#  Noncollapsed degeneration of Einstein $4$-manifolds I
Refined description of the formation of singularities

Tristan OZUCH  École Normale Supérieure, Université PSL

###### Abstract

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to
compactify the set of normalized Einstein metrics with a lower bound on the
volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one
has to add singular spaces called Einstein orbifolds, and the singularities
form as blow-downs of Ricci-flat ALE spaces.

This raises some natural issues, in particular: can all Einstein orbifolds be
Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more
precisely the smooth Einstein metrics close to a given singular one?

In this first paper, we prove that Einstein manifolds sufficiently close, in
the Gromov-Hausdorff sense, to an orbifold are actually close to a gluing of
model spaces in suitable weighted Hölder spaces. The proof consists in
controlling the metric in the neck regions thanks to the construction of
optimal coordinates.

This refined convergence is the cornerstone of our subsequent work on the
degeneration of Einstein metrics or, equivalently, on the desingularization of
Einstein orbifolds in which we show that all Einstein metrics Gromov-Hausdorff
close to an Einstein orbifold are the result of a gluing-perturbation
procedure. This procedure turns out to be generally obstructed, and this
provides the first obstructions to a Gromov-Hausdorff desingularization of
Einstein orbifolds.

###### Contents

  1. 1 Trees of singularities and naïve desingularizations
    1. 1.1 Einstein orbifolds and Ricci-flat ALE orbifolds
    2. 1.2 Naïve desingularizations of an orbifold by trees of singularities
    3. 1.3 Coordinates on a naïve desingularization
    4. 1.4 Coordinates on an Einstein metric
  2. 2 First coordinates
    1. 2.1 $\epsilon$-regularity in the neck regions
    2. 2.2 Construction of the first coordinates
    3. 2.3 Consequences of the existence of the first coordinates.
  3. 3 Controls on constant mean curvature hypersurfaces in low curvature regions
  4. 4 Perturbation of hypersurfaces with almost constant second fundamental form
  5. 5 Construction of a foliation of the neck regions by constant mean curvature hypersurfaces
    1. 5.1 Local foliation by constant mean curvature hypersurfaces
    2. 5.2 Global foliation of the neck regions.
  6. 6 Construction of coordinates and control on the metric
    1. 6.1 Coordinates based on our constant mean curvature foliation
    2. 6.2 Expression of the metric in these coordinates
    3. 6.3 Control of the metric in these coordinates
    4. 6.4 Proximity between the Einstein metric and the naïve desingularization in weighted $C^{k}$-norm
  7. A Optimal $C^{k,\alpha}$-proximity to a round sphere

## Introduction

A fundamental question in geometry and topology is the following: given a
_topology_ (a differentiable manifold, $M$), is there an optimal _geometry_ (a
Riemannian metric, $g$) with this topology?

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

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

where $\operatorname{\textup{Ric}}$ is the Ricci curvature. These metrics are
considered optimal for the homogeneity of their Ricci curvature and as
critical points of the Einstein-Hilbert functional with fixed volume:
$g\mapsto\int_{M}\operatorname{\textup{R}}_{g}dvol_{g}$, where
$\operatorname{\textup{R}}$ is the scalar curvature, which is the spatial and
directional mean value of the sectional curvatures.

In dimension $2$ and $3$ these metrics have constant sectional curvatures and
are therefore well understood as they only have $3$ different local behavior
depending on the sign of the curvature (spherical, flat and hyperbolic). Their
understanding was crucial for $2$-dimensional geometry and topology thanks to
the uniformization Theorem and in $3$-dimensional geometry topology thanks to
Thurston’s geometrization.

In dimension $4$, Einstein metrics are moreover optimal as minimizers of the
$L^{2}$-norm of Riemann curvature tensor:
$g\mapsto\int_{M}|\operatorname{\textup{Rm}}_{g}|^{2}dvol_{g}$. Indeed, By
Chern-Gauss-Bonnet formula, we have

$\int_{M}|\operatorname{\textup{Rm}}_{g}|^{2}dv_{g}=8\pi^{2}\chi(M)+\int_{M}|\operatorname{\textup{Ric}}^{0}_{g}|^{2}dv_{g},$

where $\chi$ is the Euler characteristic and where
$\int_{M}|\operatorname{\textup{Ric}}^{0}_{g}|^{2}dv_{g}\geqslant 0$ vanishes
if and only if $g$ is Einstein. Notice that in this case, the quantity
$\int_{M}|\operatorname{\textup{Rm}}_{g}|^{2}dv_{g}=8\pi^{2}\chi(M)$ is purely
topological.

But from dimension $4$, the Einstein condition also becomes flexible as it
does not imply that the sectional curvatures are constant anymore. It is
actually so flexible that Einstein metrics can develop singularities. One
major goal for $4$-dimensional geometry is therefore to understand the set of
Einstein metrics and how they can degenerate, that is to compactify the set of
Einstein metrics. In this paper, we will be interested in Einstein
$4$-manifolds which are noncollapsed, which means that they have their volume
bounded from below. The basic tool for this kind of compactification question
is Gromov’s compactness theorem [Gro81] from which the usual goals are to
obtain informations about the possible limit spaces, and to understand if the
convergence happens in a stronger sense than for the Gromov-Hausdorff
distance.

The description of the limit spaces was given by Anderson and Bando-Kasue-
Nakajima.

######  Theorem 0.1 ([And89, BKN89]).

Let $(M^{4}_{i},g_{i})_{i\in\mathbb{N}}$ be a sequence of Einstein
$4$-manifolds satisfying,

  1. 1.

the diameters of the $(M^{4}_{i},g_{i})$ are uniformly bounded,

  2. 2.

the volumes of the $(M^{4}_{i},g_{i})$ are uniformly bounded from below,

  3. 3.

the integrals
$\mathcal{E}_{i}:=\int_{M^{4}_{i}}|\textup{Rm}(g_{i})|_{g_{i}}^{2}dv_{g_{i}}$
are uniformly bounded.

Then, there exists a subsequence $(M^{4}_{i},g_{i})$ converging to an Einstein
orbifold with isolated singularities (see Definition 1.1),
$(M^{4}_{\infty},g_{\infty})$.

Moreover, for any singular point $p_{\infty}\in M_{\infty}$, there exists a
sequence $(p_{i})_{i\in\mathbb{N}}$ of points of $M_{i}^{4}$ with
$\frac{1}{t_{i}}:=|\textup{Rm}(g_{i})|(p_{i})\to\infty$ such that
$\left(M^{4}_{i},\frac{g_{i}}{t_{i}},p_{i}\right)$ converges in the pointed
Gromov-Hausdorff sense to a Ricci-flat asymptotically locally Euclidean (ALE)
manifold, see Definition 1.2.

######  Remark 0.2.

The first hypothesis prevents the formation of cusps, and the second prevents
our space to collapse to a lower dimensional space. The third hypothesis,
which is actually a consequence of the two others by [CN15, Theorem 1.13], is
a topological condition by Chern-Gauss-Bonnet formula.

This theorem leaves several questions open, in particular the following one.

######  Question 0.3 ([And10, 7.I.]).

"It has long been an open question whether Einstein orbifold metrics can be
resolved to smooth Einstein metrics in the Gromov-Hausdorff topology. The idea
here would be to reverse the process of formation of orbifold singularities
described in [Theorem 0.1]"

For example, a natural question is whether or not _any_ Einstein orbifold can
be desingularized by Einstein metrics, say with an expected topology.

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

A natural technique to desingularize Einstein orbifolds is a gluing-
perturbation procedure: we glue Ricci-flat ALE manifolds to the singularities
of the orbifold to obtain an approximate Einstein metric, and then try to
perturbate it into an actual Einstein metric. The existence of such
desingularizations in the _real_ Einstein context, and therefore a partial
answer to Anderson’s question was proven by Biquard for nondegenerate (i.e.
which does not have any $L^{2}$-infinitesimal Einstein deformation)
asymptotically hyperbolic Einstein manifolds with one singularity
$\mathbb{R}^{4}/\mathbb{Z}_{2}$. This is realized by a particular gluing
procedure of an Eguchi-Hanson metric to the singularity, see [Biq13, Theorem
0.1].

Strikingly, this particular desingularization is only possible if an
_obstruction_ is satisfied: denoting $\mathbf{R}$ the Riemannian curvature
seen as an endomorphism on $2$-forms, the orbifold must satisfy
$\det\mathbf{R}=0$ at its singular point. This obstruction is restrictive, and
for example not satisfied by hyperbolic orbifolds (which are rigid) which
therefore cannot be desingularized by this particular procedure.

### Description of the degeneration in suitable weighted Hölder spaces

The main goal of this series of paper is to prove that the obstruction of
[Biq13] holds under much less assumptions. The least satisfying one is that
the metrics of the sequence have to come from a particular gluing-perturbation
procedure. It even turns out that a convergence in the function spaces of
[Biq13] cannot be true in general if one drops any of the following
assumption:

  1. 1.

the orbifold is rigid (it does not admit infinitesimal Einstein deformations),

  2. 2.

there is only one singular point,

  3. 3.

the singularity is $\mathbb{R}^{4}/\mathbb{Z}_{2}$.

The reasons are the following:

  1. 1.

the convergence speed towards the orbifold could be arbitrarily slow compared
to the singularity formation,

  2. 2.

different singularities may form at different speed,

  3. 3.

there might be trees of singularities forming.

In this paper, we will prove the cornerstone result of our program which
states that if an Einstein metric is sufficiently close to an Einstein
orbifold in the Gromov-Hausdorff sense, then it is actually close to a glued
metric in some weighted Hölder norms bounded on $2$-tensors decaying in the
neck regions. These spaces are necessarily different from [Biq13] by the above
reasons. The main theorem is the following.

######  Theorem 0.4.

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 the result of a gluing-perturbation procedure
detailed in [Ozu19b].

Notice that we are exactly in the context of Theorem 0.1, and that no
assumption is made about the possible singularity models, the number of
singular points or the orbifold metric. The goal of the present paper is the
construction of suitable coordinates in which we will compare our metrics and
prove a decay in the neck regions.

Theorem 0.1 actually implies that there is a satisfying
$C^{\infty}$-convergence on compact regions of the orbifold and of the Ricci-
flat ALE spaces (without their singular points) appearing. Our main concern
here will be to analyze the neck regions linking them, and to prove that there
is convergence in a weighted $C^{\infty}$-sense. The main idea is to foliate
these neck regions by constant mean curvature hypersurfaces thanks to which we
will construct coordinates in which we will finally control our Einstein
metrics.

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

In the compact case, we do not expect to desingularize general Einstein
orbifolds by smooth Einstein metrics as there are infinitely many global
(expected) obstructions, but we can identify obstructions to such a
desingularization. In particular, we will identify the only local obstruction,
$\det\mathbf{R}=0$ at the singular points, to desingularizing orbifolds for a
large class of manifolds.

There is a well-known family of Ricci-flat ALE spaces called _gravitational
instantons_ which have been classified by [Kro89] and the Kähler Ricci-flat
ALE spaces which are quotients of gravitational instantons have been
classified in [Suv11]. It is a famous conjecture that all Ricci-flat ALE
spaces are Kähler. One of our main results is an obstruction to the
desingularization by this conjecturally exhaustive list of candidates.

######  Theorem 0.5 ([Ozu19b]).

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 every singularity blow-up is a Kähler Ricci-flat ALE orbifold.

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

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

The condition on the singularity model can be reformulated as a topological
assumption by [Nak90] and we have the following example.

######  Example 0.6 ([Ozu19b]).

Consider $\mathbb{S}^{4}/\mathbb{Z}_{2}$ the Einstein orbifold with two
$\mathbb{R}^{4}/\mathbb{Z}_{2}$ singularities obtained as the quotient of
$\mathbb{S}^{4}$ by $\\{\pm\\}$ in a global geodesic chart. Then, there exists
a differentiable manifold $M^{4}$, such that for any $1\leqslant p<\infty$, we
can construct a sequence of metrics $(M,g_{i})$ with both

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

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 with

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

and

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

Let us finally mention one more application in the context of spin manifolds.

######  Theorem 0.7 ([Ozu19b]).

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

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

### Outline of the paper

In Section 1, we start by giving the principal needed definitions. We then
precise the convergence in Theorem 0.1 thanks to the introduction of glued
metrics which we call _naïve desingularizations_. The goal of this article is
to prove that they approximate Einstein metrics which are Gromov-Hausdorff
close to an orbifold in the sense of some weighted Hölder norms. We finally
identify the neck regions in Einstein manifolds which are Gromov-Hausdorff
close to an orbifold. The rest of the paper will be focused on these regions,
and in Section 2, we construct some first coordinates in these neck regions by
$\epsilon$-regularity.

In Section 3, we motivate the use of constant mean curvature hypersurfaces and
prove that they are well controlled by the ambient geometry.

We develop a perturbation technique in Section 4 which we use in Section 5 to
foliate the neck regions by constant mean curvature hypersurfaces by
perturbation of the spheres of the first coordinates of Section 2.

In Section 6, we finally construct coordinates based on the foliation of
Section 5 and use them to control the metric at the scale of the curvature
which decays by the results of [Ban90].

### Acknowledgements

I would like to thank my PhD advisor, Olivier Biquard, for introducing me to
the questions motivating this series of articles and for his constant support
and encouragement.

##  1 Trees of singularities and naïve desingularizations

Let us start by giving some definitions and explain why we can reduce our
situation to the study of neck regions.

###  1.1 Einstein orbifolds and Ricci-flat ALE orbifolds

Theorem 0.1 states that the limit singular spaces one can get are Einstein
orbifolds with isolated singularities.

######  Definition 1.1 (Orbifold (with isolated singularities)).

We will say that a metric space $(M_{o},g_{o})$ is an orbifold of dimension
$n\in\mathbb{N}$ if there exists $\epsilon_{0}>0$ and a finite number of
points $(p_{k})_{k}$ of $M_{o}$ called _singular_ such that we have the
following properties:

  1. 1.

the space $(M_{o}\backslash\\{p_{k}\\}_{k},g_{o})$ is a manifold of dimension
$n$,

  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
$\mathbb{S}^{n-1}$, $\Gamma_{k}\subset SO(n)$, and a diffeomorphism
$\Phi_{k}:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{n}/\Gamma_{k}\to
U_{k}\subset M_{o}$ for which, $\Phi_{k}^{*}g_{o}$ is smooth on
$\mathbb{R}^{n}$ and for any $l\in\mathbb{N}$, there exists $C_{l}>0$ such
that

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

Theorem 0.1 only describes the formation of singularities at the scale of the
maximum of the curvature where a Ricci-flat ALE _manifold_ appears as a blow
up, but there might actually be other singularities forming at different
scales. They are modeled on Ricci-flat ALE _orbifolds_. The relevant model
spaces for us are therefore Einstein orbifolds and Ricci-flat ALE orbifolds
with isolated singularities.

