,t}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}\leqslant
Ct_{\max}\|h\|_{C^{0}_{0}(g^{D})}.$

∎

######  Lemma 3.6.

Let $0<\beta<1$, $0<\alpha<1$ and $(M,g^{D})$ a naïve desingularization of a
compact or ALE Einstein orbifold by a tree of singularities. Then, there
exists $\tau_{D}>0$ and $\epsilon_{D}>0$ and $C_{D}>0$, only depending on
$\beta$ and the metric $g_{o}$ and $g_{b_{j}}$, such that for
$t_{\max}<\tau_{D}$, and any metric $g$ satisfying
$\|g-g^{D}_{t}\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\leqslant\epsilon_{D}$, the
operator

$\tilde{\delta}_{g}\delta_{g}^{*}:\tilde{\mathbf{K}}_{o}^{\perp}\cap
r_{D}C^{3,\alpha}_{\beta,*}(g^{D})\to\tilde{\mathbf{K}}_{o}^{\perp}\cap
r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})$

is invertible and we have for any vector field $X\perp\tilde{\mathbf{K}}_{o}$
on $M$,

$\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}(g^{D})}\leqslant
C_{D}\|\tilde{\delta}_{g}\delta_{g}^{*}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}.$

###### Proof.

Let $0<\epsilon<\epsilon_{D}^{2-\beta}<\epsilon_{0}$ for $\epsilon_{D}$ and
$\epsilon$ which we will choose small enough along the proof, and assume that
$t_{\max}<\epsilon^{4}$ and therefore, on each annulus
$\mathcal{A}_{k}:=\mathcal{A}_{k}(\epsilon)$ between $N_{k}$ and $N_{j}$ or
$N_{k}$ and $M_{o}$ (in which case, we will fix $T_{o}=1$), the existence of a
diffeomorphism

$\Phi_{k}:A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}})\subset\mathbb{R}^{4}/\Gamma_{k}\to\mathcal{A}_{k}(\epsilon)\subset
M,$

such that for all $0<\beta<1$, there exists $C>0$, for which we have

$\big{\|}\Phi^{*}_{k}g^{D}-g_{e}\big{\|}_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}))}\leqslant
C\epsilon^{2-\beta}<C\epsilon_{D}.$ (16)

Until the end of the proof, we will use the notation

$A_{k}:=A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}).$

We are looking for $\tau_{D}>0$ small enough so that if $t_{\max}<\tau_{D}$,
there exists $C_{D}>0$, only depending on the constants $\epsilon_{o}>0$,
$C_{o}>0$, $\epsilon_{j}$, $C_{j}>0$, $\epsilon_{e}$ and $C_{e}$ of
Proposition 3.3 and of Proposition 2.15, such that we have for all
$X\perp\tilde{\mathbf{K}}_{o}$

$\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}(g^{D})}\leqslant
C_{D}\|\tilde{\delta}_{g^{D}}\delta_{g^{D}}^{*}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}.$

According to the estimate (15), for $t_{\max}$ small enough, it is enough to
have

$\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}(g^{D})}\leqslant\frac{C_{D}}{2}\|\delta_{g^{D}}\delta_{g^{D}}^{*}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}.$

The diffeomorphisms $\Phi_{k}:A_{k}\to\mathcal{A}_{k}$ allow us to pull the
situation back on $\mathbb{R}^{4}$, where the ratio of the annuli $A_{k}$ is
$\epsilon^{2}t_{k}^{-1/2}$ which is arbitrarily large for $t_{\max}$
arbitrarily small. According to Proposition 2.15 and thanks to the controls of
Proposition 2.23, there exists $\epsilon_{e}>0$ such that if we have
$\|\Phi_{k}^{*}g^{D}-g_{e}\|_{C^{2,\alpha}_{\beta,*}(A_{k})}<\epsilon_{e}$
(which is satisfied for $\epsilon_{D}$ small enough according to (16)), then,
there exist linear vector fields $X_{k}$ of the kernel of
$\delta_{e}\delta^{*}_{e}$ such that the vector fields
$\chi_{\mathcal{A}_{k}}X_{k}$ in the annuli $\mathcal{A}_{k}$ satisfy

$\displaystyle\|\Phi_{k}^{*}X-X_{k}\|_{r_{e}C^{3,\alpha}_{0}(A(T_{j}^{1/2}t_{k}^{1/4}))}\leqslant$
$\displaystyle\;C_{e}T_{j}^{\frac{1}{2}}t_{k}^{\frac{1}{4}}\Big{(}\epsilon^{-\beta}t_{k}^{\frac{\beta}{4}}\big{\|}\delta_{g_{e}}\delta_{g_{e}}^{*}\Phi_{k}^{*}X\big{\|}_{r^{-1}_{e}C^{1,\alpha}_{\beta}(A_{k})}$
$\displaystyle+2\epsilon^{-1}t_{k}^{\frac{1}{4}}\|\Phi_{k}^{*}X-X_{k}\|_{r_{e}C^{3,\alpha}_{\beta}(A_{k})}\Big{)}$
$\displaystyle\leqslant$
$\displaystyle\;2C_{e}T_{j}^{\frac{1}{2}}t_{k}^{\frac{1}{4}}\Big{(}\epsilon^{-\beta}t_{k}^{\frac{\beta}{4}}\big{\|}\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{)}_{|\mathcal{A}_{k}}\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$
$\displaystyle+2\epsilon^{-1}t_{k}^{\frac{1}{4}}\|(X-\chi_{\mathcal{A}_{k}}X_{k})_{\mathcal{A}_{k}}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}\Big{)}$
(17)

on
$A(\sqrt{T_{j}}t_{k}^{1/4}):=A_{e}((1/2)\sqrt{T_{j}}t_{k}^{1/4},4\sqrt{T_{j}}t_{k}^{1/4})$.
Let us then consider the decomposition

$X=X_{*}+\sum_{k}\chi_{\mathcal{A}_{k}}X_{k}$ (18)

for the rest of the proof.

The objective is now to show that there exists a constant $C_{D}>0$ such that

$\|X_{*}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}+\sum_{k}\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}\leqslant
C_{D}\|\delta_{g^{D}}\delta_{g^{D}}^{*}X\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}.$

In order to do this, we will reduce our situation to $M_{o}$ and to the
$N_{j}$ where such controls have been shown in Proposition 3.3.

On $M_{o}^{t/16}$, $g^{D}-g_{o}$ is supported in $M_{o}^{t/16}\backslash
M_{o}^{16t}$, where on each annulus $\mathcal{A}_{k}$,
$\frac{1}{2}t_{k}^{1/4}<r_{D}<2t_{k}^{1/4}$, and for all $l\in\mathbb{N}$,
there exists $C_{l}>0$ such that in these regions, we have

$t_{k}^{\frac{l}{4}}|\nabla^{l}(g^{D}-g_{o})|_{g_{o}}\leqslant
C_{l}t_{k}^{\frac{1}{2}}.$ (19)

Since $M_{o}^{16t}\subset M_{o}^{t}\subset M_{o}^{t/16}$, there exists a cut-
off function $\chi_{M_{o}^{t/16}}$ supported in $M_{o}^{t/16}$ such that
$\chi_{M_{o}^{t/16}}\equiv 1$ on $M_{o}^{t}$ and such that for all
$l\in\mathbb{N}$, there exists $C_{l}>0$ for which in each $\mathcal{A}_{k}$,

$t_{k}^{\frac{l}{4}}|\nabla^{l}\chi_{M_{o}^{t/16}}|_{g^{D}}\leqslant C_{l}.$
(20)

We then define a vector field $X_{o}$ on $M_{o}$ by

$X_{o}:=\chi_{M_{o}^{t/16}}X_{*}+\sum_{k\in
K_{o}}\chi_{B_{o}(p_{k},\epsilon)}X_{k},$

where $K_{o}$ is the set of $k$ such that the annulus $\mathcal{A}_{k}$ has a
nonempty intersection with $M_{o}^{t}$.