######  Definition 1.2 (ALE orbifold (with isolated singularities)).

An ALE orbifold of dimension $n\in\mathbb{N}$, $(N,g_{b})$ 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 Riemannian manifold of dimension
$n$,

  2. 2.

for each singular point $p_{k}$ of $N$, there exists $U_{k}$, a neighborhood
of $p_{k}$ in $N$, $\Gamma_{k}\subset SO(n)$, a finite subgroup acting freely
on the sphere $\mathbb{S}^{n-1}$, and a diffeomorphism
$\Psi_{k}:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{n}/\Gamma_{k}\to
U_{k}\subset M_{o}$ for which $\Psi_{k}^{*}g_{b}$ is smooth and 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
$\Psi_{\infty}:(\mathbb{R}^{n}/\Gamma_{\infty})\backslash
B_{e}(0,\epsilon_{0}^{-1})\to N\backslash K$ such that we have

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

###  1.2 Naïve desingularizations of an orbifold by trees of singularities

By [Ban90], under the assumptions of Theorem 0.1, at each singular point, a
tree of Ricci-flat orbifolds form.

######  Definition 1.3 (Tree of singularities and desingularization pattern).

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 also assume that there is a one to one
"gluing" 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
_tree of singularities_ or a _desingularization pattern_ for $(M_{o},g_{o})$.

Let us define metrics which will mimic the above degeneration of Einstein
manifolds, they will be called _naïve desingularizations_ and the main goal of
this paper is to show that degenerations of Einstein manifolds are well
approximated by these naïve desingularizations.

Let us start by defining the naïve gluing of an ALE space $(N,g_{b})$ to a
singularity of an Einstein orbifold $(M_{o},g_{o})$. Recall the constant
$\epsilon_{0}>0$ of Definitions 1.1 and 1.2.

Let $0<2\epsilon<\epsilon_{0}$, $t>0$, and $(M_{o},g_{o})$ be an orbifold and
$\Phi:B_{e}(0,\epsilon_{0})\subset\mathbb{R}^{4}/\Gamma\to U$ a local chart
around a singular point $p\in M_{o}$ satisfying
$\Phi^{*}g_{o}=g_{e}+\mathcal{O}(r_{e}^{2})$. 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
$\Psi_{\infty}^{*}g_{b}=g_{e}+\mathcal{O}(r_{e}^{-4})$. Consider finally
$\chi$ a $C^{\infty}$-function from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$
supported on $[0,2]$ and equal to $1$ on $[0,1]$ and for all $s>0$, define
$\phi_{s}:x\in\mathbb{R}^{4}/\Gamma\to sx\in\mathbb{R}^{4}/\Gamma$.

######  Definition 1.4 (Naïve gluing of an ALE space to an orbifold
singularity).

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

$g_{o}\\#_{p,t}g_{b}=\chi(t^{-1/4}r_{e})\Phi^{*}g_{o}+\big{(}1-\chi(t^{-1/4}r_{e})\big{)}\phi_{t^{1/2},*}\Psi_{\infty}^{*}g_{b}$

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

More generally, it is possible to desingularize iteratively by trees of Ricci-
flat ALE orbifolds.

Let $0<2\epsilon<\epsilon_{0}$, $(M_{o},g_{o})$ be an Einstein orbifold and
$(N_{j},g_{b_{j}})$ be a tree of ALE Ricci-flat such that
$M=M_{o}\\#_{j}N_{j}$ is the result of a desingularization pattern $D$, and
let $0<t_{j}<\epsilon^{4}$ be _relative gluing scales_.

######  Definition 1.5 (Naïve desingularization of an orbifold by a tree or
Ricci-flat orbifolds).

The _naïve desingularization_ metric $g^{D}_{t}$ is then the result of the
following iterative construction.

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
$(N_{k}\\#N_{j},g_{b_{k}}\\#_{p_{j},t_{j}}g_{b_{j}},S_{k}\backslash\\{p_{j}\\})$
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 $N_{j}(2\epsilon)$,
the metric is $T_{j}g_{b_{j}}$.

######  Remark 1.6.

Our construction depends on a gauge choice: the diffeomorphisms used to glue
the infinity of our ALE spaces to orbifold singularities can be composed with
any isometry of $\mathbb{R}^{4}/\Gamma_{k}$. There is therefore a gluing gauge
$\phi_{k}\in Isom(\mathbb{R}^{4}/\Gamma_{k})$ at each point. These choices are
equivalent to gluing different ALE spaces and they lead to different metrics.
We will see that in certain cases only some of these choices can be
perturbated to Einstein metrics.

###  1.3 Coordinates on a naïve desingularization

The above metric has some well identified regions in which the metric is that
of the orbifold, of one of the Ricci-flat ALE spaces or close to an annulus of
a flat cone

######  Definition 1.7 (Coordinates on the model spaces: $M_{o}(\epsilon)$,
$N_{j}(\epsilon)$ and $A_{k}(t,\epsilon)$).

With the notations of definitions 1.1 and 1.2, for
$0<\epsilon\leqslant\epsilon_{0}$, we will denote

  * •

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

  * •

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

  * •

$A_{j}(t,\epsilon):=A_{e}(\epsilon^{-1}\sqrt{T_{j}}\sqrt{t_{j}},\epsilon\sqrt{T_{j}})\subset\mathbb{R}^{4}/\Gamma_{k}$.

We see that these sets naturally embed in our manifold $M$ thanks to the above
naïve desingularization construction.

######  Remark 1.8.

Those are the compact sets of the orbifolds minus their singular points which
appear in [Ban90, Theorem A]. They are exhaustive as $\epsilon$ tends to zero.

They moreover form a covering of $M=M_{o}\\#_{j}N_{j}$ adapted to a
degeneration at relative scales $(t_{j})_{j}$:

$M=\bigcup_{j}\big{(}N_{j}(\epsilon)\cup A_{j}(t,\epsilon)\big{)}\cup
M_{o}(\epsilon).$ (1)

The goal will be to identify each of these regions with a region of an
Einstein manifold close to an Einstein orbifold in the Gromov-Hausdorff sense.

###  1.4 Coordinates on an Einstein metric

Let $(M,g)$ be an Einstein metric sufficiently close to an Einstein orbifold
$(M_{o},g_{o})$.

######  Definition 1.9 (Manifold $\epsilon$-approximated by a tree of
singularities).

For $\epsilon>0$, and $l\in\mathbb{N}$, we will say that a Riemannian manifold
$(M,g)$ is $\epsilon$-approximated in $C^{l}$-norm by a naïve
desingularization $(M,g^{D}_{t})$ of $(M_{o},g_{o})$ if there exists a
diffeomorphism $\Phi:M\to M$ such that if we denote
$\Phi(M_{o}(\epsilon))=\mathcal{M}_{o}(\epsilon)\subset M$ and
$\Phi(N_{j}(\epsilon))=\mathcal{N}_{j}(\epsilon)\subset M$ and
$\mathcal{A}_{k}(t,\epsilon)$ the region of
$M\backslash\mathcal{M}_{o}(32\epsilon)\cup\bigcup_{j}\mathcal{N}_{j}(32\epsilon)$
with nonempty intersection with
$\Psi_{\infty}\big{(}B_{e}(0,\epsilon^{-1})\big{)}\Big{)}\subset\mathcal{N}_{k}(\epsilon)$
we have:

  1. 1.

on $\mathcal{M}_{o}(\epsilon)$ in $M$, we have

$\left\|\Phi^{*}g-g_{o}\right\|_{C^{l}(g_{o})}=\left\|\Phi^{*}g-g^{D}\right\|_{C^{l}(g^{D})}\leqslant\epsilon,$

  2. 2.

on the zone $\mathcal{N}_{j}(\epsilon)$ in $M$, we have

$\left\|\frac{\Phi^{*}g}{T_{j}}-g_{b_{j}}\right\|_{C^{l}(g_{b_{j}})}=\left\|\Phi^{*}g-g^{D}\right\|_{C^{l}(g^{D})}\leqslant\epsilon,$

  3. 3.

and on $\mathcal{A}_{k}(t,\epsilon)$, we have

$\int_{\mathcal{A}_{k}(t,\epsilon)}|\operatorname{\textup{Rm}}|^{2}dv_{g}<\epsilon^{2}.$

######  Proposition 1.10.

Under the assumptions of Theorem 0.1, given a sequence $(M_{i},g_{i})$ of
Einstein manifolds converging in the Gromov-Hausdorff sense to an Einstein
orbifold $(M_{o},g_{o})$, there exist naïve desingularizations
$(M_{i},g^{D}_{t_{i}})$ and a subsequence such that for all $\epsilon>0$ and
$l\in\mathbb{N}$, for $i$ large enough, the manifold $(M_{i},g_{i})$ is
$\epsilon$-approximated in $C^{l}$-norm by $(M_{i},g^{D}_{t_{i}})$.

###### Proof.

Let $(M_{i},g_{i})\to(M_{o},g_{o})$ be a sequence of manifolds satisfying the
assumptions of Theorem 0.1. Up to taking a subsequence, for any singular point
of $M_{o}$, there exists a sequence of scales $T_{i}>0$ and of points $p_{i}$
of $M_{i}^{4}$ of high curvature
$|\textup{Rm}(g_{i})|(p_{i})\rightarrow\infty$ such that:
$\left(M^{4}_{i},\frac{g_{i}}{T_{i}},p_{i}\right)$ converging smoothly on
compact subsets to $(N,g_{b},p)$ a Ricci-flat ALE orbifold without its
singular points. The convergence is moreover smooth at a bounded from below
distance from the singular points since we are considering Einstein metrics
with bounded curvature and harmonic radii bounded from below.

Let $\epsilon>0$ and $l\in\mathbb{N}$, by considering the compact
$N_{j}(\epsilon)$, there exists $i_{j}\in\mathbb{N}$ such that for all
$i>i_{0}$, there is a diffeomorphism on its image $\Phi_{i}:N(\epsilon)\to
M_{i}$ with
$\Big{\|}\frac{\Phi_{i}^{*}g_{i}}{T_{i}}-g_{b_{j}}\Big{\|}_{C^{l}}\leq\epsilon$
and similarly, there exists a rank $i_{o}$ from which we have the
$C^{l}$-closeness on $M_{o}(\epsilon)$. Since there are only finitely many
scales in each tree of singularities, we can take the maximum of the ranks
$i_{j}$ and $i_{o}$ to obtain the result.

The control of the $L^{2}$-norm of the Riemannian curvature is proven in
[Ban90]. ∎

The main objective of this paper will be to construct good coordinates in the
neck regions $\mathcal{A}_{k}(t,\epsilon)$. These regions
$\mathcal{A}_{k}(t,\epsilon)$ between $N_{k}$ and $N_{j}$ are included in
metric annuli
$A\Big{(}\frac{1}{8}T_{k}^{\frac{1}{2}}\epsilon^{-1},8T_{j}^{\frac{1}{2}}\epsilon\Big{)}$
centered at $p_{k}$, and in
$A\Big{(}\frac{1}{8}t_{k}^{\frac{1}{2}}\epsilon^{-1},8\epsilon\Big{)}$ if
$N_{k}$ is directly glued to $M_{o}$ (in this case, $t_{k}=T_{k}$ because it
is the shallowest scale). Let us note that the volume growth is almost
Euclidean on them.

######  Lemma 1.11.

For all $0<\delta<1$, there exists $\epsilon>0$ such that if a manifold
$(M,g)$ is $\epsilon$-approximated by a tree of singularities $g^{D}_{t}$ in
$C^{0}$-norm for $t<\epsilon^{4}$ with
$\operatorname{\textup{Ric}}(g)\geqslant-3g$, then, if
$\mathcal{A}_{k}(t,\epsilon)\subset
A\Big{(}\frac{1}{8}T_{k}^{\frac{1}{2}}\epsilon^{-1},8T_{j}^{\frac{1}{2}}\epsilon\Big{)}=A(\rho_{1},\rho_{2})$,
we have

$\Big{|}\frac{\operatorname{\textup{Vol}}_{g}(B_{g}(\delta^{-1}\rho_{2}))}{\operatorname{\textup{Vol}}_{g}(B_{g}(\delta\rho_{1}))}-\frac{(\delta^{-1}\rho_{2})^{4}}{(\delta\rho_{1})^{4}}\Big{|}\leqslant\delta.$

###### Proof.

Let us consider the case of an annular region $\mathcal{A}_{k}(t,\epsilon)$
between a Ricci-flat ALE $(N,g_{b})$ and an orbifold $(M_{o},g_{o})$, which
are asymptotic to $\mathbb{R}^{4}/\Gamma$, at scale $0<t<\epsilon^{4}$ for
$\epsilon>0$ which we will choose small enough. By definition of an
$\epsilon$-approximated metric, the ball $(B(\delta^{-1}\rho_{2}),g)$ is
arbitrarily close in the Gromov-Hausdorff sense to the ball of radius
$\delta^{-1}\epsilon$ of $(M_{o},g_{o})$ for $\epsilon$ arbitrarily small, and
by definition of an orbifold metric, as $\delta^{-1}\epsilon\to 0$, we have

$\frac{\operatorname{\textup{Vol}}_{g_{o}}(B_{g_{o}}(\delta^{-1}\epsilon))}{(\delta^{-1}\epsilon)^{4}}\to\frac{\omega_{4}}{|\Gamma|},$

the volume of the unit ball of $\mathbb{R}^{4}/\Gamma$.

Similarly, the ball $(B_{g/t}(\delta\epsilon^{-1}),g/t)$ is arbitrarily close
in the Gromov-Hausdorff sense to the ball of radius $\delta\epsilon^{-1}$ of
$(N,g_{b})$ whose volume satisfies for $\delta^{-1}\epsilon\to 0$

$\frac{\operatorname{\textup{Vol}}_{g_{b}}(B_{g_{b}}(\delta\epsilon^{-1}))}{(\delta\epsilon^{-1})^{4}}\to\frac{\omega_{4}}{|\Gamma|}.$

As a consequence, by the continuity of volume for the Gromov-Hausdorff
distance between manifolds with $\operatorname{\textup{Ric}}\geqslant-3g$,
[Col97], we have the stated result for $\epsilon$ small enough. ∎

The goal is now to construct a diffeomorphism from a flat cone of
$\big{(}\mathbb{R}^{n}/\Gamma,g_{e}\big{)}$ to our intermediate annulus.
Thanks to the controls on the curvature of [Ban90, Proposition 3], we expect
that a "natural" coordinate system must also have good enough controls. Here
we choose to foliate our annuli by constant mean curvature hypersurfaces
before constructing coordinates relying on them.

##  2 First coordinates

The main difficulty here in constructing our coordinates when compared to the
context of [BKN89] is that we do not have an asymptotic behavior of our metric
on which we can base our construction. We will need to start by using a first
set of coordinates which is only partially satisfying. The main goal of this
section is the proof of the following proposition.

######  Proposition 2.1.

For all $\delta>0$, $D_{0}>0$, $v_{0}>0$, $E>0$ and $l\in\mathbb{N}$, there
exists $\epsilon_{2}>0$ such that if an Einstein manifold satisfies:

  * •

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

  * •

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

  * •

its Ricci curvature is bounded below by $-3$,

  * •

there exists an annulus $A(\rho_{1},\rho_{2})$ such that $\rho_{1}<4\rho_{2}$
satisfies:

$\int_{A(\rho_{1},\rho_{2})}|\operatorname{\textup{Rm}}|^{2}\leqslant\epsilon_{2}^{2},$

and such that $\int_{B(\rho_{2})}|\operatorname{\textup{Rm}}|^{2}\leqslant E$,

  * •

the volume growth in the annulus is almost Euclidean

$\Big{|}\frac{\operatorname{\textup{Vol}}(B(\rho_{2}))}{\operatorname{\textup{Vol}}(B(\epsilon_{2}\rho_{1}))}-\frac{\rho_{2}^{4}}{(\epsilon_{2}\rho_{1})^{4}}\Big{|}\leqslant\epsilon_{2}.$

Then, there exists a subset
$\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}$ close to the annulus
$A\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}$ in the following sense,

$A\Big{(}(2+\delta)\rho_{1},\Big{(}\frac{1}{2}-\delta\Big{)}\rho_{2}\Big{)}\subset\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}\subset
A\Big{(}(2-\delta)\rho_{1},\Big{(}\frac{1}{2}+\delta\Big{)}\rho_{2}\Big{)},$

such that there exists $\Gamma$ a finite subgroup of $SO(4)$ acting freely on
$\mathbb{S}^{3}$ and a diffeomorphism
$\Phi:A_{e}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}\to\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)},$
satisfying for all $\rho\in\Big{[}2\rho_{1},\frac{1}{4}\rho_{2}\Big{]}$,
denoting
$\Phi_{\rho}:=\Phi\circ\phi_{\rho}:A_{e}(1,2)\to\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)},$
where $\phi_{\rho}$ is the homothetic transformation of ratio $\rho$ on
$\mathbb{R}^{4}/\Gamma$:

$\Big{\|}\frac{\Phi_{\rho}^{*}g}{\rho^{2}}-g_{e}\Big{\|}_{C^{l}(A_{e}(1,2))}\leqslant\delta.$

###  2.1 $\epsilon$-regularity in the neck regions

######  Definition 2.2 (Hölder spaces).

The $C^{k,\alpha}$-norms will be taken at the scale of the injectivity radius.
Which means that we define the $C^{k,\alpha}$-norm of a function $f$ in the
following way:

$\|f\|_{C^{k,\alpha}}:=\|f\|_{C^{0}}+...+\|\nabla_{g}^{k}f\|_{C^{0}}+[\nabla_{g}^{k}f]_{\alpha},$

where

$[h]_{\alpha}=\sup_{\\{(x,y)|d(x,y)<\textup{inj}\\}}\Big{|}\frac{h(x)-h(y)}{d(x,y)^{\alpha}}\Big{|}_{g}.$

We will also sometimes use the _harmonic radius_ which is defined at $x\in M$,
as the supremum of the $r>0$ for which there exists a diffeomorphism
$\Phi:B(x,r)\to\mathbb{R}^{n}$, $\Phi(y)=(h_{1}(y),...,h_{n}(y))$ such that
for all $i\in\\{1,...,n\\}$,

$\Delta_{g}h_{i}=0,$

and denoting $g_{ij}$ the push forward of $g$ by $\Phi$,

$\|g_{ij}-\delta_{ij}\|_{C^{0}}+r^{1+\alpha}\|g_{ij}\|_{C^{1+\alpha}}\leqslant\frac{1}{100}.$

######  Lemma 2.3.

Under the assumptions of Proposition 2.1, there exists a constant $r_{0}>0$
only depending on $D_{0}$ and $v_{0}$, such that for all points of
$A\big{(}2\rho_{1},\frac{1}{2}\rho_{2}\big{)}$ at distance
$2\rho_{1}<\rho<\frac{1}{2}\rho_{2}$ from the center, the injectivity radius
and the harmonic radius are bounded below by $r_{0}\rho$.

###### Proof.

Let us consider a point $x$ at distance
$\rho\in[2\rho_{1},\frac{1}{2}\rho_{2}]$ from the center of the annulus and
the ball $B(x,\frac{\rho}{2})$. Thanks to Bishop-Gromov inequality, the volume
of $B(x,\frac{\rho}{2})$ is bounded from below by $C_{1}\rho^{4}$ for a
constant $C_{1}>0$ only depending on $v_{0}$ and $D_{0}$. To control the
curvature, we use the following $\epsilon$-regularity theorem of [And89,
BKN89]:

######  Lemma 2.4.

In an Einstein manifold $(M,g)$ satisfying the assumptions of Proposition 2.1,
there exists a constant $\epsilon=\epsilon(C_{0})$ such that if at a point
$x\in M$ and for $\rho>0$ we have
$\operatorname{\textup{Vol}}_{g}(B_{g}(x,\rho))>C_{0}\rho^{4}$ and

$\int_{B(x,2\rho)}|\operatorname{\textup{Rm}}(g)|^{2}dv(g)<\epsilon^{2},$ (2)

then, we have

$|\operatorname{\textup{Rm}}(g)|\leqslant\frac{1}{\rho^{2}}$ (3)

on $B(x,\rho)$.

Therefore for $\rho\in[4\rho_{1},\frac{1}{4}\rho_{2}]$, up to shrinking the
constant $\epsilon_{2}$ of the statement to be smaller than
$\epsilon(16C_{1})>0$, the ball $B(x,\frac{\rho}{2})$ satisfies (2). Hence,
the curvature at distance $\rho$ is smaller than $\frac{1}{\rho^{2}}$ by (3).

By [CGT82, Theorem 4.3], these controls on the curvature and the volume at
scale $\rho$ imply that there exists a constant $r_{1}>0$ depending on the
volume of the ball of radius $\rho$ and on a bound on the curvature at this
scale such that at any point at distance $\rho$ from the center of the
annulus, the injectivity radius is bounded below by $r_{1}\rho$. Since the
curvature is bounded, there also exists $0<r_{0}<r_{1}$ depending on $r_{1}$
and the bound on the curvature such that the harmonic radius is bounded below
by $r_{0}\rho$. ∎

###  2.2 Construction of the first coordinates

Let us start by constructing first coordinates which we will improve later.

The crucial tool to prove this result is the almost volume cone theorem of
Cheeger and Colding.

######  Lemma 2.5 ([CC96, Theorem 4.85]).

For all $\delta>0$, there exists $\kappa>0$ such that for any $r>0$, if
$(M^{n},g)$ satisfies, $\operatorname{\textup{Ric}}(g)\geqslant-(n-1)\kappa
r^{-2}$ and there exists a point $p\in M$ at which,

$\frac{\operatorname{\textup{Vol}}(B(p,\kappa r))}{v_{-\kappa r^{-2}}(\kappa
r)}-\frac{\operatorname{\textup{Vol}}(B(p,2r))}{v_{-\kappa
r^{-2}}(2r)}<\kappa,$

where $v_{-\kappa r^{-2}}(s)$ is the volume of the ball of radius $s$ in the
simply connected space with sectional curvatures constant equal to $-\kappa
r^{-2}$, then, there exists a metric cone $(C(X),d_{C(X)},0)$ such that
$diam(X)\leqslant\pi$ and,

$d_{GH}\Big{(}\big{(}B(p,r),p\big{)},\big{(}B_{C(X)}(0,r),0\big{)}\Big{)}\leq\delta
r.$

We can refine this result in the case of $4$-dimensional Einstein manifolds

######  Lemma 2.6 (Almost volume cone in a $4$-dimensional Einstein manifold).

For all $\delta>0$, $l\in\mathbb{N}$, $v_{0}>0$, and $E>0$, there exists
$\kappa>0$ such that for all $r>0$, if $(M^{4},g)$ satisfies
$\operatorname{\textup{Ric}}(g)=\Lambda g$ with, $|\Lambda|\leqslant 3\kappa
r^{-2}$ and if there exists a point $p\in M$ for which,

$\operatorname{\textup{Vol}}(B(p,r))\geqslant v_{0}r^{4},$
$\frac{\operatorname{\textup{Vol}}(B(p,\kappa r))}{v_{-\kappa r^{-2}}(\kappa
r)}-\frac{\operatorname{\textup{Vol}}(B(2r))}{v_{-\kappa r^{-2}}(2r)}<\kappa,$

and

$\int_{B(p,2r)}|\operatorname{\textup{Rm}}|^{2}dv\leqslant E,$

then, there exists a finite subgroup $\Gamma\subset SO(4)$ (with order bounded
in terms of $v_{0}$), and a diffeomorphism
$\Phi:A_{e}(\delta,(1-\delta))\to\hat{A}(\delta r,(1-\delta)r)$, where,

$A(2\delta r,(1-2\delta)r)\subset\hat{A}(\delta r,(1-\delta)r)\subset
A\Big{(}\frac{1}{2}\delta r,\Big{(}1-\frac{1}{2}\delta\Big{)}r\Big{)},$ (4)

such that on the annulus $A_{e}(\delta r,(1-\delta)r)$, we have:

$\Big{\|}\frac{\Phi^{*}g}{r^{2}}-g_{e}\Big{\|}_{C^{l}(g_{e})}\leqslant\delta.$
(5)

######  Remark 2.7.

The assumptions on the Riemannian curvature and the volume are there to bound
the order of $\Gamma$. They prevent our Einstein manifolds to be too close to
$\mathbb{R}^{4}/\Gamma_{i}$ where the order of $\Gamma_{i}$ goes to infinity.
In such a situation, the sequence could converge to the cone $\mathbb{R}^{+}$.

These hypotheses are moreover redundant by [CN15, Theorem 1.13], the
hypotheses on the Ricci curvature and the volume imply a bound on the
$L^{2}$-norm of the curvature.

###### Proof.

By rescaling we can restrict our attention to $r=1$ and assume towards a
contradiction that there exist $\delta>0$, $E>0$ and $v_{0}>0$, and a sequence
of counterexamples $(M^{4}_{i},g_{i},p_{i})$ satisfying
$\operatorname{\textup{Vol}}(B_{i}(p_{i},1))>v_{0}>0$,
$\int_{B_{i}(p_{i},2)}|\operatorname{\textup{Rm}}_{i}|^{2}dv_{i}\leqslant E$
and sequences of real numbers $\kappa_{i}\to 0$ and $\Lambda_{i}$, with
$|\Lambda_{i}|\leqslant 3\kappa_{i}$ for which
$\operatorname{\textup{Ric}}(g_{i})=\Lambda_{i}g_{i}$ and
$\frac{\operatorname{\textup{Vol}}(B_{i}(p_{i},\kappa_{i}))}{v_{-\kappa_{i}}(\kappa_{i})}-\frac{\operatorname{\textup{Vol}}(B_{i}(p_{i},2))}{v_{-\kappa_{i}}(2)}<\kappa_{i}$
but such that for any $\Gamma$ finite subgroup of $SO(4)$, there does not
exist a diffeomorphism $\Phi:A_{e}(\delta,(1-\delta))\to\hat{A}(\delta
r,(1-\delta)r)$ where $\hat{A}(\delta r,(1-\delta)r)$ satisfies (4), and such
that we have the control (5).

Since $\operatorname{\textup{Vol}}(B_{i}(p_{i},1))>v_{0}>0$ and
$\operatorname{\textup{Ric}}(g_{i})\geqslant-3$, a subsequence of the balls
$B_{i}(p_{i},1)$ converges to a metric space $(Y,d_{Y},p)$ in the Gromov-
Hausdorff sense. By Lemma 2.5, up to taking a subsequence, for all $i$, we can
assume that $\kappa_{i}$ is small enough to have

$d_{GH}\Big{(}\big{(}B_{i}(p,1),p_{i}\big{)},\big{(}B_{C(X_{i})}(0,1),0\big{)}\Big{)}\leqslant\frac{1}{i},$

for a particular metric cone $(C(X_{i}),d_{C(X_{i})},0)$ with
$diam(X_{i})\leqslant\pi$. By the triangular inequality, the sequence
$\big{(}B_{C(X_{i})}(0,1),0\big{)}$ also converges to $(Y,d_{Y},p)$ in the
Gromov-Hausdorff sense. Now, since the $C(X_{i})$ with the distances
$d_{C(X_{i})}\big{(}(r,x),(s,y)\big{)}:=\sqrt{r^{2}+s^{2}-2rs\cos(d_{X_{i}}(x,y))}$
converge, we deduce by fixing $r$ and $s$ that the $(X_{i},d_{X_{i}})$
converge to $(X,d_{X})$, and therefore that the limit is a metric cone,
$(Y,d_{Y},p)=(C(X),d_{C(X)},0)$.

By the compactness theorem for noncollapsed Einstein metrics with bounded
$L^{2}$-norm for the curvature [BKN89, And89], $(Y,d_{Y},p)$ is an Einstein
orbifold. The limit of $(B_{i}(p_{i},1),g_{i},p_{i})$ in the Gromov-Hausdorff
sense is therefore both the radius $1$ ball of a metric cone and of an
Einstein orbifold of dimension $4$. Since an Einstein orbifold has bounded
curvature, the cone has to be flat. The limit is therefore
$\mathbb{R}^{4}/\Gamma_{\infty}$ for $\Gamma_{\infty}$ a finite subgroup of
$SO(4)$.

Now, the convergence of Einstein metrics is smooth on compact subsets of the
complement of the singularities by [BKN89], and there is only one here at the
singular point of $\mathbb{R}^{4}/\Gamma_{\infty}$, hence, there exists a
diffeomorphism $\Phi:A_{e}(\delta,(1-\delta))\to\hat{A}(\delta r,(1-\delta)r)$
satisfying (4) and (5), which is a contradiction. ∎

We can now use these results to construct some coordinates in our annulus.