By construction, $X_{*}=X_{o,*}$ on $M_{o}^{t}$ and we therefore have the
following control. Denoting $X_{o,*}:=\chi_{M_{o}^{t/16}}X_{*}$, we have

$\|X_{o,*}\|_{r_{o}C^{3,\alpha}_{\beta}(g_{o})}\geqslant\|X_{o,*}\|_{r_{o}C^{0}_{\beta}(g_{o})}\geqslant\|(X_{*})_{|M_{o}^{t}}\|_{r_{D}C^{0}_{\beta}(g^{D})}.$
(21)

On $M_{o}^{\frac{t}{16}}$, we have,

$\displaystyle\delta_{g^{D}}\delta_{g^{D}}^{*}X=\delta_{g_{o}}\delta_{g_{o}}^{*}X_{o}+\delta_{g_{o}}\delta_{g_{o}}^{*}(X-X_{o})+\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}-\delta_{g_{o}}\delta_{g_{o}}^{*}\big{)}X.$

Since the cut off functions are bounded in norm $C^{3,\alpha}_{0}(g^{D})$ and
$C^{3,\alpha}_{0}(g_{o})$ by (20), and since their derivatives are supported
in $M_{o}^{t/16}\backslash M_{o}^{t}$, we have the following lower bound
thanks to (17) and to (19): for $C>0$ depending on the cut off function, we
have

$\displaystyle\big{\|}\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{)}_{|M_{o}^{t/16}}\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\geqslant$
$\displaystyle\;\|\delta_{g_{o}}\delta_{g_{o}}^{*}X_{o}\|_{r^{-1}_{o}C^{1,\alpha}_{\beta}(g_{o})}$
$\displaystyle-C\big{\|}(X_{*})_{|M_{o}^{t/16}\backslash
M_{o}^{t}}\big{\|}_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}$ $\displaystyle-
CC_{2}\sum_{k\in K_{o}}t_{k}^{\frac{1}{2}}\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}}$
$\displaystyle\geqslant$
$\displaystyle\;\|\delta_{g_{o}}\delta_{g_{o}}^{*}X_{o}\|_{r^{-1}_{o}C^{1,\alpha}_{\beta}(g_{o})}$
$\displaystyle-2CC_{e}\Big{(}\big{\|}\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{)}_{|\mathcal{A}_{k}}\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$
$\displaystyle+\sum_{k\in
K_{o}}\epsilon^{\beta-1}t_{k}^{\frac{1-\beta}{4}}\|(X-\chi_{\mathcal{A}_{k}(\epsilon)}X_{k})_{\mathcal{A}_{k}}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}\Big{)}$
$\displaystyle-CC_{2}\sum_{k\in
K_{o}}t_{k}^{\frac{1}{2}}\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}},$ (22)

because the cut off is only on $X_{*}$ and on the annulus. Hence, the error
between $X$ and $X_{o}$ on $M_{o}^{t/16}$ is equivalent to
$|\Phi_{k}^{*}(X-\chi_{\mathcal{A}_{k}(\epsilon)}X_{k})|_{g_{e}}$ on
$A_{e}((1/2)\sqrt{T_{j}}t_{k}^{1/4},4\sqrt{T_{j}}t_{k}^{1/4})$ which is
estimated in (17), and the error between $g^{D}$ and $g_{o}$ is estimated in
(19). Now, when $t_{\max}\to 0$, we have

$\frac{\|\pi_{\mathbf{K}_{o}^{\perp}}X_{o}\|_{r_{o}C^{3,\alpha}_{\beta,*}(g_{o})}}{\|X_{o}\|_{r_{o}C^{3,\alpha}_{\beta,*}(g_{o})}}\to
0$

because $X\perp\tilde{\mathbf{K}}_{o}$. Proposition 3.3 therefore yields, for
$t_{\max}$ small enough,

$\|X_{o}\|_{r_{o}C^{3,\alpha}_{\beta,*}(g_{o})}\leqslant
2C_{o}\|\delta_{g_{o}}\delta_{g_{o}}^{*}X_{o}\|_{r^{-1}_{o}C^{1,\alpha}_{\beta}(g_{o})}.$
(23)

And therefore, thanks to (21) and (22), for $t_{\max}$ small enough, and
denoting $C^{\prime}:=CC_{2}+2C_{o}CC_{e}\epsilon^{\beta-1}$ and
$\gamma(t_{\max}):=\sum_{k}t_{\max}^{\frac{1}{2}}+t_{\max}^{\frac{1-\beta}{4}}$,
we have

$\displaystyle\|(X_{*})_{|M_{o}^{t}}\|_{r_{D}C^{0}_{\beta}(g^{D})}+\sum_{k\in
K_{o}}$
$\displaystyle\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}-C^{\prime}\gamma(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}$
$\displaystyle\leqslant$
$\displaystyle\|X_{o}\|_{r_{o}C^{3,\alpha}_{\beta,*}(g_{o})}-C^{\prime}\gamma(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}$
$\displaystyle\leqslant$ $\displaystyle
C_{o}\|\delta_{g_{o}}\delta_{g_{o}}^{*}X_{o}\|_{r^{-1}_{o}C^{1,\alpha}_{\beta}(g_{o})}-C^{\prime}\gamma(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}$
$\displaystyle\leqslant$ $\displaystyle
2C_{o}(1+CC_{e})\big{\|}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})},$
(24)

where we successively used (21), (23) and (22).

Indeed, on an orbifold $(M_{o},g_{o})$, the vector fields of the decomposition
$X_{o}=X_{o,*}+\sum_{k\in K_{o}}\chi_{B_{o}(p_{k},\epsilon)}X_{k}$ reaching
the infimum of the definition of the norm
$\|.\|_{r_{o}C^{3,\alpha}_{\beta,*}}$ are determined by the limit of
$r_{o}^{-1}X_{o}$ at each singular point according to Remark 2.21. Here, the
infimum is therefore reached with the $X_{k}$ of the decomposition (18).

We next consider the vector field $X_{1}:=X-\sum_{k\in
K_{o}}\chi_{\mathcal{A}_{k}}X_{k}$ which satisfies for a constant $C>0$,

$\displaystyle\|\delta_{g^{D}}\delta_{g^{D}}^{*}X_{1}\|_{r_{D}^{-1}C^{1,\alpha}_{\beta}(g^{D})}\leqslant$
$\displaystyle\;C\big{(}\|\delta_{g^{D}}\delta_{g^{D}}^{*}X\|_{r_{D}^{-1}C^{1,\alpha}_{\beta}(g^{D})}+\gamma(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}\big{)}$
(25)

thanks to the control (24) of $\sum_{k\in
K_{o}}\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}$.

Given $j\in K_{o}$, the Ricci-flat ALE orbifold $(N_{j},g_{b_{j}})$ is glued
to $M_{o}$ and we can extend the vector field $X_{1}=X_{*}+\sum_{k\notin
K_{o}}\chi_{\mathcal{A}_{l}}X_{l}$ to $N_{j}$ by

$X_{j}:=\chi_{N_{j}^{t/16}}X_{*}+\sum_{l\in
K_{j}}\chi_{B_{j}(p_{l},\epsilon)}X_{l},$

where $K_{j}$ is the set of $k\neq j$ such that $\mathcal{A}_{k}$ has a
nonempty intersection with $N_{j}^{t}$.

######  Remark 3.7.

By considering $X_{1}$ instead of $X$, we do not have a linear vector field of
the kernel of $\delta_{e}\delta^{*}_{e}$ to extend at at infinity of $N_{j}$.
The vector field $X_{j}$ is therefore well controlled in
$r_{b_{j}}C^{3,\alpha}_{\beta,*}$.

The difference $\frac{g^{D}}{T_{j}}-g_{b_{j}}$ is supported in
$N_{j}^{t/16}\backslash N_{j}^{16t}$, where we have
$\frac{1}{2}\sqrt{T_{j}}t_{k}^{\frac{1}{4}}<r_{D}=\sqrt{T_{j}}r_{b_{j}}<2\sqrt{T_{j}}t_{k}^{\frac{1}{4}}$
at the singular points and
$\frac{1}{2}\sqrt{T_{j}}t_{j}^{-\frac{1}{4}}<r_{D}=\sqrt{T_{j}}r_{b_{j}}<2\sqrt{T_{j}}t_{j}^{-\frac{1}{4}}$
at infinity, and more precisely, there exists for all $l\in\mathbb{N}$,
$C_{l}>0$ such that we have

$t_{k}^{\frac{l}{4}}\Big{|}\nabla^{l}\Big{(}\frac{g^{D}}{T_{j}}-g_{b_{j}}\Big{)}\Big{|}_{g_{b_{j}}}\leqslant
C_{l}t_{k}^{\frac{1}{2}},$ (26)

and

$t_{j}^{-\frac{l}{4}}\Big{|}\nabla^{l}\Big{(}\frac{g^{D}}{T_{j}}-g_{b_{j}}\Big{)}\Big{|}_{g_{b_{j}}}\leqslant
C_{l}t_{j}^{\frac{1}{2}}.$ (27)

Denoting $X_{j,*}:=\chi_{N_{j}^{t/16}}X_{*}$, we have

$\|X_{j,*}\|_{r_{b_{j}}C^{3,\alpha}_{\beta}(g_{b_{j}})}\geqslant\|X_{j,*}\|_{r_{b_{j}}C^{0}_{\beta}(g_{b_{j}})}\geqslant\|X_{*}\|_{r_{D}C^{0}_{\beta}(g^{D})},$
(28)

and thanks to (17) and the inequalities (26) and (27), writing

$\displaystyle\delta_{g^{D}}\delta_{g^{D}}^{*}X_{1}:=\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}X_{j}+\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}(X_{1}-X_{j})+\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}-\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}\big{)}X_{1},$

we have

$\displaystyle\big{\|}\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}X_{1}\big{)}_{|N_{j}^{t/16}}\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}\geqslant$
$\displaystyle\;\|\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}X_{j}\|_{r^{-1}_{b_{j}}C^{1,\alpha}_{\beta}(g_{b_{j}})}$
$\displaystyle-2CC_{e}\Big{(}\|\big{(}\delta_{g^{D}}\delta_{g^{D}}^{*}X_{1}\big{)}_{|\mathcal{A}_{k}}\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$
$\displaystyle+\sum_{k\in
K_{j}}\epsilon^{\beta-1}t_{k}^{\frac{1-\beta}{4}}\|(X_{*})_{\mathcal{A}_{k}}\|_{r_{D}C^{3,\alpha}_{\beta}(g^{D})}\Big{)}$
$\displaystyle-
CC_{2}t_{k}^{\frac{1}{2}}\|X_{1}\|_{r_{D}C^{3,\alpha}_{\beta,*}}-CC_{2}t_{j}^{\frac{1}{2}}\|X_{1}\|_{r_{D}C^{3,\alpha}_{\beta,*}},$
(29)

where we remark that
$\|X_{1}\|_{r_{D}C^{3,\alpha}_{\beta,*}}\leqslant\|X_{*}\|_{r_{D}C^{3,\alpha}_{\beta}}+\sum_{k\in
K_{j}}\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}$.

Proposition 3.3 then yields

$\|X_{j}\|_{r_{j}C^{3,\alpha}_{\beta,*}(g_{b_{j}})}\leqslant
C_{o}\|\delta_{g_{b_{j}}}\delta_{g_{b_{j}}}^{*}X_{j}\|_{r^{-1}_{j}C^{1,\alpha}_{\beta}(g_{b_{j}})},$

and thanks to the control (25), we then have for a constant $C>0$ only
depending on the metrics $g_{o}$ and $g_{b_{j}}$, the control

$\displaystyle\|(X_{*})_{|N_{j}^{t}}\|_{r_{D}C^{0}_{\beta}(g^{D})}+\sum_{k\in
K_{j}}\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}-C\gamma$
$\displaystyle(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}$
$\displaystyle\leqslant$
$\displaystyle\;C\big{\|}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})},$
(30)

similar to (24).

Iterating this to the other Ricci-flat ALE orbifolds of the tree of
singularities, we get controls similar to (30) on all the $N_{j}$ which, with
(24) on $M_{o}$, give the following control on the whole manifold only
depending on $g_{o}$ and the $g_{b_{j}}$

$\displaystyle\|X_{*}\|_{r_{D}C^{0}_{\beta}(g^{D})}+\sum_{k}$
$\displaystyle\|X_{k}\|_{r_{e}C^{0}_{0}(g_{e})}-C\gamma(t_{\max})\|X\|_{r_{D}C^{0}_{\beta,*}(g^{D})}$
$\displaystyle\leqslant
C\big{\|}\delta_{g^{D}}\delta_{g^{D}}^{*}X\big{\|}_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$

and the stated result for $g=g^{D}$ because
$M=M_{0}^{t}\cup\bigcup_{j}N_{j}^{t}$ for $t_{\max}$ small enough.

To get the estimate for another metric $g$ close to $g^{D}$, we just use
Proposition 2.23 to ensure that for
$\|g-g^{D}\|_{C^{2,\alpha}_{\beta,*}(g^{D})}$ arbitrarily small,
$\tilde{\delta}_{g}\delta^{*}_{g}$ is arbitrarily close to
$\delta_{g^{D}}\delta^{*}_{g^{D}}$ for the operator norm on
$r_{D}C^{3,\alpha}_{\beta,*}(g^{D})$.

Finally, notice that
$(\tilde{\delta}_{g^{D}})^{*}=\delta^{*}_{g^{D}}\pi_{\tilde{\mathbf{K}}_{o}^{\perp}}$,
and therefore that $\tilde{\delta}_{g^{D}}\delta^{*}_{g^{D}}$ is self adjoint
on $\tilde{\mathbf{K}}_{o}^{\perp}$. Its injectivity implies its surjectivity
by integration by parts on the compact manifold $(M,g^{D})$. ∎

We can finally prove the main result of the section by fixed point theorem.

######  Proposition 3.8.

Let $0<\beta<1$, and $(M,g^{D})=(M,g^{D}_{t})$ a naïve desingularization of a
compact or ALE Einstein orbifold, $(M_{o},g_{o})$. Then, there exists
$\epsilon_{D},\tau_{D},C_{D}>0$ which only depends on the metrics $g_{o}$ and
$g_{b_{j}}$ of $D$ such that for $t_{\max}\leqslant\tau_{D}$ and for any
metric $g$ satisfying
$\|g-g^{D}\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\leqslant\epsilon_{D}$, there
exists a unique vector field $X\perp\tilde{\mathbf{K}}_{o}$ on $M$ for which,

$\tilde{\delta}_{g^{D}}(\exp_{X}^{*}g)=0,$

where $\exp_{X}$ is the diffeomorphism $\exp_{X}:x\in
M\mapsto\exp_{x}^{g^{D}}(X(x))$.