###### Proof of Proposition 2.1.

Let $\delta>0$, $D_{0}>0$, $v_{0}>0$, $E>0$, $\Lambda\in[-3,3]$ and
$l\in\mathbb{N}$, and a $4$-dimensional manifold, $(M,g)$ satisfying:

  * •

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

  * •

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

  * •

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

  * •

there exists $p\in M$, and $\rho_{2}>0$, such that we have
$\int_{B(p,\rho_{2})}|\operatorname{\textup{Rm}}|^{2}\leqslant E$.

The assumptions on the Ricci curvature, the volume and the diameter imply by
Bishop-Gromov inequality that there exists a constant
$C_{0}=C_{0}(D_{0},v_{0})>0$ such that for any ball of radius $\rho\leqslant
D_{0}$, we have
$\frac{1}{C_{0}}\rho^{4}\leqslant\operatorname{\textup{Vol}}(B(\rho))\leqslant
C_{0}\rho^{4}$.

Let $l\in\mathbb{N}$, and denote $0<\kappa_{0}<\frac{1}{4}$ the constant of
Lemma 2.6 associated to $\delta$, $l$, $E$ and $\frac{1}{C_{0}}$. Let us look
for $\epsilon_{2}>0$ small enough so that for any
$0<\rho_{1}\leq\epsilon_{2}\rho_{2}$ for which we have:

$\int_{A(\rho_{1},\rho_{2})}|\operatorname{\textup{Rm}}|^{2}\leqslant\epsilon_{2}^{2},$

and

$\Big{|}\frac{\operatorname{\textup{Vol}}(B(\rho_{2}))}{\operatorname{\textup{Vol}}(B(\epsilon_{2}\rho_{1}))}-\frac{\rho_{2}^{4}}{(\epsilon_{2}\rho_{1})^{4}}\Big{|}\leqslant\epsilon_{2},$

then, the conclusion of Proposition 2.1 holds with $\delta>0$. Since
$\operatorname{\textup{Ric}}=\Lambda g$ and since
$\frac{1}{C_{0}}\rho^{4}\leqslant\operatorname{\textup{Vol}}(B(\rho))\leqslant
C_{0}\rho^{4}$, we have

$\displaystyle\epsilon_{2}^{2}$
$\displaystyle\;\geqslant\int_{A(\rho_{1},\rho_{2})}|\operatorname{\textup{Rm}}|^{2}dv$
$\displaystyle\;\geqslant
6\int_{A(\rho_{1},\rho_{2})}|\operatorname{\textup{Ric}}|^{2}dv$
$\displaystyle\;\geqslant
24\Lambda^{2}\operatorname{\textup{Vol}}(A(\rho_{1},\rho_{2}))$
$\displaystyle\;\geqslant
24\Lambda^{2}\Big{(}\frac{1}{C_{0}}\rho_{2}^{4}-C_{0}\rho^{4}_{1}\Big{)}$
$\displaystyle\;\geqslant
24\Lambda^{2}\rho_{2}^{4}\Big{(}\frac{1}{C_{0}}-C_{0}\epsilon_{2}^{4}\Big{)},$

hence, if
$\Big{(}\frac{1}{C_{0}}-C_{0}\epsilon_{2}^{4}\Big{)}>\frac{1}{24C_{0}}$, that
is, $\epsilon_{2}^{4}<\frac{23}{24C_{0}^{2}}$, then we have

$|\Lambda|\leqslant C_{0}^{\frac{1}{2}}\epsilon_{2}\rho_{2}^{-2}.$

Therefore, for any $\rho\in[\rho_{1}\frac{\rho_{2}}{2}]$, if
$\epsilon_{2}^{4}<\frac{23}{24C_{0}^{2}}$, the ball of radius $\rho$ satisfies
the hypotheses of Lemma 2.6 with $\kappa=C_{0}^{\frac{1}{2}}\epsilon_{2}$. We
can finally choose $\epsilon_{2}>0$ even smaller so that
$C_{0}^{\frac{1}{2}}\epsilon_{2}<\kappa_{0}$ to conclude. ∎

###  2.3 Consequences of the existence of the first coordinates.

The diffeomorphism of Proposition 2.1 allows us to pull back our Einstein
metric on a flat annulus
$A_{e}(\rho_{1},\rho_{2})\subset\mathbb{R}^{4}/\Gamma$, where the Einstein
metric $g$ is $C^{\infty}$-close to the flat metric $g_{e}$. This will let us
construct hypersurfaces of almost constant second fundamental form for our
Einstein metric.

######  Note 2.8.

Here and in the following, we will use the convention that our normal vectors
point outwards, and we define the second fundamental form as

$A^{\Sigma}(X,Y):=\langle X,\nabla_{Y}N^{\Sigma}\rangle,$

which ensures that spheres in the Euclidean space have positive second
fundamental form.

######  Lemma 2.9.

For any $l\in\mathbb{N}$, there exists $C>0$ and $\delta_{0}>0$ such that if a
metric $g$ on $A_{e}\big{(}\frac{1}{4},4\big{)}$ satisfies,

$\|g-g_{e}\|_{C^{l+1}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}\leqslant\delta_{0},$

we have, for all $s\in\big{[}\frac{1}{4},4\big{]}$, the sphere of radius $s$
of $\mathbb{R}^{n}/\Gamma$, $\mathcal{S}_{s}$, has a second fundamental form
for $g$, $A_{g}^{\mathcal{S}_{s}}$, which satisfies

$\Big{\|}\frac{A_{g}^{\mathcal{S}_{s}}}{s}-\frac{g^{\mathcal{S}_{s}}}{s^{2}}\Big{\|}_{C^{l}\big{(}\frac{g^{\mathcal{S}_{s}}}{s^{2}}\big{)}}\leqslant
C\|g-g_{e}\|_{C^{l+1}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}.$

We moreover have

$\Big{\|}\frac{g^{\mathcal{S}_{s}}}{s^{2}}-g^{\mathbb{S}^{n-1}/\Gamma}\Big{\|}_{C^{l+1}(g^{\mathbb{S}^{n-1}/\Gamma})}\leqslant
C\|g-g_{e}\|_{C^{l+1}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}.$

###### Proof.

The second fundamental form involves one derivative of the metric which is
controlled in $C^{l+1}$-norm by assumption. Since for $g_{e}$, the
hypersurfaces $\mathcal{S}_{s}$ have a second fundamental form constant equal
to $\frac{g^{\mathbb{S}^{n-1}/\Gamma}}{s^{2}}$, we can conclude that we indeed
control their second fundamental form

$\Big{\|}\frac{A_{g}^{\mathcal{S}_{s}}}{s}-\frac{g^{\mathcal{S}_{s}}}{s^{2}}\Big{\|}_{C^{l}\big{(}\frac{g^{\mathcal{S}_{s}}}{s^{2}}\big{)}}\leqslant
C\|g-g_{e}\|_{C^{l+1}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}.$

In a similar fashion, we get the control on the metric. ∎

Given a smooth and orientable hypersurface, $\Sigma$, we define for
$x\in\Sigma$ and $s\in\mathbb{R}$, $F^{g}(x,s):=\gamma^{g}_{x}(s)$, where
$\gamma^{g}_{x}$ is the geodesic for $g$ such that $\gamma^{g}_{x}(0)=x$, and
${\gamma^{\prime}}_{x}^{g}(0)$ is the normal to $\Sigma$ at $x$. We define the
_normal injectivity radius of $\Sigma$_, denoted
$\textup{inj}_{g}^{\perp}(\Sigma)$ as the supremum of the constants $r>0$ for
which the map $F^{g}:\Sigma\times[-r,r]\to M$ is a diffeomorphism on its
image.

######  Lemma 2.10.

For all $l\in\mathbb{N}$ and $\epsilon>0$, there exists $C>0$ and $\delta>0$
such that if a metric $g$ satisfies
$\|g-g_{e}\|_{C^{l+2}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}\leqslant\delta,$
then for any hypersurface $\Sigma\subset A_{e}\big{(}\frac{1}{2},2\big{)}$
satisfying $\|A^{\Sigma}_{g_{e}}\|_{C^{l+1}}\leqslant 10$, we have the two
following properties:

  * •

the normal injectivity radii of $\Sigma$ for $g$
$\textup{inj}_{g}^{\perp}(\Sigma)$, and $g_{e}$,
$\textup{inj}_{g_{e}}^{\perp}(\Sigma)$ satisfy

$\Big{|}\frac{\textup{inj}_{g}^{\perp}(\Sigma)}{\textup{inj}_{g_{e}}^{\perp}(\Sigma)}-1\Big{|}\leqslant\epsilon.$

  * •

the two normal exponential maps are also close to each other for
$\Sigma\times\big{[}-(1-\epsilon)\textup{inj}_{g_{e}}^{\perp}(\Sigma),(1-\epsilon)\textup{inj}_{g_{e}}^{\perp}(\Sigma)\big{]}$:
there exists a vector field $X$ on $\mathbb{R}^{n}/\Gamma$,
$\|X\|_{C^{l+1}(g_{e})}<\epsilon$, for which we have,

$F^{g}\circ(F^{g_{e}})^{-1}=\phi_{X},$

where $\phi_{X}:p\mapsto\exp^{g_{e}}_{p}(X(p))$.

###### Proof.

The curves $F^{g}(x,.)=\gamma^{g}_{x}(.)$ and
$F^{g_{e}}(x,.)=\gamma^{g_{e}}_{x}(.)$ are respectively the solutions of the
geodesic equation for $g$ and $g_{e}$,

$\left\\{\begin{aligned}
\nabla_{\gamma_{x}^{\prime}}\gamma_{x}^{\prime}&\;=0,\\\
\gamma_{x}(0)&\;=x,\\\ \gamma_{x}^{\prime}(0)&\;=N_{x}.\end{aligned}\right.$
(6)

where $N^{g}_{x}$ and $N^{g_{e}}_{x}$ are the outwards normal to $\Sigma$ for
the two metrics. By assumption, the metrics $g$ and $g_{e}$ are
$C^{l+2}$-close. This implies that $N_{x}^{g}$ and $N^{g_{e}}_{x}$ are
$C^{l+2}$-close, and that $\nabla^{g}$ and $\nabla^{g_{e}}$ are
$C^{l+1}$-close.

By continuity with respect to the initial conditions and parameters of order
$2$ differential equations, given a fixed hypersurface $\Sigma$, for all
$\epsilon>0$, there exists $\delta>0$ such that if
$\|g-g_{e}\|_{C^{l+2}}\leqslant\delta$, then, the difference between the
normal geodesics is small in the following $C^{l}$-sense since we work in
local charts. There exists a vector field $X$ defined along each
$\gamma^{g_{e}}_{x}$ such that

$\gamma^{g}_{x}(s)=\phi_{X}(\gamma^{g_{e}}_{x}(s)):=\exp_{\gamma^{g_{e}}_{x}(s)}\big{(}X(\gamma^{g_{e}}_{x}(s))\big{)},$

and $\|X\|_{C^{l+1}(g_{e})}<\epsilon$. Hence, the equality
$\gamma^{g}_{x}(s)=\phi_{X}(\gamma^{g_{e}}_{x}(s))$ rewrites:

$F^{g}=\phi_{X}\circ F^{g_{e}}.$

∎

In particular, we deduce that if a hypersurface $\Sigma^{\prime}$ is a small
normal perturbation of $\Sigma$ for the metric $g$, then, it is also a small
normal perturbation of $\Sigma$ for the metric $g_{e}$, and conversely.

######  Corollary 2.11.

For all $\epsilon_{0}$ $l\in\mathbb{N}$, there exists $C>0$ and
$\delta_{0}>0$, such that for any metric $g$ satisfying,

$\|g-g_{e}\|_{C^{l+2}\big{(}A_{e}\big{(}\frac{1}{4},4\big{)}\big{)}}\leqslant\delta_{0},$

and for any hypersurface $\Sigma\subset A_{e}\big{(}\frac{1}{2},2\big{)}$
satisfying $\|A^{\Sigma}_{g_{e}}\|_{C^{l+1}}\leqslant 10$, we have: for any
function $w:\Sigma\to\mathbb{R}$ with $\|w\|_{C^{l}}\leqslant\epsilon_{0}$,
and denoting $\Sigma(w):=\\{F(x,w(x)),x\in\Sigma\\}$,

  1. 1.

there exists $w^{\prime}:\Sigma\to\mathbb{R}$ with
$\|w^{\prime}\|_{C^{l}}\leqslant 2\epsilon_{0}$ such that

$\Sigma_{g_{e}}(w)=\Sigma_{g}(w^{\prime}),$

  2. 2.

there exists $w^{\prime\prime}:\Sigma\to\mathbb{R}$ with
$\|w^{\prime\prime}\|_{C^{l}}\leqslant 2\epsilon_{0}$ such that

$\Sigma_{g}(w)=\Sigma_{g_{e}}(w^{\prime\prime}).$

###### Proof.

According to Lemma 2.10, we have $F^{g}=\phi_{X}\circ F^{g_{e}}$, where $X$ is
arbitrarily small in $C^{l+1}$-norm for $\delta_{0}$ arbitrarily small. Adding
the assumption $\|A^{\Sigma}_{g_{e}}\|_{C^{l+1}}\leqslant 10$, ensures that
for $\epsilon_{0}>0$, the metrics $g_{e}:=(F^{g_{e}})^{*}g_{e}$ and
$g_{p}:=(g_{e})_{|\Sigma}+ds^{2}$ on
$\Sigma\times[-2\epsilon_{0},2\epsilon_{0}]$, are arbitrarily close in
$C^{l}$-norm for $\epsilon_{0}$ arbitrarily small. This implies that

$\Big{[}(F^{g_{e}})^{-1}\circ\phi_{X}\circ F^{g_{e}}\Big{]}=\phi^{e}_{X^{e}},$

where $X^{e}$ is the pull-back of $X$ by $F^{g_{e}}$, and
$\phi^{e}_{X^{e}}(p):=\exp^{g_{e}}_{p}(X^{e}(p))$. Hence, we have
$\|X^{e}\|_{C^{l+1}(g_{e})}\leqslant\epsilon_{0}$ by the proximity of the
metrics $g_{e}$ and $g_{p}$,
$\phi^{e}_{X^{e}}=(\phi^{\Sigma}_{X_{\Sigma}^{p}},\phi^{\mathbb{R}}_{X_{\mathbb{R}}^{p}})$,
where
$\|X_{\Sigma}^{p}\|_{C^{l}(g^{\Sigma})}+\|X_{\mathbb{R}}^{p}\|_{C^{l}(ds^{2})}\leqslant
2\epsilon_{0}$.