We moreover have, $\|X\|_{r_{D}C^{3,\alpha}_{\beta,*}(g^{D})}\leqslant
C_{D}\|\tilde{\delta}_{g^{D}}(g-g^{D})\|_{r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})}$,
and therefore, there exists $\delta:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with
$\lim_{0}\delta=0$ such that we have

$\|\exp_{X}^{*}g-g\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\leqslant\delta\big{(}\|g-g^{D}\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\big{)}.$

###### Proof.

Let us fix $g$ a metric on $M$, such that
$\|g-g^{D}\|_{C^{2,\alpha}_{\beta,*}}\leqslant\epsilon$ for $\epsilon>0$ which
we will choose small enough along the proof and define the operator
$F_{g}:r_{D}C^{3,\alpha}_{\beta,*}(g^{D})\to
r^{-1}_{D}C^{1,\alpha}_{\beta}(g^{D})$ which to a vector field $X$ associates

$F_{g}(X):=\tilde{\delta}_{(\exp_{g^{D}}X)_{*}g^{D}}g.$

The objective is therefore to find $X$ such that $F_{g}(X)=0$, which will
imply that $\tilde{\delta}_{g^{D}}(\exp_{g^{D}}X)^{*}g=0$ because for any
diffeomorphism $\phi$,
$\phi_{*}(\delta_{g^{D}}\phi^{*}g)=\delta_{\phi_{*}g^{D}}g$ (by applying
$\phi_{*}$ to $g^{D}$ which is $C^{\infty}$, we do not loose regularity). The
map $g\mapsto F_{g}$ is linear, and the linearization of the operator
$F_{g^{D}}$ around zero is $\tilde{\delta}_{g^{D}}\delta^{*}_{g^{D}}$ which is
invertible between the orthogonals of $\tilde{\mathbf{K}}_{o}$ according to
Lemma 3.6.

There remains to control the nonlinear terms in our norms, which is a
consequence of the behavior of our norms with respect to the usual operations,
see Remark 2.10. We can therefore finally put our metrics in gauge with
respect to each other thanks to a fixed point theorem with explicit constant
below, Lemma 3.9. ∎

######  Lemma 3.9.

Let $\Phi:E\to F$, be a smooth map between Banach spaces and let
$Q:=\Phi-\Phi(0)-d_{0}\Phi$.

Assume that there exist $q>0$, $r_{0}>0$ and $c>0$ such that:

  1. 1.

for all $x$ and $y$ in $B(0,r_{0})$, we have the following control on the
nonlinear terms

$\|Q(x)-Q(y)\|\leqslant q(\|x\|+\|y\|)\|x-y\|.$

  2. 2.

the linearization $d_{0}\Phi$ is an isomorphism, and more precisely, we have

$\|(d_{0}\Phi)^{-1}\|\leq c.$

If $r\leqslant\min\Big{(}r_{0},\frac{1}{2qc}\Big{)}$ and
$\|\Phi(0)\|\leqslant\frac{r}{2c}$, then, the equation $\Phi(x)=0$ admits a
unique solution in $B(0,r)$.

###  3.3 Einstein metrics in gauge

Let us now come back to Einstein metrics which can be characterized thanks to
the Bianchi identity as the zero set of

$E(g):=\operatorname{\textup{Ric}}(g)-\frac{\textup{R}(g)}{2}g+\frac{n-2}{2n}\overline{R}(g)g$

on a compact manifold $M$, where
$\overline{R}:=\frac{1}{\operatorname{\textup{Vol}}(M,g)}\int_{M}\operatorname{\textup{R}}(g)dv_{g}$.
Notice that $\delta_{g}E(g)=0$, again by the Bianchi identity.

The equation $E(g)=0$ is invariant by the action of diffeomorphisms and by
scaling, we will therefore restrict our attention to deformations which are
transverse to these actions in order to obtain an operator whose linearization
is elliptic. More precisely, we will fix the volume and fix a gauge thanks to
the reduced divergence-free condition.

It turns out that we can characterize the zeros of $E$ in reduced divergence-
free gauge as the zeros of $\mathbf{\Phi}$ defined by

$\mathbf{\Phi}(g):=E(g)+\delta_{g}^{*}\tilde{\delta}_{g^{D}}g.$

Indeed, if we have $E(g)=0$ and $\tilde{\delta}_{g^{D}}g=0$, then we have
$\mathbf{\Phi}(g)=0$. And conversely, if $\mathbf{\Phi}(g)=0$, then since
$E(g)$ is divergence-free (for $g$) by the Bianchi identity, we get
$\tilde{\delta}_{g}\mathbf{\Phi}(g)=(\tilde{\delta}_{g}\delta_{g}^{*})\tilde{\delta}_{g^{D}}g$,
and since for $g$ close enough to $g^{D}$,
$(\tilde{\delta}_{g}\delta_{g}^{*})$ is invertible on the image of
$\tilde{\delta}_{g^{D}}$ by Lemma 3.6, we finally have
$\tilde{\delta}_{g^{D}}g=0$. In a $C^{2,\alpha}_{\beta,*}(g^{D})$-neighborhood
of $g^{D}$ the zero set of $\mathbf{\Phi}$ is exactly the set of Einstein
metrics in reduced divergence-free gauge with respect to $g^{D}$.

######  Corollary 3.10.

Let $D_{0},v_{0}>0$, $l\in\mathbb{N}$, and $\beta=\beta(v_{0},D_{0})>0$
obtained in Corollary 2.7. Then, for all $\epsilon>0$, there exists
$\delta=\delta(\epsilon,D_{0},v_{0},l)>0$ such that if $(M,g^{\mathcal{E}})$
is an Einstein manifold satisfying

  * •

the volume is bounded below by $v_{0}>0$,

  * •

the diameter is bounded above by $D_{0}$,

  * •

the Ricci curvature is bounded $|\operatorname{\textup{Ric}}|\leq 3$.

and such that for an Einstein orbifold $(M_{o},g_{o})$,

$d_{GH}\big{(}(M,g^{\mathcal{E}}),(M_{o},g_{o})\big{)}\leqslant\delta,$

then, there exists a naïve desingularization $(M,g^{D})$ of $(M_{o},g_{o})$ by
a tree of singularities, and a diffeomorphism $\psi:M\to M$ such that

$\big{\|}\psi^{*}g^{\mathcal{E}}-g^{D}\big{\|}_{C^{l}_{\beta,*}(g^{D})}\leqslant\epsilon,$

and

$\tilde{\delta}_{g^{D}}(\psi^{*}g^{\mathcal{E}})=0.$

In particular, we have

$\mathbf{\Phi}(\psi^{*}g^{\mathcal{E}})=0.$

##  4 Resolution of the Einstein equation modulo obstructions

We will now show that it is always possible to produce metrics which are
Einstein modulo some obstructions (the cokernel of the linearization of the
Einstein operator) in our weighted Hölder spaces. The main result of the
section is Theorem 4.9 which allows us to perturbate any naïve
desingularization $g^{D}$ to an Einstein modulo obstructions metric and in
particular according to [Ozu19a] we produce all Einstein metrics close to an
Einstein orbifold in the Gromov-Hausdorff sense by this procedure.