Now, for all $(x,s)\in\Sigma\times[-2\epsilon_{0},2\epsilon_{0}]$, we have

$\displaystyle F^{g}(x,s)$
$\displaystyle\;=F^{g_{e}}\circ\Big{[}(F^{g_{e}})^{-1}\circ\phi_{X}\circ
F^{g_{e}}\Big{]}(x,s),$
$\displaystyle\;=F^{g_{e}}\circ(\phi^{\Sigma}_{X_{\Sigma}^{p}},\phi^{\mathbb{R}}_{X_{\mathbb{R}}^{p}})(x,s)$
$\displaystyle\;=F^{g_{e}}\big{(}\phi^{\Sigma}_{X_{\Sigma}^{p}}(x),\phi^{\mathbb{R}}_{X_{\mathbb{R}}^{p}}(s)\big{)}.$

In particular, for all $w:\Sigma\to[-2\epsilon_{0},2\epsilon_{0}]$, we have

$F^{g}(x,w(x))=\big{(}\phi^{\Sigma}_{X_{\Sigma}^{p}}(x),w^{\prime}(\phi^{\Sigma}_{X_{\Sigma}^{p}}(x))\big{)},$

where $w^{\prime}:=\phi^{\mathbb{R}}_{X_{\mathbb{R}}^{p}}\circ
w\circ\big{(}\phi^{\Sigma}_{X_{\Sigma}^{p}}\big{)}^{-1}$. By the controls of
$X_{\mathbb{R}}^{p}$ and $X_{\Sigma}^{p}$ which we can assume as small as
wanted by choosing $\delta_{0}$ and $\epsilon_{0}$ small enough, we get the
stated result. ∎

##  3 Controls on constant mean curvature hypersurfaces in low curvature
regions

Let us motivate the use of constant mean curvature hypersurfaces. We control
both the way they are embedded in our manifold and their intrinsic geometry
thanks to the ambient curvature which we will control in our neck regions by
[Ban90, Proposition 3].

######  Note 3.1.

All along this section, we will denote constant mean curvature hypersurfaces
with a tilde to insist on their particularity.

######  Proposition 3.2.

For all $k\in\mathbb{N}$ and $0<\alpha<1$, there exists $\epsilon>0$ and $C>0$
for which, if $\tilde{\Sigma}$ is a hypersurface of a manifold $(M^{n},g)$
satisfying:

  1. 1.

its mean curvature is constant equal to $n-1$,

  2. 2.

there exists $\Gamma$ a finite subgroup of $SO(n)$, and a diffeomorphism
$\Phi:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}$ such that we have

$\left\|\Phi^{*}g^{\tilde{\Sigma}}-g^{\mathbb{S}^{n-1}/\Gamma}\right\|_{C^{1,\alpha}(\mathbb{S}^{n-1}/\Gamma)}\leqslant\epsilon.$

  3. 3.

the ambient curvature is small on $\tilde{\Sigma}$,

$\|\operatorname{\textup{Rm}}(g)\|_{C^{k}(g)}\leqslant\epsilon,$

then:

  * •

the second fundamental form is almost constant at the scale of curvature,

$\big{\|}A^{\tilde{\Sigma}}-g^{\tilde{\Sigma}}\big{\|}_{C^{k,\alpha}}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{k}(g)},$

  * •

there exists a diffeomorphism $\Psi:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}$
such that,

$\big{\|}\Psi^{*}g^{\tilde{\Sigma}}-g^{\mathbb{S}^{n-1}/\Gamma}\big{\|}_{C^{k+1,\alpha}(\mathbb{S}^{n-1}/\Gamma)}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{k}(g)}.$

Let us precise the notations we will use for this proof:

  * •

$\nabla^{\tilde{\Sigma}}$: the Levi-Civita connection of the hypersurface,

  * •

$\nabla$: the Levi-Civita connection of the ambient space,

  * •

$N^{\tilde{\Sigma}}$: the outwards normal to the hypersurface,

  * •

$A^{\tilde{\Sigma}}(X,Y):=\langle X,\nabla_{Y}N^{\tilde{\Sigma}}\rangle$: the
second fundamental form of the hypersurface which will be seen as a $2$-tensor
of the tangent space to $\tilde{\Sigma}$,

  * •

$H^{\tilde{\Sigma}}$: the mean curvature of the hypersurface,

  * •

for $B$ a $1$-form on $\tilde{\Sigma}$ taking values in $T^{*}\tilde{\Sigma}$,
we define

$(d^{\nabla^{\tilde{\Sigma}}}B)(X,Y):=\nabla^{\tilde{\Sigma}}_{X}B(Y,.)-\nabla^{\tilde{\Sigma}}_{Y}B(X,.),$

  * •

we will denote $\delta^{\nabla^{\tilde{\Sigma}}}$ the formal adjoint of
$d^{\nabla^{\tilde{\Sigma}}}$.

We have the following classical formulas for the second formula for any
hypersurface $\Sigma$: for all vector fields $X,Y,Z$ on $\Sigma$,

$d^{\nabla^{\Sigma}}A^{\Sigma}(X,Y,Z)=g\big{(}\operatorname{\textup{Rm}}(X,Y)N^{\Sigma},Z\big{)},$

and

$\delta^{\nabla^{\Sigma}}A^{\Sigma}(X)=\operatorname{\textup{Ric}}\big{(}N^{\Sigma},X\big{)}-dH(X).$

Denoting $A_{0}^{\tilde{\Sigma}}$ the trace-free part of $A^{\tilde{\Sigma}}$
for $g^{\tilde{\Sigma}}$, if $\tilde{\Sigma}$ is a constant mean curvature
hypersurface, by the above formulas, we even have

$d^{\nabla^{\tilde{\Sigma}}}A_{0}^{\tilde{\Sigma}}(X,Y,Z)=g\big{(}\operatorname{\textup{Rm}}(X,Y)N^{\tilde{\Sigma}},Z\big{)},$
(7)

and

$\delta^{\nabla^{\tilde{\Sigma}}}A_{0}^{\tilde{\Sigma}}(X)=\operatorname{\textup{Ric}}\big{(}N^{\tilde{\Sigma}},X\big{)}.$
(8)

The right hand sides being well controlled, we can control
$A_{0}^{\tilde{\Sigma}}$ by noticing that the operator
$d^{\nabla^{\tilde{\Sigma}}}+\delta^{\nabla^{\tilde{\Sigma}}}$ is elliptic.

######  Lemma 3.3.

The operator $d^{\nabla^{\tilde{\Sigma}}}+\delta^{\nabla^{\tilde{\Sigma}}}$
acting on symmetric $2$-tensors is elliptic. Moreover, for any metric $g$
sufficiently $C^{0}$-close to $g^{\mathbb{S}^{n-1}}$, the operator
$d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}$ is
injective on the $2$-tensors which are traceless for $g$.

###### Proof.

The symbol of the operator $d^{\nabla}+\delta^{\nabla}$ applied to $\xi\in
T\tilde{\Sigma}^{*}$ is the sum of the exterior product by $\xi$ and the
interior product by the vector field associated to $\xi$ by the metric. These
two parts take values in different spaces, and the kernel of the first one is
the set of forms proportional to $\xi$ while for the second it is the set of
those orthogonal to $\xi$. The kernel is therefore reduced to $\\{0\\}$ and
the symbol of the operator is invertible, hence, $d^{\nabla}+\delta^{\nabla}$
is elliptic.

According to [Bes87, 12.69], the associated Laplacian:
$-\hat{\Delta}:=\delta^{\nabla}d^{\nabla}+d^{\nabla}\delta^{\nabla}$ takes the
following form for a symmetric $2$-tensor,

$-\hat{\Delta}^{\tilde{\Sigma}}h=(\nabla^{\tilde{\Sigma}})^{*}\nabla^{\tilde{\Sigma}}h-\mathring{\operatorname{\textup{R}}}^{\tilde{\Sigma}}h+h\circ\operatorname{\textup{Ric}}^{\tilde{\Sigma}},$

where $\mathring{\operatorname{\textup{R}}}(h)$ is the action of curvature on
symmetric $2$-tensors: for an orthonormal basis $(e_{i})_{i}$,

$\mathring{\operatorname{\textup{R}}}(h)(X,Y)=\sum_{i}h\big{(}\operatorname{\textup{Rm}}(e_{i},X)Y,e_{i}\big{)},$

or
$\mathring{\operatorname{\textup{R}}}(h)_{ij}:=\operatorname{\textup{Rm}}_{iklj}h_{mn}g^{km}g^{ln}$
in local coordinates. Hence, on the unit sphere of dimension $n-1$, we have
$\operatorname{\textup{Rm}}^{\mathbb{S}^{n-1}}_{iklj}=g_{ij}g_{kl}-g_{il}g_{kj}$,
and therefore
$\mathring{\operatorname{\textup{R}}}^{\mathbb{S}^{n-1}}(h)_{ij}=tr(h)g_{ij}-h_{ij}$,
and $\alpha\circ\operatorname{\textup{Ric}}^{\mathbb{S}^{n-1}}=(n-2)\alpha$.
This gives

$-\hat{\Delta}^{\mathbb{S}^{n-1}}h=(\nabla^{\mathbb{S}^{n-1}})^{*}\nabla^{\mathbb{S}^{n-1}}h+(n-1)h-tr(h)\mathrm{Id}.$

Let $h_{0}$ be a $C^{1,\alpha}$ symmetric $2$-tensor whose trace for
$g^{\mathbb{S}^{n-1}}$ vanishes and which is in the kernel of the operator.
The trace of $h_{0}$ vanishes so we have,

$(\nabla^{\mathbb{S}^{n-1}})^{*}\nabla^{\mathbb{S}^{n-1}}h_{0}+(n-1)h_{0}=0.$

Integrating by parts against $h_{0}$ yields

$\|\nabla h_{0}\|_{L^{2}}^{2}+(n-1)\|h_{0}\|_{L^{2}}^{2}=0,$ (9)

and finally $h_{0}=0$. The operator
$d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}$ is
therefore injective on symmetric $2$-tensors whose trace for
$g^{\mathbb{S}^{n-1}}$ vanishes.

Let us now consider a metric $g$ close to $g^{\mathbb{S}^{n-1}}$, that is,
assume that there exists $\epsilon>0$ which we will choose small enough such
that

$\|g-g^{\mathbb{S}^{n-1}}\|_{C^{0}}\leq\epsilon.$

We want to show that the operator
$d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}$ is also
invertible on tensors whose trace vanishes for $g$. Let $h$ be a symmetric
$2$-tensor which is traceless for $g$. By proximity of the metrics, for
$\epsilon$ small enough, there exists $C=C(n)$ for which

$|tr^{\mathbb{S}^{n-1}}h|_{\mathbb{S}^{n-1}}\leqslant
C\epsilon|h|_{\mathbb{S}^{n-1}}.$ (10)

We still have

$-\hat{\Delta}^{\mathbb{S}^{n-1}}h=(\nabla^{\mathbb{S}^{n-1}})^{*}\nabla^{\mathbb{S}^{n-1}}h+(n-1)h-tr^{\mathbb{S}^{n-1}}(h)\textup{Id},$

and thanks to (10), if
$\big{(}d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}\big{)}h=0$,
then

$0\geqslant\|\nabla^{\mathbb{S}^{n-1}}h\|^{2}+(n-1-C\epsilon)\|h\|^{2},$

which gives $h=0$ if $\epsilon$ is small enough to have $n-1>C\epsilon$. We
finally conclude that
$d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}$ is
injective on symmetric $2$-tensors whose trace for $g$ vanishes. ∎

######  Corollary 3.4.

For all $n\in\mathbb{N}$, there exists $\epsilon>0$ such that if
$\tilde{\Sigma}$ is a hypersurface of a manifold $(M^{n},g)$ satisfying:

  1. 1.

its mean curvature is constant equal to $n-1$,

  2. 2.

there exists $\Gamma$, a finite subgroup of $SO(n)$, and a diffeomorphism
$\Phi:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}$ such that we have:

$\left\|\Phi^{*}g^{\tilde{\Sigma}}-g^{\mathbb{S}^{n-1}/\Gamma}\right\|_{C^{1,\alpha}(\mathbb{S}^{n-1}/\Gamma)}\leqslant\epsilon,$

  3. 3.

on $\tilde{\Sigma}$,

$\|\operatorname{\textup{Rm}}(g)\|_{C^{k}(g)}\leqslant\epsilon,$

then, for all $p\geqslant 1$ and $l\in\mathbb{N}$, there exists $C>0$ such
that for every symmetric $2$-tensor $u$ on $\tilde{\Sigma}$ whose trace with
respect to $g^{\tilde{\Sigma}}$ vanishes, we have,

$\|u\|_{W^{l+1,p}}\leqslant
C\Big{(}\big{\|}\delta^{\nabla^{\tilde{\Sigma}}}u\big{\|}_{W^{l,p}}+\big{\|}d^{\nabla^{\tilde{\Sigma}}}u\big{\|}_{W^{l,p}}\Big{)}$

###### Proof.

Thanks to the hypothesis 2, there exists
$\Phi:\tilde{\Sigma}\to\mathbb{S}^{n-1}$ such that we have the following
control: for all $p\geqslant 1$, there exists a constant $C>0$ such that for
all $2$-tensor $u$ we have

$\Big{\|}\Phi^{*}\big{(}d^{\nabla^{\tilde{\Sigma}}}+\delta^{\nabla^{\tilde{\Sigma}}}\big{)}u-\big{(}d^{\nabla^{\mathbb{S}^{n-1}}}+\delta^{\nabla^{\mathbb{S}^{n-1}}}\big{)}u\Big{\|}_{L^{p}}\leqslant
C\epsilon\left\|u\right\|_{W^{1,p}}.$

The conclusion comes then from estimates for injective Fredholm operators
which are consequences of the open mapping theorem between Banach spaces. ∎

###### Proof of Proposition 3.2.

Let $0<\alpha<1$, and $p>\frac{n}{1-\alpha}$. Then, $W^{1,p}$ embeds
continuously in $C^{\alpha}$, which means that there exists
$C_{1}=C_{1}(p,\alpha,n)>0$ such that for all $2$-tensor $u$, we have:

$\|u\|_{C^{\alpha}}\leq C_{1}\|u\|_{W^{1,p}}.$

Define on $\mathbb{S}^{n-1}$ the operators,
$P:=\delta^{\nabla^{\mathbb{S}^{n-1}}}+d^{\nabla^{\mathbb{S}^{n-1}}}$ and
$P^{\prime}:=\Phi^{*}\big{(}\delta^{\nabla^{\tilde{\Sigma}}}+d^{\nabla^{\tilde{\Sigma}}}\big{)}$
between the Banach spaces $W^{1,p}$ and $L^{p}$. According to Lemma 3.3, $P$
is injective on traceless $2$-tensors. Thanks to the expressions (7) and (8)
of the operator $\delta^{\nabla}+d^{\nabla}$ applied to
$A^{\tilde{\Sigma}}_{0}$, by Corollary 3.4, we have,