We have seen in Corollary 3.10 that up to a diffeomorphism, any Einstein
metric $g$ close to $(M_{o},g_{o})$ in the Gromov-Hausdorff sense is a
solution of

$\mathbf{\Phi}(g):=E(g)+\delta_{g}^{*}\tilde{\delta}_{g^{D}}g=0.$

To study this equation, we will naturally start by studying its linearization
on volume preserving deformations, that is, on $2$-tensors $h$ satisfying
$\int_{M}\textup{tr}_{g}hdv_{g}=0$, this yields the formula

$\displaystyle\bar{P}_{g}(h):=$
$\displaystyle\;d_{g}\mathbf{\Phi}(h)=\frac{1}{2}\Big{(}\nabla^{*}_{g}\nabla_{g}h-2\delta_{g}^{*}\delta_{g}h+2\delta^{*}_{g}\tilde{\delta}_{g^{D}}h-\nabla_{g}^{2}\textup{tr}_{g}h-2\mathring{\operatorname{\textup{R}}}_{g}(h)$
$\displaystyle+\operatorname{\textup{Ric}}_{g}\circ
h+h\circ\operatorname{\textup{Ric}}_{g}-\operatorname{\textup{R}}_{g}h+(\Delta_{g}\textup{tr}_{g}h)g-(\delta_{g}\delta_{g}h)g+\langle\operatorname{\textup{Ric}}_{g},h\rangle_{g}g$
$\displaystyle+\frac{1}{2}\overline{\operatorname{\textup{R}}}(g)h-\frac{1}{2\operatorname{\textup{Vol}}(g)}\int_{M}\Big{\langle}\operatorname{\textup{Ric}}(g)-\frac{\operatorname{\textup{R}}(g)}{2},h\big{\rangle}_{g}dv_{g}\Big{)}\Big{)},$

in dimension $4$. If $g$ is an Einstein metric and $h$ is a trace-free and
divergence-free $2$-tensor, $\bar{P}$ reduces to

$P:=\frac{1}{2}\nabla^{*}\nabla-\mathring{\operatorname{\textup{R}}}_{b},$

which is simpler to study.

###  4.1 Kernel and cokernel of the linearization on model spaces

##### Exceptional values for $P_{e}:=\frac{1}{2}\nabla_{e}^{*}\nabla_{e}$ on
$(\mathbb{R}^{4}/\Gamma,g_{e})$ and gauge constraints.

As described in the proof of Proposition 2.13, the elements of the kernel of
$P_{e}$ on $\mathbb{R}^{4}/\Gamma$ are sums of homogeneous $2$-tensors whose
coefficients in an orthonormal basis of $\mathbb{R}^{4}$ are homogeneous of
order $k$ or $-2-k$ for $k\in\mathbb{N}$.

However, some of these tensors cannot appear in our developments because they
are not trace-free or in divergence-free gauge.

######  Lemma 4.1.

On $\mathbb{R}^{4}/\Gamma$ for $\Gamma\neq\\{e\\}$, there is no harmonic
homogeneous $2$-tensor whose coefficients are of order $-2$ or $-3$ in
divergence-free gauge.

###### Proof.

A harmonic homogeneous $2$-tensor $h$ with coefficients of order $-2$ or $-3$
satisfying $tr(h)=0$ on $\mathbb{R}^{4}/\Gamma$ pulls back on $\mathbb{R}^{4}$
as a matrix whose coefficients are homogeneous harmonic functions.

More precisely, a homogeneous element of the kernel of
$\nabla_{e}^{*}\nabla_{e}$ of order $-2$ is of the form
$\frac{H_{0}}{r_{e}^{2}}$ for $H_{0}$ a constant $2$-tensor on
$\mathbb{R}^{4}$. Such a $2$-tensor can only be divergence-free if $H_{0}=0$
because we have

$\delta_{e}\Big{(}\frac{H_{0}}{r_{e}^{2}}\Big{)}=2\frac{H_{0}(\partial_{r_{e}},.)}{r_{e}^{3}},$

and, if $x\in\mathbb{R}^{4}$ is an eigenvector of $H_{0}$ of eingenvalue
$\lambda$, since $\partial_{r_{e}}(x)=\frac{x}{|x|}$, and we then have
$\delta_{e}\Big{(}\frac{H_{0}}{r_{e}^{2}}\Big{)}_{x}\big{(}\frac{x}{|x|}\big{)}=-2\frac{H_{0}(\frac{x}{|x|},\frac{x}{|x|})}{r_{e}^{3}(x)}=-2\frac{\lambda}{r_{e}^{3}(x)}$
and this vanishes if an only if $\lambda=0$.

According to the description of [CT94, Proposition 4.65], the traceless
harmonic $2$-tensors decaying as $r_{e}^{-3}$, are Lie derivatives of the
metric with respect to a vector field dual of a $1$-form of the type
$2r_{e}^{-1}d_{S}\phi\;+\;r_{e}^{-2}\phi dr_{e}$ (see Section 3.1) where
$\phi$ is an eingenfunction of the Laplacian of eingenvalue $-3$. Such a non-
zero eingenfunction does not exist on non trivial quotients of the sphere
$\mathbb{S}^{3}$ because there is no non-zero linear function on
$\mathbb{R}^{4}$ which is invariant by $\Gamma$. ∎

##### Kernel of the operator $P$ on the model spaces.

Let us start by describing the kernel $P$ on our model spaces.

######  Lemma 4.2 (Decaying kernel of $P$).

Let $(N,g_{b})$ be a Ricci-flat ALE orbifold, and denote
$P_{b}:=\frac{1}{2}\nabla^{*}_{b}\nabla_{b}-\mathring{\operatorname{\textup{R}}}_{b}$,
and $\mathbf{O}(g_{b})$, the kernel of $P_{g_{b}}$ on
$C^{2,\alpha}_{\beta,*}(g_{b})$.

The elements of $\mathbf{O}(g_{b})$ decay at least like $r_{b}^{-4}$ at
infinity, and for all $\mathbf{o}_{b}\in\mathbf{O}(g_{b})$, we have the
following development coordinates at infinity,

$\mathbf{o}_{b}=O^{4}+\mathcal{O}(r_{b}^{-5}),$

with $O^{4}\sim r_{b}^{-4}$ a harmonic homogeneous.

Let also $(M_{o},g_{o})$ be a compact Einstein orbifold, we denote
$\mathbf{O}(g_{o})$ the kernel of $P_{o}$ on $C^{2,\alpha}_{\beta,*}$ for all
$0<\beta<1$. an element $\mathbf{o}_{o}\in\mathbf{O}(g_{o})$ has a development

$\mathbf{o}_{o}=O_{0}+O_{2}+\mathcal{O}(r_{o}^{3}),$

for harmonic homogeneous $2$-tensors $O_{i}\sim r_{o}^{i}$.

###### Proof.

Let us consider $\mathbf{o}\in\mathbf{O}(g_{b})$, for which
$P_{b}\mathbf{o}=0$, and $\mathbf{o}=\mathcal{O}(r_{b}^{-\beta})$ for some
$\beta>0$. Such a $2$-tensor is actually necessarily traceless and divergence-
free. Indeed, we have
$\delta_{g_{b}}P_{g_{b}}=\frac{1}{2}\nabla_{g_{b}}^{*}\nabla_{g_{b}}\delta_{g_{b}},$
and
$\textup{tr}_{g_{b}}P_{g_{b}}=\frac{1}{2}\nabla_{g_{b}}^{*}\nabla_{g_{b}}\textup{tr}_{g_{b}}.$
Therefore, if $P_{g_{b}}h=0$ for $h=\mathcal{O}(\rho^{-\delta})$ for some
$\delta>0$, then $\delta_{g_{b}}h=0$, and $\textup{tr}_{g_{b}}h=0$ by the
maximum principle. We deduce from Lemma 4.1 that $\mathbf{o}$ decays at least
like $\rho^{-4}$ and its principal term is a harmonic $2$-tensor.