$\displaystyle\big{\|}A_{0}^{\tilde{\Sigma}}\big{\|}_{W^{1,p}}$
$\displaystyle\;\leqslant
C\left(\big{\|}\delta^{\nabla^{\tilde{\Sigma}}}A_{0}^{\tilde{\Sigma}}\big{\|}_{L^{p}}+\big{\|}d^{\nabla^{\tilde{\Sigma}}}A_{0}^{\tilde{\Sigma}}\big{\|}_{L^{p}}\right)$
$\displaystyle\;=C\left(\big{\|}\operatorname{\textup{Rm}}(.,.)N^{\tilde{\Sigma}}\big{\|}_{L^{p}}+\big{\|}\operatorname{\textup{Ric}}(N^{\tilde{\Sigma}})\big{\|}_{L^{p}}\right)$

Since the hypersurface has bounded volume (actually close to the volume of
$\mathbb{S}^{n-1}/\Gamma$), there exists $C_{2}=C_{2}(p,n)>0$ such that we
have

$\|\operatorname{\textup{Ric}}(N^{\tilde{\Sigma}})\|_{L^{p}}\leqslant
C_{2}\|\operatorname{\textup{Rm}}(.,.)N^{\tilde{\Sigma}}\|_{C^{0}},$

and

$\|\operatorname{\textup{Rm}}(.,.)N^{\tilde{\Sigma}}\|_{L^{p}}\leqslant
C_{2}\|\operatorname{\textup{Rm}}(.,.)N^{\tilde{\Sigma}}\|_{C^{0}}.$

Hence, by continuous embedding of $W^{1,p}$ in $C^{\alpha}$, we have

$\|A_{0}^{\tilde{\Sigma}}\|_{C^{\alpha}}\leqslant
2C_{1}C_{2}C\|\operatorname{\textup{Rm}}(.,.)N^{\tilde{\Sigma}}\|_{C^{0}},$

which is the stated result for $k=0$.

Let us finally explain how to control the higher levels of regularity by
induction. Assume that we have the following controls for $l\in\mathbb{N}$, on
the hypersurface $\tilde{\Sigma}$: there exists $C>0$ such that,

  1. 1.

$\|\operatorname{\textup{Rm}}(g)\|_{C^{l+1}(g)}\leqslant\epsilon$,

  2. 2.

$\big{\|}A^{\tilde{\Sigma}}-g^{\tilde{\Sigma}}\big{\|}_{C^{l,\alpha}}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{l}(g)}$,

  3. 3.

there exists a diffeomorpihsm $\Psi:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}$
with

$\left\|\Psi^{*}g^{\tilde{\Sigma}}-g^{\mathbb{S}^{n-1}/\Gamma}\right\|_{C^{l+1,\alpha}(\mathbb{S}^{n-1}/\Gamma)}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{l}(g)}.$

Gauss equation gives:

$\displaystyle g(\operatorname{\textup{Rm}}^{\tilde{\Sigma}}(X,Y)Z,W)=$
$\displaystyle\;\;g(\operatorname{\textup{Rm}}(X,Y)Z,W)$
$\displaystyle\;+A^{\tilde{\Sigma}}(X,Z)A^{\tilde{\Sigma}}(Y,W)-A^{\tilde{\Sigma}}(Y,Z)A^{\tilde{\Sigma}}(X,W).$

And controls 1 and 2 above imply therefore that there exists
$C=C(\alpha,l,n)>0$ which gives

$\big{\|}\operatorname{\textup{Ric}}(g^{\tilde{\Sigma}})-(n-1)g^{\tilde{\Sigma}}\big{\|}_{C^{l,\alpha}}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{l+1}(g)},$

and we can use Proposition 3.5 proven in the appendix.

######  Proposition 3.5.

Let $(M,g_{0})$ be an $n$-dimensional manifold with sectional curvatures equal
to $1$, and $k\in\mathbb{N}$. Then, there exists $\delta>0$ and $C>0$, such
that if a metric $g$ on $M$ satisfies:

$\|g-g_{0}\|_{C^{k+1,\alpha}}\leqslant\delta$

then, there exists a diffeomorphism $\Phi:M\to M$, such that we have

  * •

if $k\geqslant 1$,

$\|\Phi^{*}g-g_{0}\|_{C^{k+1,\alpha}}\leqslant
C\big{\|}\operatorname{\textup{Ric}}(g)-(n-1)g\big{\|}_{C^{k-1,\alpha}},$

  * •

or else, if $k=0$,

$\|\Phi^{*}g-g_{0}\|_{C^{1,\alpha}}\leqslant
C\big{\|}\operatorname{\textup{Ric}}(g)-(n-1)g\big{\|}_{C^{0}}.$

Hence, in our case, there exists a diffeomorphism
$\tilde{\Psi}_{\tilde{\Sigma}}:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}$ with

$\left\|\tilde{\Psi}_{\tilde{\Sigma}}^{*}g^{\tilde{\Sigma}}-g^{\mathbb{S}^{n-1}/\Gamma}\right\|_{C^{l+2,\alpha}(\mathbb{S}^{n-1}/\Gamma)}\leqslant
C\|\operatorname{\textup{Rm}}(g)\|_{C^{l+1}(g)}.$

This diffeomorphism combined to the controls in $C^{l+1}$-norm on the
curvature imply by elliptic regularity (just like in the proof with $k=0$)
that there exists a constant $C^{\prime}=C^{\prime}(l+1,\alpha,n)>0$ such that

$\big{\|}A^{\tilde{\Sigma}}-g^{\tilde{\Sigma}}\big{\|}_{C^{l+1,\alpha}}\leqslant
C^{\prime}\|\operatorname{\textup{Rm}}(g)\|_{C^{l+1}(g)}.$

We therefore have the property for $l+1$ and we can iterate. ∎

##  4 Perturbation of hypersurfaces with almost constant second fundamental
form

During all of our construction, without loss of generality by rescaling, we
will always reduce our construction to the case where the mean curvature is
constant equal to $n-1$. Let us first show that we can perturbate a
hypersurface with almost constant second fundamental form to a constant mean
curvature hypersurface.

######  Proposition 4.1.

For all $n\in\mathbb{N}$ and $0<r_{0}<\frac{1}{6}$, there exists $C>0$ and
$0<\epsilon<\frac{r_{0}}{C}$ such that for any hypersurface $\Sigma$
diffeomorphic to $\mathbb{S}^{n-1}/\Gamma$, for $\Gamma\neq\\{e\\}$, of a
manifold $(M,g)$ satisfying:

  1. 1.

the normal injectivity radius of $\Sigma$ and the injectivity radius of the
points of $\Sigma$ in $M$ are bounded below by $3r_{0}$,

  2. 2.

$\big{\|}A^{\Sigma}-g^{\Sigma}\big{\|}_{C^{\alpha}}\leqslant\epsilon$,

  3. 3.

on the annulus $\\{x,d^{M}_{g}(x,\Sigma)<3r_{0}\\}$, we have
$\|\operatorname{\textup{Rm}}(g)\|_{C^{2}(g)}<\epsilon$,

Then, there exists a unique function $w$ satisfying
$\|w\|_{C^{2,\alpha}}\leqslant C\epsilon$ and such that we have

$H^{\Sigma(w)}\equiv n-1,$

where we denoted $\Sigma(w):=\\{\gamma_{x}(w(x)),x\in\Sigma\\}$, where
$\gamma_{x}$ is the geodesic with $\gamma_{x}(0)=x$, and
$\gamma^{\prime}_{x}(0)$ is the unit outwards normal to $\Sigma$. We moreover
have the following control,

$\|w\|_{C^{2,\alpha}}\leqslant
C\big{\|}H^{\Sigma}-(n-1)\big{\|}_{C^{\alpha}}.$

######  Remark 4.2.

Recall that we defined our Hölder norms at the scale of the injectivity
radius, see Definition 2.2. In our applications, the controls on the
injectivity radius will be consequences of Lemma 2.3, and the different
curvature controls will be consequences of the previous section.

In general, the construction of spheres with constant mean curvature can be
tricky (see for example [Ye96, PX09]) because it is achieved by constructing a
right-inverse for the operator $\Delta_{\mathbb{S}^{n-1}}+(n-1)$, where
$\Delta_{\mathbb{S}^{n-1}}$ is the Laplace-Beltrami operator (with nonpositive
eigenvalues) which is not invertible. In our case, our hypersurfaces are not
close to $\mathbb{S}^{n-1}$, but to $\mathbb{S}^{n-1}/\Gamma$ for
$\Gamma\neq\\{e\\}$. The crucial difference it makes is that $-(n-1)$ is not
an eigenvalue anymore (see [CEV17] for the use of the same remark in the
context of asymptotically conical manifolds). Indeed, the eigenfunctions with
eigenvalue $-(n-1)$ are the restrictions of linear functions of
$\mathbb{R}^{n}$ to the sphere which cannot be $\Gamma$-invariant unless it
vanishes.

######  Remark 4.3.

Geometrically, this comes from the fact that given
$\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$, the translations provide constant
mean curvature perturbations of the spheres. In the case of
$\mathbb{S}^{n-1}/\Gamma\subset\mathbb{R}^{n}/\Gamma$, the center cannot move
as it is the fixed point of the group action.

#### Normal perturbations of the hypersurface.

Consider normal perturbations of the hypersurface of the form
$\Sigma(w):=\\{\gamma_{x}(w(x)),x\in\Sigma\\}$. The computations of [PX09]
adapted to our situation yield the following development of the mean
curvature.

######  Lemma 4.4.

For any $0<r_{0}<\frac{1}{6}$ and any smooth hypersurface $\Sigma\subset M$
whose injectivity radius and normal injectivity radius are larger than
$3r_{0}$, there exists $\epsilon>0$ such that for all $w:\Sigma\to\mathbb{R}$
such that $\|w\|_{C^{2,\alpha}}\leqslant\epsilon$, the mean curvature of
$\Sigma(w)$ has the development:

$H(\Sigma(w))=H(\Sigma)-J_{\Sigma,M}w+Q_{\Sigma,M}(w),$

where

$J_{\Sigma,M}:=\Delta_{\Sigma}+|A^{\Sigma}|^{2}+\emph{Ric}^{M}(N,N)$

is the Jacobi operator of the hypersurface, and where $Q_{\Sigma,M}$ is such
that for a constant $C>0$ depending on the $C^{2}$-norm of the curvature and
the $C^{1}$-norm of the second fundamental form and $r_{0}$, we have :

$\|Q_{\Sigma,M}(w)-Q_{\Sigma,M}(w^{\prime})\|_{C^{\alpha}}\leqslant
C\|w-w^{\prime}\|_{C^{2,\alpha}}\big{(}\|w\|_{C^{2,\alpha}}+\|w^{\prime}\|_{C^{2,\alpha}}\big{)}.$

Proposition 4.1 is then a consequence of the following lemma applied to
$\Sigma$.

######  Lemma 4.5.

Under the assumptions of Proposition 4.1, there exists $\epsilon_{1}>0$ such
that there exists a unique $w$ satisfying
$\|w\|_{C^{2,\alpha}}\leqslant\epsilon_{1}$ and the equation

$H(\Sigma(w))=n-1.$

We moreover have the control:

$\|w\|_{C^{2,\alpha}}\leqslant C\left\|H(\Sigma)-(n-1)\right\|_{C^{\alpha}}.$

###### Proof.

This is a consequence of the following quantitative version of the inverse
function theorem proven by Banach fixed point theorem.

######  Lemma 4.6.

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 $B(0,r)$.

By considering the operator $\Phi:C^{2,\alpha}(\Sigma)\to
C^{0,\alpha}(\Sigma)$ defined by

$\Phi:w\in C^{2,\alpha}(\Sigma)\mapsto H(\Sigma(w))\in C^{0,\alpha}(\Sigma),$

we are under the assumptions of Lemma 4.6, with $\Phi(0)=H(\Sigma)$,
$d_{0}\Phi=J_{\Sigma,M}$:

  1. 1.

$-(n-1)$ is not an eigenvalue of $\Delta_{\mathbb{S}^{n-1}}$, hence
$J_{\mathbb{S}^{n-1}/\Gamma,\mathbb{R}^{n}/\Gamma}$ is invertible with bounded
inverse. Therefore, the operator $J_{\Sigma,M}$ which is arbitrarily close
(for $\epsilon$ arbitrarily small) to
$J_{\mathbb{S}^{n-1}/\Gamma,\mathbb{R}^{n}/\Gamma}$ for the operator norm from
$C^{2,\alpha}(\Sigma)$ to $C^{\alpha}(\Sigma)$ has a bounded inverse,

  2. 2.

there exists $C>0$ depending on $r_{0},\alpha$ and $\epsilon$ :

$\|Q_{\Sigma,M}(w)-Q_{\Sigma,M}(w^{\prime})\|_{C^{\alpha}}\leqslant
C\|w-w^{\prime}\|_{C^{2,\alpha}}(\|w\|_{C^{2,\alpha}}+\|w^{\prime}\|_{C^{2,\alpha}}),$

  3. 3.

$\left\|H(\Sigma)-(n-1)\right\|_{C^{\alpha}}$ is arbitrarily small.

This unique solution actually satisfies $C^{\prime}>0$,

$\|w\|_{C^{2,\alpha}}\leqslant C^{\prime}\|H(\Sigma)-1\|_{C^{\alpha}}.$

∎

For $\Sigma$ satisfying the assumptions of Proposition 4.1 let us define
$\tilde{\Sigma}:=\Sigma(w)$, where $w$ is the unique solution given by Lemma
4.5. This is a constant mean curvature perturbation of $\Sigma$ which proves
Proposition 4.1.

##  5 Construction of a foliation of the neck regions by constant mean
curvature hypersurfaces

###  5.1 Local foliation by constant mean curvature hypersurfaces

######  Proposition 5.1 (Local foliation).

For all $n\in\mathbb{N}$ and $0<r_{0}<\frac{1}{6}$, there exists $C>0$ and
$0<\epsilon<\frac{r_{0}}{C}$ such that if a hypersurface $\tilde{\Sigma}_{1}$
diffeomorphic to $\mathbb{S}^{n-1}/\Gamma$, for $\Gamma\neq\\{e\\}$, of a
manifold $(M,g)$ satisfies:

  * •

the normal injectivity radius of $\tilde{\Sigma}_{1}$ and the injectivity
radius of the points $\tilde{\Sigma}_{1}$ in $M$ are bounded below by
$3r_{0}$,

  * •

$H^{\tilde{\Sigma}_{1}}\equiv n-1$, and
$\|A^{\tilde{\Sigma}_{1}}-g^{\tilde{\Sigma}_{1}}\|_{C^{\alpha}}<\epsilon$,

  * •

on the annulus $\\{x,d^{M}_{g}(x,\Sigma)<3r_{0}\\}$, we have
$\|\operatorname{\textup{Rm}}\|_{C^{2}}\leqslant\epsilon$.

Then, there exists a family of hypersurfaces $\tilde{\Sigma}_{s}$,
$s\in\Big{[}\frac{1}{1+2r_{0}},1+2r_{0}\Big{]}$ with constant mean curvature
$\frac{n-1}{s}$ foliating a region of the annulus
$\tilde{A}\Big{(}\frac{1}{1+2r_{0}},1+2r_{0}\Big{)}$ satisfying:

$\left\\{x,\;d^{M}_{g}(x,\Sigma)<r_{0}\right\\}\subset\tilde{A}\Big{(}\frac{1}{1+2r_{0}},1+2r_{0}\Big{)}\subset\left\\{x,\;d^{M}_{g}(x,\Sigma)<3r_{0}\right\\}.$

The construction we propose consists in perturbating normally the level sets
of the distance function to the hypersurface $\tilde{\Sigma}_{1}$ in constant
mean curvature hypersurfaces by 4.1.

Let us define the equidistant hypersurfaces
$\Sigma_{s}:=\\{\gamma_{x}(s-1),x\in\tilde{\Sigma}_{1}\\}$ where $\gamma_{x}$
is the geodesic outwards normal to the hypersurface $\tilde{\Sigma}$ starting
at $x$. These hypersurfaces foliate the annulus
$\\{x,d^{M}_{g}(x,\Sigma)<3r_{0}\\}$ by definition of the normal injectivity
and their geometry is well controlled around $\tilde{\Sigma}_{1}$.

######  Lemma 5.2.

For all $n\in\mathbb{N}$ and $0<r_{0}<\frac{1}{6}$, there exists $C>0$ and
$0<\epsilon<\frac{r_{0}}{C}$ such that if a hypersurface $\tilde{\Sigma}_{1}$
diffeomorphic to $\mathbb{S}^{n-1}/\Gamma$, $\Gamma\neq\\{e\\}$, of a manifold
$(M,g)$ satisfies:

  * •

the normal injectivity radius of $\tilde{\Sigma}_{1}$ and the injectivity
radius of points of $\tilde{\Sigma}_{1}$ in $M$ are bounded below by $3r_{0}$,

  * •

$H^{\tilde{\Sigma}_{1}}\equiv n-1$, and
$\|A^{\tilde{\Sigma}_{1}}-g^{\tilde{\Sigma}_{1}}\|_{C^{\alpha}}<\epsilon$,

  * •

on the annulus $\\{x,d^{M}_{g}(x,\Sigma)<3r_{0}\\}$, we have
$\|\operatorname{\textup{Rm}}\|_{C^{2}}\leqslant\epsilon$,

then,

$\displaystyle\Big{\|}\frac{A^{\Sigma_{s}}}{s}-\frac{g^{\Sigma_{s}}}{s^{2}}\Big{\|}_{C^{\alpha}\big{(}\frac{g^{\Sigma_{s}}}{s^{2}}\big{)}}$
$\displaystyle\;\leqslant
C\big{(}\|A^{\tilde{\Sigma}_{1}}-g^{\tilde{\Sigma}_{1}}\|_{C^{\alpha}}+|s-1|\|\operatorname{\textup{Rm}}\|_{C^{0}}+|s-1|^{2}\|\operatorname{\textup{Rm}}\|_{C^{2}}\big{)}$
$\displaystyle\;\leqslant C(1+r_{0}^{2})\epsilon,$ (11)

and

$\Big{\|}\frac{H^{\Sigma_{s}}}{s}-\frac{n-1}{s^{2}}\Big{\|}_{C^{\alpha}\big{(}\frac{g^{\Sigma_{s}}}{s^{2}}\big{)}}\leqslant
C\big{(}|s-1|\|\operatorname{\textup{Ric}}\|_{C^{0}}+|s-1|^{2}\|\operatorname{\textup{Rm}}\|_{C^{2}}\big{)}\leqslant
C\big{(}r_{0}+r_{0}^{2}\big{)}\epsilon.$ (12)

###### Proof.

Given a hypersurface $\Sigma_{1}$ whose normal injectivity radius is larger
than $3r_{0}>0$, we define a vector field $N$ on the region
$\\{x,d^{M}_{g}(x,\Sigma)<3r_{0}\\}$ by setting $N(x)$, for $x\in\Sigma_{s}$,
as the outwards normal vector of the hypersurface $\Sigma_{s}$.

Denoting, $S(X):=\nabla_{X}N$, for $X$ a vector field on $M$, we have

$(\nabla_{N}S)(X)=-(S^{2}(X)+\operatorname{\textup{Rm}}(X,N)N).$ (13)

Assuming $\|S_{|\Sigma_{1}}-\mathrm{Id}\|_{C^{\alpha}}<\epsilon$ for
$\epsilon>0$ small enough, we can follow $S$ along the normal geodesics, and
integrating the previous Riccati equation (13), we find that there exits $C>0$
such that for $|s-1|\leqslant r_{0}\leqslant\frac{1}{6}$,

$\Big{\|}\frac{S_{|\Sigma_{s}}}{s}-\frac{\mathrm{Id}}{s^{2}}\Big{\|}_{C^{\alpha}}\leqslant\|S_{|\Sigma_{1}}-\mathrm{Id}\|_{C^{\alpha}}+C|s-1|\|\operatorname{\textup{Rm}}\|_{C^{0}}+C|s-1|^{2}\|\operatorname{\textup{Rm}}\|_{C^{2}}.$

Denoting $\kappa_{i}(s)$ the eigenvalues of the second fundamental form of
$\Sigma_{s}$, and $(e_{i}(s))$ an associated orthonormal basis of
eigenvectors, if we project (13) on $e_{i}(s)$, we get:

$\kappa_{i}^{\prime}(s)=-\left(\kappa_{i}(s)^{2}+g\big{(}\operatorname{\textup{Rm}}(N^{s},e_{i}(s))e_{i}(s),N^{s}\big{)}\right)$

and finally, by summing on $i$,
$H^{\prime}(s)=-\big{(}|A|^{2}(s)+\operatorname{\textup{Ric}}(N^{s},N^{s})\big{)}$.
With the initial condition $H(1)\equiv n-1$, we finally have the second stated
estimate. ∎

###### Proof of Proposition 5.1.

Thanks to the controls (12), and (11), we can use Lemma 4.5 to perturbate
normally each hypersurface $\Sigma_{s}$ to a constant mean curvature
hypersurface $\tilde{\Sigma}_{s}$. Hence, there exists a family of
hypersurfaces
$(\tilde{\Sigma}_{s})_{s\in\left[\frac{1}{1+2r_{0}},1+2r_{0}\right]}$ with
mean curvature constant equal to $\frac{1}{s}$ for each $s$. These
hypersurfaces are defined from a _unique_ function $w_{s}$ from
$\tilde{\Sigma}_{1}$ to $\mathbb{R}$ by Proposition 4.1,

$\tilde{\Sigma}_{s}=\Sigma(s(1+w_{s})):=\\{\gamma_{x}(s(1+w_{s}(x))),x\in\tilde{\Sigma}_{1}\\},$

we moreover have the control

$\|w_{s}\|_{C^{2,\alpha}}\leqslant C\epsilon(|s-1|+|s-1|^{2}).$

Let us now show that this family actually foliates the annulus. More
precisely, we will show that given $s$ and
$s^{\prime}\in[\frac{1}{1+2r_{0}},1+2r_{0}]$, close enough and such that
$s<s^{\prime}$, $\tilde{\Sigma}_{s^{\prime}}$ is a normal perturbation of
$\tilde{\Sigma}_{s}$ by a _strictly positive_ function
$w_{s}^{s^{\prime}}:\tilde{\Sigma}_{s}\to\mathbb{R}$. They are indeed normal
perturbation of each other because they are small (in $C^{2,\alpha}$-norm)
perturbations of the equidistant hypersurfaces from $\tilde{\Sigma}_{1}$.

By Lemma 4.5, for $s$ and $s^{\prime}$ close enough, the function
$w_{s}^{s^{\prime}}$ satisfies $\|w_{s}^{s^{\prime}}\|_{C^{2,\alpha}}\leqslant
2|s^{\prime}-s|$ and is a solution to

$-J_{\tilde{\Sigma}_{s},M}(w_{s}^{s^{\prime}})=\frac{n-1}{s^{\prime}}-\frac{n-1}{s}+Q_{\tilde{\Sigma}_{s},M}(w_{s}^{s^{\prime}}),$

where there exists $C=C(\epsilon,n,\alpha,r_{0})>0$ independent of $s$ such
that for any functions $w$, and $w^{\prime}$, we have
$\|Q_{\tilde{\Sigma}_{s},M}(w)-Q_{\tilde{\Sigma}_{s},M}(w^{\prime})\|_{C^{\alpha}}\leqslant
C\|w-w^{\prime}\|_{C^{2,\alpha}}(\|w\|_{C^{2,\alpha}}+\|w^{\prime}\|_{C^{2,\alpha}})$.

We therefore get
$\|Q_{\tilde{\Sigma}_{s},M}(w_{s}^{s^{\prime}})\|_{C^{\alpha}}\leqslant
4C|s^{\prime}-s|^{2}$, and since $s\tilde{\Sigma}_{s}$ is close to
$\mathbb{S}^{n-1}/\Gamma$, for $|s^{\prime}-s|$ small enough,
$J_{\tilde{\Sigma}_{s},M}$ is close to $\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1$,
and we have:

$\Big{\|}-\big{(}\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1\big{)}(w_{s}^{s^{\prime}}-(s^{\prime}-s))\Big{\|}_{C^{\alpha}}\leqslant
C\big{(}(s^{\prime}-s)^{2}+\epsilon(s^{\prime}-s)\big{)}.$

Since $-\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1$ is invertible, there exists
$C=C(\alpha,n)>0$ such that

$\|w_{s}^{s^{\prime}}-(s^{\prime}-s)\|_{C^{2,\alpha}}\leqslant
C\big{(}(s^{\prime}-s)^{2}+\epsilon(s^{\prime}-s)\big{)}.$

Finally, we have
$w_{s}^{s^{\prime}}=(s^{\prime}-s)+\mathcal{O}\big{(}(s^{\prime}-s)^{2}+\epsilon(s^{\prime}-s)\big{)}$,
and in particular, for $s^{\prime}-s>0$ and $\epsilon$ small enough (depending
on $n$ and $\alpha$), $w_{s}^{s^{\prime}}>0$.

This implies that for all $s,s^{\prime}\in[\frac{1}{1+2r_{0}},1+2r_{0}]$, the
hypersurfaces $\tilde{\Sigma}_{s}$ and $\tilde{\Sigma}_{s^{\prime}}$ are all
disjoint and foliate the annulus. Indeed, we have just seen that for
$s<s^{\prime}$ and $|s-s^{\prime}|$ small enough, the hypersurface
$\tilde{\Sigma}_{s^{\prime}}$ is strictly included in one side of
$\tilde{\Sigma}_{s}$. Recalling that for all $s$,
$\tilde{\Sigma}_{s}=\tilde{\Sigma}_{1}(s(1+w_{s}))$, this means that for all
$x$, the function $s\mapsto(s(1+w_{s}(x)))$ is strictly increasing, and
therefore that we indeed have a foliation of the annulus. ∎

######  Remark 5.3.

It was not directly possible to ensure that two hypersurfaces
$\tilde{\Sigma}_{s}$ and $\tilde{\Sigma}_{s^{\prime}}$ do not intersect for
$s,s^{\prime}\in[\frac{1}{1+2r_{0}},1+2r_{0}]$ from our previous controls.
Indeed, the control $\|w_{s}\|_{C^{2,\alpha}}\leqslant
C(\epsilon|s-1|+\epsilon|s-1|^{2})$ is not enough to rule out
$w_{s}(x)=w_{s^{\prime}}(x^{\prime})$ when $s-1$ is much larger than
$s-s^{\prime}$.

###  5.2 Global foliation of the neck regions.

Let us come back to our applications and consider an Einstein manifold $(M,g)$
close to an Einstein orbifold $(M_{o},g_{o})$ in the Gromov-Hausdorff sense.
According to Proposition 1.10, this manifold is $\epsilon$-approximated by a
tree of Ricci-flat ALE orbifolds desingularizing the orbifold. We have
satisfying coordinates on the regions $\mathcal{M}_{o}(\epsilon)$ and
$\mathcal{N}_{k}(\epsilon)$ and want to find good coordinates in the annuli
$\mathcal{A}_{k}(t,\epsilon)$. They are included in metric annuli
$A(\rho_{1},\rho_{2})$ for
$\rho_{1}:=\frac{1}{8}\epsilon^{-1}T_{k}^{\frac{1}{2}}$ and
$\rho_{2}:=8\epsilon T_{j}^{\frac{1}{2}}$ on which we have the following:

  1. 1.

the curvature is controlled by [Ban90, Proposition 3]: for all
$l\in\mathbb{N}$, there exists $C>0$ such that at distance
$\rho\in\big{[}\frac{\rho_{1}}{2},2\rho_{2}]$ from $p$, we have

$\rho^{2+l}\big{|}\nabla^{l}\operatorname{\textup{Rm}}\big{|}\leqslant\eta(\rho):=C\epsilon\left[\left(\frac{\rho_{1}}{\rho}\right)^{\beta_{1}}+\left(\frac{\rho}{\rho_{2}}\right)^{\beta_{2}}\right],$
(14)

  2. 2.

by Proposition 2.1 together with Lemma 1.11, there exists $\delta(\epsilon)$
such that $\delta(\epsilon)\to 0$ when $\epsilon\to 0$ and a diffeomorphism
$\Phi:A_{e}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}\to\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}$,
where
$A\Big{(}2\rho_{1}+\delta,\frac{1}{2}\rho_{2}-\delta\Big{)}\subset\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}\subset
A\Big{(}2\rho_{1}-\delta,\frac{1}{2}\rho_{2}+\delta\Big{)}$, such that for all
$\rho\in[2\rho_{1},\frac{1}{4}\rho_{2}]$, we have,

$\Big{\|}\frac{\Phi^{*}_{\rho}g}{\rho^{2}}-g_{e}\Big{\|}_{C^{l}(A_{e}(1,2))}\leqslant
C\delta(\epsilon),$

  3. 3.

by Lemma 2.9, there exists therefore a foliation of
$\hat{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}$ by hypersurfaces
$\Sigma_{s}$ whose second fundamental form is almost constant in
$C^{\alpha}$-norm,

  4. 4.

by Lemma 2.3 and Lemma 2.10 there exists $r_{0}>0$ such that the injectivity
radius at each point $x$ is bounded below by $r_{0}d(x,p)$, and the normal
injectivity radius to a $\Sigma_{s}$ by Lemma 2.9 is bounded below by
$r_{0}s$.