In the same way in the neighborhood of a singularity of an orbifold
$(M_{o},g_{o})$ or of a Ricci-flat ALE orbifold $(N,g_{b})$, since there is no
harmonic $2$-tensor with linear growth because of the action of the nontrivial
group $\Gamma$, an element of the kernel admits a development

$\mathbf{o}=O_{0}+O_{2}+\mathcal{O}(r_{b}^{3}),$

where $O_{0}$ and $O_{2}$ are harmonic homogeneous $2$-tensors in $r_{o}^{0}$
and $r_{o}^{2}$ respectively. ∎

##### Estimates on the inverses.

Just like for the operator $\delta\delta^{*}$ in Proposition 3.3, the
operators $P_{g_{o}}$ and $P_{g_{b_{j}}}$ are injective on the orthogonal of
their respective kernels.

######  Lemma 4.3.

Let $(N,g_{b})$ be a Ricci-flat ALE orbifold and $(M_{o},g_{o})$ an Einstein
orbifold, and $0<\beta<1$.

Then, the operators

$P_{b}:\mathbf{O}(g_{b})^{\perp}\cap C^{2,\alpha}_{\beta,*}(g_{b})\to
r^{-2}_{b}C^{\alpha}_{\beta}(g_{b}),$

and

$P_{o}:\mathbf{O}(g_{o})^{\perp}\cap C^{2,\alpha}_{\beta,*}(g_{o})\to
r^{-2}_{o}C^{\alpha}_{\beta}(g_{o})$

are injective and there exists $C_{o}>0$ and $C_{b}>0$, such that we have for
all $2$-tensor $h_{b}\perp\mathbf{O}(g_{b})$ on $N$ and
$h_{o}\perp\mathbf{O}(g_{o})$ on $M_{o}$,

$\|h_{b}\|_{C^{2,\alpha}_{\beta,*}(g_{b})}\leqslant
C_{b}\|P_{b}h_{b}\|_{r^{-2}_{b}C^{\alpha}_{\beta}(g_{b})},$ (31)

and

$\|h_{o}\|_{C^{2,\alpha}_{\beta,*}(g_{o})}\leqslant
C_{o}\|P_{o}h_{o}\|_{r^{-2}_{o}C^{\alpha}_{\beta}(g_{o})}.$ (32)

Moreover, their respective cokernels are $\mathbf{O}(g_{b})$ and
$\mathbf{O}(g_{o})$.

###### Proof.

By standard theory of elliptic operators between weighted Hölder spaces, the
operators

$P_{o}:C^{2,\alpha}_{-\beta}(g_{o})\to r^{-2}_{o}C^{\alpha}_{-\beta}(g_{o}),$

and

$P_{b}:r_{b}^{-\beta}C^{2,\alpha}_{0}(g_{b})\to
r^{-2-\beta}_{b}C^{\alpha}_{0}(g_{b})$

are Fredholm for $0<\beta<1$ because we avoid the exceptional values close to
zero: $-2$ and $1$. Let us study their kernels and cokernels.

Let us start by the case of an Einstein orbifold $(M_{o},g_{o})$ and notice
that $\ker_{C^{2,\alpha}_{-\beta}(g_{o})}P_{g_{o}}\subset\mathbf{O}(g_{o})$
because there is no exceptional value between $0$ and $-\beta$. The kernel of
$P_{o}:C^{2,\alpha}_{-\beta}(g_{o})\to r^{-2}_{o}C^{\alpha}_{-\beta}(g_{o})$
is therefore equal to $\mathbf{O}(g_{o})$. Since $P_{o}$ is self adjoint and
since we are strictly between two exceptional values, its cokernel is the
kernel of $P_{o}$ on $r^{-2}_{o}C^{\alpha}_{\beta}(g_{o})$ which is also
reduced to $\mathbf{O}(g_{o})$ by a similar argument.

Similarly, the kernel of $P_{b}:r_{b}^{-\beta}C^{2,\alpha}_{0}(g_{b})\to
r^{-2-\beta}_{b}C^{\alpha}_{0}(g_{b})$ is $\mathbf{O}(g_{b})$, and its
cokernel is the kernel of $P_{b}$ on $r^{-2+\beta}_{b}C^{\alpha}_{0}(g_{b})$
which is reduced to $\mathbf{O}(g_{b})$.

For the cokernels of

$P_{b}:\mathbf{O}(g_{b})^{\perp}\cap C^{2,\alpha}_{\beta,*}(g_{b})\to
r^{-2}_{b}C^{\alpha}_{\beta}(g_{b}),$

and

$P_{o}:\mathbf{O}(g_{o})^{\perp}\cap C^{2,\alpha}_{\beta,*}(g_{o})\to
r^{-2}_{o}C^{\alpha}_{\beta}(g_{o}),$

like in the end of the proof of Proposition 3.3, we use the fact that

$P_{o}\big{(}C^{2,\alpha}_{\beta,*}(g_{o})\big{)}=P_{o}\big{(}C^{2,\alpha}_{-\beta}(g_{o})\big{)}\cap
r^{-2}_{o}C^{\alpha}_{\beta}(g_{o}),$

and

$P_{b}\big{(}C^{2,\alpha}_{\beta,*}(g_{b})\big{)}=P_{b}\big{(}r_{b}^{-\beta}C^{2,\alpha}_{0}(g_{b_{j}})\big{)}\cap
r^{-2}_{b}C^{\alpha}_{\beta}(g_{b}).$

∎

##### Approximation of kernels and cokernels on a naïve desingularization.

We wish to solve the equation $\operatorname{\textup{Ric}}(g)=\Lambda g$ for a
metric $g$ close to $g^{D}$ modulo the kernel and the cokernel of the
linearization of the Einstein operator. We will use approximate kernels and
cokernels defined as the truncated infinitesimal deformations of each model
space on the tree of singularities.

Let $\mathbf{o}_{o}\in\mathbf{O}(g_{o})$ and
$\mathbf{o}_{j}\in\mathbf{O}(g_{b_{j}})$, and define
$\mathbf{o}_{o}=\mathbf{o}_{o,*}+\sum_{k}\chi_{B_{o}(p_{k},\epsilon_{0})}\mathbf{o}_{o,k}$
and
$\mathbf{o}_{j}=\mathbf{o}_{j,*}+\sum_{k}\chi_{B_{j}(p_{k},\epsilon_{0})}\mathbf{o}_{j,k}$
their respective decompositions as a $2$-tensor of $C^{2,\alpha}_{\beta}$ and
constant $2$-tensors truncated in the neighborhoods of the singular points. We
define on $M$ the following $2$-tensors

$\tilde{\mathbf{o}}_{o,t}:=\chi_{M_{o}^{16t}}\mathbf{o}_{o,*}+\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\mathbf{o}_{o,k},$

and

$\tilde{\mathbf{o}}_{j,t}:=\chi_{N_{j}^{16t}}\mathbf{o}_{j,*}+\sum_{k}\chi_{\mathcal{A}_{k}}\mathbf{o}_{j,k}.$

######  Remark 4.4.

We have $\tilde{\mathbf{o}}_{o,t}=\mathbf{o}_{o}$ on $M_{o}^{16t}$, and
$\tilde{\mathbf{o}}_{j,t}=\mathbf{o}_{j}$ on $N_{j}^{16t}$.

######  Definition 4.5 (Space of truncated obstructions).

Let $(M,g^{D}_{t})$ be a naïve desingularization of a Einstein orbifold
$(M_{o},g_{o})$. On $M$, we will denote

$\tilde{\mathbf{O}}(g^{D}):=\Big{\\{}\tilde{\mathbf{o}}_{o,t}+\sum_{j}\tilde{\mathbf{o}}_{j,t},\;\mathbf{o}_{o}\in\mathbf{O}(g_{o}),\;\mathbf{o}_{j}\in\mathbf{O}(g_{b_{j}})\Big{\\}},$

the space of _truncated obstructions_.

######  Remark 4.6.

For $\beta<2$, we have

$\Big{\|}\tilde{\mathbf{o}}_{o,t}+\sum_{j}T_{j}\tilde{\mathbf{o}}_{j,t}\Big{\|}_{C^{0}_{\beta,*}(g^{D})}\approx\sup\big{(}\|\mathbf{o}_{o}\|_{C^{0}_{\beta,*}(g_{o})},\|\mathbf{o}_{j}\|_{C^{0}_{\beta,*}(g_{b_{j}})}\big{)},$
$\Big{\|}\tilde{\mathbf{o}}_{o,t}+\sum_{j}\tilde{\mathbf{o}}_{j,t}\Big{\|}_{r_{D}^{-2}C^{\alpha}_{\beta}(g^{D})}\approx\sup\big{(}\|\mathbf{o}_{o}\|_{C^{0}_{\beta,*}(g_{o})},\|\mathbf{o}_{j}\|_{C^{0}_{\beta,*}(g_{b_{j}})}\big{)},$

and

$\Big{\|}\tilde{\mathbf{o}}_{o,t}+\sum_{j}\tilde{\mathbf{o}}_{j,t}\Big{\|}_{L^{2}(g^{D})}\approx\sup\big{(}\|\mathbf{o}_{o}\|_{L^{2}(g_{o})},\|\mathbf{o}_{j}\|_{L^{2}(g_{b_{j}})}\big{)}.$

We would like to produce Einstein metrics in divergence-free gauge with
respect to $g^{D}$. But the point is that it is not always possible because
the space $\tilde{\mathbf{O}}(g^{D})$ is an (approximate) obstruction space.
We will show that we can perturbate $g^{D}+v$ for parameters $t>0$ and
$v\in\tilde{\mathbf{O}}(g^{D})$ small enough to obtain a metric
$\hat{g}_{v}=\hat{g}_{\phi,t,v}$ which will be in gauge with respect to
$g^{D}+v$ and solution of:

$\mathbf{\Phi}(\hat{g}_{v})\in\tilde{\mathbf{O}}(g^{D}),$

hence the term _modulo obstructions_.

##### Control of the inverse of the linearization.

We can first show that the linearization is invertible and that we can control
its inverse _independently of the gluing scales_ thanks to Lemma 4.3.

Let us start by showing that the operator
$\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}P_{g^{D}}$ is close to $P_{g^{D}}$ for
a sufficiently degenerate tree of singularities.

######  Lemma 4.7.

There exists $C>0$ such that for any $2$-tensor $h\in
C^{2,\alpha}_{\beta,*}(g^{D})$, we have

$\big{\|}\big{(}\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}P_{g^{D}}-P_{g^{D}}\big{)}h\big{\|}_{r_{D}^{-2}C^{\alpha}_{\beta}(g^{D})}\leqslant
Ct_{\max}^{\frac{1}{2}}\|h\|_{C^{2,\alpha}_{\beta,*}(g^{D})}.$

###### Proof.

The proof is similar to the proof of the estimate (15).

On $M_{o}$, for $\mathbf{o}_{o}\in\mathbf{O}(g_{o})$, we have
$P_{g_{o}}(\mathbf{o}_{o})=0$. We can then integrate by parts by using the
fact that $P_{g^{D}}$ is self adjoint

$\displaystyle\Big{|}\int_{M}\langle
P_{g^{D}}h,\tilde{\mathbf{o}}_{o,t}\rangle_{g^{D}}dv_{g^{D}}\Big{|}=$
$\displaystyle\;\Big{|}\int_{M}\langle
h,P_{g^{D}}(\tilde{\mathbf{o}}_{o,t})\rangle_{g^{D}}dv_{g^{D}}\Big{|},$

and we can decompose, since $g^{D}=g_{o}$ on $M_{o}^{16t}$,

$\displaystyle P_{g^{D}}(\tilde{\mathbf{o}}_{o,t})=$
$\displaystyle\;P_{g_{o}}\mathbf{o}_{o}+P_{g_{o}}((\chi_{M_{o}^{t}}-1)\mathbf{o}_{o,*})$
$\displaystyle+P_{g_{o}}\Big{(}\sum_{k}(\chi_{\mathcal{A}_{k}(\epsilon_{0})}-\chi_{B_{o}(p_{k},\epsilon_{0})})\mathbf{o}_{o,k}\Big{)}\Big{)}+\big{(}P_{g_{o}}-P_{g^{D}}\big{)}\Big{(}\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\mathbf{o}_{o,k}\Big{)}$

to get, thanks to the controls of $\chi_{M_{o}^{r}}$ the following estimates
(compare with the proof of (15))

$|P_{g_{o}}((\chi_{M_{o}^{t}}-1)\mathbf{o}_{o,*})|_{g_{o}}\leqslant\;C\|\mathbf{o}_{o}\|_{C^{0}_{0}(g_{o})}\mathbb{1}_{\\{2t_{k}^{\frac{1}{4}}<r_{D}<4t_{k}^{\frac{1}{4}}\\}}t_{k}^{-\frac{1}{2}},$
$\Big{|}P_{g_{o}}\Big{(}\sum_{k}(\chi_{\mathcal{A}_{k}(\epsilon_{0})}-\chi_{B_{o}(p_{k},\epsilon_{0})})\mathbf{o}_{o,k}\Big{)}\Big{)}\Big{|}\leqslant
C\|\mathbf{o}_{o}\|_{C^{0}_{0}(g_{o})}\mathbb{1}_{\\{\epsilon_{0}^{-1}t_{k}^{\frac{1}{2}}<r_{D}<2\epsilon_{0}^{-1}t_{k}^{\frac{1}{2}}\\}}t_{k}^{-1},$

and

$\Big{|}\big{(}P_{g_{o}}-P_{g^{D}}\big{)}\Big{(}\sum_{k}\chi_{\mathcal{A}_{k}(\epsilon_{0})}\mathbf{o}_{o,k}\Big{)}\Big{|}\leqslant
C\|\mathbf{o}_{o}\|_{C^{0}_{0}(g_{o})}\mathbb{1}_{\\{\epsilon_{0}^{-1}t_{k}^{\frac{1}{2}}<r_{D}<2t_{k}^{\frac{1}{4}}\\}}t_{k}^{2}r_{D}^{-6}.$

Finally, since
$\operatorname{\textup{Vol}}\big{(}A_{e}\big{(}t_{k}^{\frac{1}{4}},2t_{k}^{\frac{1}{4}}\big{)}\big{)}\approx
t_{k}$,
$\operatorname{\textup{Vol}}\big{(}A_{e}\big{(}\epsilon_{0}^{-1}\sqrt{t_{k}},2\epsilon_{0}^{-1}\sqrt{t_{k}}\big{)}\big{)}\approx
t_{k}^{2}$, and, since we moreover have
$\int_{\\{\epsilon_{0}^{-1}t_{k}^{\frac{1}{2}}<r_{D}<2t_{k}^{\frac{1}{4}}\\}}t_{k}^{2}r_{D}^{-6}.r_{D}^{3}\approx
t_{\max}$, we conclude that

$\Big{|}\int_{M}\langle
P_{g^{D}}h,\tilde{\mathbf{o}}_{o,t}\rangle_{g^{D}}dv_{g^{D}}\Big{|}\leqslant
Ct_{k}^{\frac{1}{2}}\|h\|_{C^{0}_{0}(g^{D})}\|\mathbf{o}_{o}\|_{C^{0}_{0}(g_{o})}.$
(33)

Similarly, for the bubbles $N_{j}$, for
$\mathbf{o}_{j}\in\mathbf{O}(g_{b_{j}})$ by conformal invariance of the
$L^{2}$-norm of the $2$-tensors in dimension $4$ and since
$P_{\frac{g}{t}}=tP_{g}$ for any metric $g$ and any $t>0$, we have

$\displaystyle\int_{M}\langle
P_{g^{D}}h,\chi\mathbf{o}_{j}\rangle_{g^{D}}dv_{g^{D}}$
$\displaystyle=\int_{M}\Big{\langle}\frac{h}{T_{j}},P_{\frac{g^{D}}{T_{j}}}\tilde{\mathbf{o}}_{j}\Big{\rangle}_{\frac{g^{D}}{T_{j}}}dv_{\frac{g^{D}}{T_{j}}}$
$\displaystyle=\int_{M}\Big{\langle}\frac{h}{T_{j}},P_{g_{b_{j}}}\tilde{\mathbf{o}}_{j}\Big{\rangle}_{g_{b_{j}}}dv_{g_{b_{j}}}$

the control at the singular points is the same and at infinity we have

$P_{g_{b}}\big{(}\tilde{\mathbf{o}}_{j,t}\big{)}=P_{g_{b_{j}}}\mathbf{o}_{j}+P_{g_{b_{j}}}((\chi_{N_{j}^{t}}-1)\mathbf{o}_{j,*}),$

hence, since $\mathbf{o}_{j}=\mathcal{O}(r_{b_{j}}^{-4})$, we have

$\big{|}P_{g_{b_{j}}}(\tilde{\mathbf{o}}_{j,t})\big{|}_{g_{b_{j}}}\leqslant
C\mathbb{1}_{\\{\frac{1}{2}t_{j}^{-1/4}<r_{b_{j}}<t_{j}^{-1/4}\\}}r_{b_{j}}^{-6},$

and therefore, since the volume of the annulus of radii
$\operatorname{\textup{Vol}}\big{(}A_{e}(1/2T_{j}^{\frac{1}{2}}t_{j}^{-\frac{1}{4}},T_{j}^{\frac{1}{2}}t_{j}^{-\frac{1}{4}})\big{)}\approx
T_{j}^{2}t_{j}^{-1}$ and
$\Big{|}\chi_{N^{16t}_{j}}\frac{h}{T_{j}}\Big{|}_{g_{b_{j}}}\leqslant\|h\|_{C^{2,\alpha}_{\beta,*}(g^{D})}$,
we have

$\Big{|}\int_{M}\langle
P_{g^{D}}h,\tilde{\mathbf{o}}_{j,t}\rangle_{g^{D}}dv_{g^{D}}\Big{|}\leqslant
Ct_{\max}^{\frac{1}{2}}\|h\|_{C^{0}_{0}(g^{D})}\|\mathbf{o}_{j}\|_{C^{0}_{0}(g_{b_{j}})},$
(34)

and finally,

$\big{\|}\big{(}\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}P_{g}^{D}-P_{g}^{D}\big{)}h\big{\|}_{r_{D}^{-2}C^{\alpha}_{\beta}(g^{D})}\leqslant
Ct_{\max}^{\frac{1}{2}}\|h\|_{C^{2,\alpha}_{\beta,*}(g^{D})}.$

∎

######  Proposition 4.8.

Let $0<\beta<1$, $k\in\mathbb{N}$, $0<\alpha<1$ and let $(M,g^{D})$ be a naïve
desingularization of a compact or ALE Einstein orbifold by a tree of
singularities. Then, there exists $\tau_{D}>0$ and $\epsilon_{D}>0$ and
$C_{D}>0$ only depending on $\beta$, and of the metric $g_{o}$ and $g_{b_{j}}$
of $D$ such that for $t_{\max}<\tau_{D}$, for all metric $g$ such that
$\|g-g^{D}_{t}\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\leqslant\epsilon_{D}$, the
operator

$\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}P_{g}:\tilde{\mathbf{O}}(g^{D})^{\perp}\cap
C^{2,\alpha}_{\beta,*}(g^{D})\to\tilde{\mathbf{O}}(g^{D})^{\perp}\cap
r^{-2}_{D}C^{\alpha}_{\beta}(g^{D}),$

where $\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}$ is the orthogonal projection
for $g^{D}$ on $\tilde{\mathbf{O}}(g^{D})^{\perp}$, is invertible and such
that we have for any $2$-tensor $h\perp\tilde{\mathbf{O}}(g^{D})$ on $M$,

$\|h\|_{C^{2,\alpha}_{\beta,*}(g^{D})}\leqslant
C_{D}\|\pi_{\tilde{\mathbf{O}}(g^{D})^{\perp}}P_{g}h\|_{r^{-2}_{D}C^{\alpha}_{\beta}(g^{D})}.$

###### Proof.

The proof is similar to that of Lemma 3.6. The idea is again to extend the
$2$-tensors on the model spaces and to deduce a control on the whole tree of
singularities.

Let $0<\epsilon<\epsilon_{D}^{2-\beta}<\epsilon_{0}$ for $\epsilon_{D}$ and
$\epsilon$ which we will choose small enough along the proof, and assume that
$t_{\max}<\epsilon^{4}$ in order to have on each annulus
$\mathcal{A}_{k}:=\mathcal{A}_{k}(\epsilon)$ between $N_{k}$ and $N_{j}$, the
existence of a diffeomorphism

$\Phi_{k}:A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}})\subset\mathbb{R}^{4}/\Gamma_{k}\to\mathcal{A}_{k}\subset
M,$

such that for any $0<\beta<1$, there exists $C>0$,

$\big{\|}\Phi^{*}_{k}g^{D}-g_{e}\big{\|}_{C^{2,\alpha}_{\beta}(A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}))}\leqslant
C\epsilon^{2-\beta}<C\epsilon_{D},$

by definition of $g^{D}$. Until the end of the proof, we will denote

$A_{k}:=A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{k}},\epsilon\sqrt{T_{j}}).$

Let $h$ be a symmetric $2$-tensor on $M$. Thanks to Proposition 2.13 and to
its generalization to metrics close to $g_{e}$ by Proposition 2.23, for
$\epsilon_{D}$ and $t_{\max}$ small enough, we can choose constant and
traceless $2$-tensors $H_{k}$ on $\mathbb{R}^{4}/\Gamma_{k}$ such that we have
on
$A(\sqrt{T_{j}}t_{k}^{1/4}):=A_{e}((1/2)\sqrt{T_{j}}t_{k}^{1/4},4\sqrt{T_{j}}t_{k}^{1/4})$

$\displaystyle\|\Phi_{k}^{*}h-H_{k}\|_{C^{2,\alpha}_{0}(A(T_{j}^{1/2}t_{k}^{1/4}))}\leqslant$
$\displaystyle\;C_{e}\Big{(}\epsilon^{-\beta}t_{k}^{\frac{\beta}{4}}\big{\|}P_{g_{e}}\Phi_{k}^{*}h\big{\|}_{r^{-2}_{e}C^{\alpha}_{\beta}(A_{k})}$
$\displaystyle+4\epsilon^{-1}t_{k}^{\frac{1}{4}}\|\Phi_{k}^{*}h-H_{k}\|_{C^{2,\alpha}_{\beta}(A_{k})}\Big{)}$
$\displaystyle\leqslant$