The control (14) means that in the middle of the annulus, we have a much
better control than initially expected since the classical
$\epsilon$-regularity theorems would only give
$\rho^{2}|\operatorname{\textup{Rm}}|\leqslant
C\|\operatorname{\textup{Rm}}\|_{L^{2}(A(\rho_{1},\rho_{2}))}$. From the point
of view of analysis, the difference is substantial as it allows one to do
analysis in weighted Hölder spaces, and this will be heavily used in [Ozu19b].

######  Remark 5.4.

Analogous estimates hold in any dimension $n$ by assuming some much less
natural $L^{\frac{n}{2}}$-controls on the Riemann curvature.

The main difficulty in constructing good coordinates here is that the interior
radius, $\rho_{1}$, is neglectible when compared to the radius at which our
controls are optimal,
$\bar{\rho}:=(\rho_{1}^{\beta_{1}}\rho_{2}^{\beta_{2}})^{\frac{1}{\beta_{1}+\beta_{2}}}$
which minimizes $\eta$. Constant mean curvature hypersurfaces are convenient
in this context as they can be constructed in the entire zone _a priori_ , and
can then be controlled by Proposition 3.2 _a posteriori_.

######  Proposition 5.5.

For all $n\in\mathbb{N}$, there exists $\epsilon_{0}>0$, and $C>0$ such that
for all $0<\epsilon<\epsilon_{0}$, if $(M,g)$ is an Einstein manifold which is
$\epsilon$-approximated by a tree of singularities, in which we therefore have
$\mathcal{A}_{k}(t,\epsilon)$ an intermediate annulus included in a metric
annulus $A(4\rho_{1},\frac{1}{4}\rho_{2})$ such that
$\int_{A(\rho_{1},\rho_{2})}|\operatorname{\textup{Rm}}|^{2}dv<\epsilon^{2}$.
Then, there exists a foliation of a region
$\tilde{A}\big{(}2\rho_{1},\frac{1}{2}\rho_{2}\big{)}$ bounded by two
hypersurfaces of mean curvature respectively constant equal to
$\frac{n-1}{2\rho_{1}}$ and $\frac{n-1}{\frac{1}{2}\rho_{2}}$,

$\mathcal{A}_{k}(t,\epsilon)\subset
A\Big{(}(2+C\epsilon)\rho_{1},\Big{(}\frac{1}{2}-C\epsilon\Big{)}\rho_{2}\Big{)}\subset\tilde{A}\big{(}2\rho_{1},\frac{1}{2}\rho_{2}\big{)}\subset
A\Big{(}(2-C\epsilon)\rho_{1},\Big{(}\frac{1}{2}+C\epsilon\Big{)}\rho_{2}\Big{)},$

by hypersurfaces which we will denote $\tilde{\Sigma}_{s}$ if their mean
curvature equals $\frac{n-1}{s}$.

###### Proof.

Let us perturbate the hypersurfaces $\Sigma_{s}$, images of the spheres
$(s,\mathbb{S}^{n-1}/\Gamma)\subset\mathbb{R}^{+}\times\mathbb{S}^{n}/\Gamma$
by the diffeomorphism of Proposition 2.1 for
$s\in\big{[}2\rho_{1},\frac{1}{2}\rho_{2}\big{]}$. By Lemma 2.9, their metrics
are close to the sphere metrics: for all $l\in\mathbb{N}$, there exists $C>0$
and a diffeomorphism $\Psi_{\Sigma}:\mathbb{S}^{n-1}/\Gamma\to\Sigma$ such
that

$\left\|\frac{\Psi_{\Sigma}^{*}g^{\Sigma_{s}}}{s^{2}}-g^{\mathbb{S}^{n-1}/\Gamma}\right\|_{C^{l}(\mathbb{S}^{n-1}/\Gamma)}\leqslant
C\delta.$

They are moreover in a region where the curvature is smaller than $C\delta$
for a constant $C=C(n)>0$, the injectivity radius of their points is bounded
below by Lemma 2.3, and their normal injectivity by Lemma 2.10. Choosing
$\epsilon$ (and therefore $C\delta(\epsilon)$) small enough, we are under the
assumptions of Proposition 4.1, and for any
$s\in\big{[}2\rho_{1},\frac{1}{2}\rho_{2}\big{]}$, there exists a hypersurface
$\tilde{\Sigma}_{s}$ with mean curvature constant equal to $\frac{1}{s}$ which
is a normal perturbation:

$\tilde{\Sigma}_{s}=\Sigma_{s}(w_{s})=\\{\gamma^{s}_{x}(w_{s}(x)),x\in\Sigma_{s}\\}.$

We now need to ensure that we indeed obtained a foliation. Given
$s_{0}\in\big{[}2(1+2r_{0})\rho_{1},\frac{1}{2(1+2r_{0})}\rho_{2}\big{]}$, and
$\tilde{\Sigma}_{s_{0}}$, the hypersurface whose mean curvature is constant
equal to $\frac{n-1}{s_{0}}$ obtained as a perturbation of
$\Phi(s_{0},\mathbb{S}^{n-1}/\Gamma)$, for
$s\in\big{[}\frac{s_{0}}{(1+2r_{0})},(1+2r_{0})s_{0}\big{]}$ and denote
$\hat{\Sigma}^{s_{0}}_{s}$ the hypersurface whose mean curvature is constant
equal to $\frac{n-1}{s}$ obtained by perturbation of the hypersurface
equidistant to $\tilde{\Sigma}_{s_{0}}$ as described in Proposition 5.1.

The assumptions of Proposition 5.1 are satisfied because our coordinates are
close enough to the flat metric. Since the two hypersurfaces
$\hat{\Sigma}^{s_{0}}_{s}$ and $\tilde{\Sigma}_{s_{0}}$ have the same mean
curvature and are both small normal perturbations of $\tilde{\Sigma}_{s_{0}}$
by Lemma 2.11, they are equal by the uniqueneness of Proposition 5.1. Since
the hypersurfaces $\hat{\Sigma}_{s}^{s_{0}}$ are disjoint, so are the
hypersurfaces $\tilde{\Sigma}_{s}$. ∎

##  6 Construction of coordinates and control on the metric

To construct a diffeomorphism, we will use the constant mean curvature
hypersurface where our estimates are optimal and follow the gradient lines of
the function $s:=H(\tilde{\Sigma}_{s})$ whose level sets are the hypersurfaces
$\tilde{\Sigma}_{s}$. The main result is the following

######  Proposition 6.1.

Under the same assumptions as Proposition 5.5, there exists $\beta_{1}>0$,
$\beta_{2}>0$, $\Gamma\neq\\{e\\}$ a finite subgroup of $SO(4)$ such that for
all $l\in\mathbb{N}$, there exists $C>0$, and a diffeomorphism:

$\Phi:A_{e}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)}\to\tilde{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)},$

such that for all $\rho\in[2\rho_{1},\frac{1}{4}\rho_{2}]$, denoting
$\Phi_{\rho}:=\Phi\circ\phi_{\rho}:A_{e}(1,2)\to\tilde{A}\Big{(}2\rho_{1},\frac{1}{2}\rho_{2}\Big{)},$
where $\phi_{\rho}$ is the homothetic transformation of ratio $\rho$ on
$\mathbb{R}^{4}/\Gamma$, we have:

$\Big{\|}\frac{\Phi^{*}_{\rho}g}{\rho^{2}}-g_{e}\Big{\|}_{C^{l}(A_{e}(1,2))}\leqslant
C\epsilon\left(\Big{(}\frac{\rho_{1}}{\rho}\Big{)}^{\beta_{1}}+\Big{(}\frac{\rho}{\rho_{2}}\Big{)}^{\beta_{2}}\right).$

######  Remark 6.2.

Our proofs again work in any dimension $n\in\mathbb{N}$ for metrics with a
region close to a flat cone $\mathbb{R}^{n}/\Gamma$ (for
$\Gamma\neq\mathrm{Id}$) where the curvature is small in $L^{\frac{n}{2}}$.

###  6.1 Coordinates based on our constant mean curvature foliation

In the foliation constructed in Proposition 5.5, one particular hypersurface
is better controlled than the others at its scale. It is
$\tilde{\Sigma}_{\bar{\rho}}$ where
$\eta(\bar{\rho})=\min_{\rho_{1}<\rho<\rho_{2}}\eta(\rho)$, which satisfies
the following properties by (14) and 3.2:

  1. 1.

there exists a diffeomorphism
$\phi:\mathbb{S}^{n-1}/\Gamma\to\tilde{\Sigma}_{\bar{\rho}}$ such that: for
all $k\in\mathbb{N}$ and $0<\alpha<1$, there exists $C=C(k,\alpha,n,r_{0})>0$
for which,

$\Big{\|}\frac{\phi^{*}g^{\tilde{\Sigma}_{\bar{\rho}}}}{\bar{\rho}^{2}}-g^{\mathbb{S}^{n}/\Gamma}\Big{\|}_{C^{k,\alpha}(g^{\mathbb{S}^{n}/\Gamma})}<C\eta(\bar{\rho}),$

  2. 2.

for all $k\in\mathbb{N}$ and $0<\alpha<1$, there exists $C=C(k,\alpha,n)>0$
such that

$\Big{\|}\frac{A^{\tilde{\Sigma}_{\bar{\rho}}}}{\bar{\rho}}-\frac{g^{\tilde{\Sigma}_{\bar{\rho}}}}{\bar{\rho}^{2}}\Big{\|}_{C^{k,\alpha}\big{(}\frac{g^{\tilde{\Sigma}_{\bar{\rho}}}}{\bar{\rho}^{2}}\big{)}}\leqslant
C\eta(\bar{\rho}).$

  3. 3.

for all $k\in\mathbb{N}$, there exists $C=C(k,n)>0$ such that on the annulus
$[\frac{1}{2}\bar{\rho},2\bar{\rho}]$ we have,

$\bar{\rho}^{2+k}\big{|}\nabla^{k}\operatorname{\textup{Rm}}\big{|}_{g}\leqslant
C\eta(\bar{\rho}).$

We then define a diffeomorphism
$\Phi:[\rho_{1},\rho_{2}]\times\mathbb{S}^{n-1}/\Gamma\to\tilde{A}(\rho_{1},\rho_{2})$
in the following way: at $\bar{\rho}$,

$\Phi(\bar{\rho},x):=\phi(x),$

where $\phi$ is the diffeomorphism of the above point $1$, and for all
$s\in[\rho_{1},\rho_{2}]$,

$\partial_{s}\Phi(s,x)=u_{s}(x)N^{\tilde{\Sigma}_{s}}\big{(}\Phi(s,x)\big{)}=-\textup{grad}\Big{(}\frac{1}{s}\Big{)},$

where $u_{s}$ is such that
$\Phi(s,\mathbb{S}^{n-1}/\Gamma)=\tilde{\Sigma}_{s}$. This is a diffeomorphism
on its image since two curves following the vector field
$-\textup{grad}\big{(}\frac{1}{s}\big{)}$, which never vanishes, cannot
intersect if they are not equal.

###  6.2 Expression of the metric in these coordinates

In the coordinates given by $\Phi$, the metric has the following form,

$\Phi^{*}g(s,x)=u_{s}(x)^{2}ds^{2}+s^{2}h_{s}(x),$

where $h_{s}$ is a metric on $\mathbb{S}^{n-1}/\Gamma$.

Let us take the following notations for the rest of the section in which we
will always work on $\mathbb{R}^{n}/\Gamma$:

  * •

$\phi_{s}(x)=\Phi(s,x)$, which implies that
$s^{2}h_{s}=\phi_{s}^{*}g^{\tilde{\Sigma}_{s}}$.

  * •

$H(s)=\phi_{s}^{*}H^{\tilde{\Sigma}_{s}}\equiv\frac{n-1}{s}$,

  * •

$A(s)=\phi_{s}^{*}A^{\tilde{\Sigma}_{s}}$,

  * •

$N(s)=\phi_{s}^{*}N^{\tilde{\Sigma}_{s}}$

  * •

$f_{i}(s)=d_{(s,.)}\Phi(0,e_{i})$, where the $e_{i}$ form an orthonormal basis
of $T_{x}\mathbb{S}^{n-1}/\Gamma$,

  * •

$\nabla^{s}=\phi_{s}^{*}\nabla^{\tilde{\Sigma}_{s}}$,

  * •

$\Delta^{s}=\phi_{s}^{*}\Delta^{\tilde{\Sigma}_{s}}$, where
$\Delta^{\tilde{\Sigma}_{s}}$ is the Laplace-Beltrami operator associated to
the metric $g^{\tilde{\Sigma}_{s}}$,

  * •

$\textup{grad}^{s}$ is the gradient for the metric $s^{2}h_{s}$,

  * •

$K(s)_{i}^{j}=g\big{(}\operatorname{\textup{Rm}}_{g}(N(s),f_{i}(s))f_{j}(s),N(s)\big{)}$.

We have the following variations for the different geometric quantities.

By [HP99, Theorem 3.2], given a variation
$\partial_{s}\Phi(s,x)=u_{s}N^{\tilde{\Sigma}_{s}}$, we have the following
variation formulas:

$\partial_{s}\big{(}s^{2}h_{s}\big{)}=2u_{s}A(s),$ (15)
$\partial_{s}N(s)=\textup{grad}^{s}u_{s},$ (16)
$\partial_{s}A(s)_{ij}=-\nabla^{s}_{i}\nabla^{s}_{j}u_{s}+\big{(}\sum_{k}A(s)_{ik}A(s)_{j}^{k}+K(s)_{i}^{j}\big{)},$
(17)
$\partial_{s}H(s)=-\Big{(}\Delta^{s}+\big{|}A(s)\big{|}^{2}+\operatorname{\textup{Ric}}\big{(}N(s),N(s)\big{)}\Big{)}u_{s}.$
(18)

###  6.3 Control of the metric in these coordinates

Let us finally compare our Einstein metric to the flat metric in these
coordinates.

######  Proposition 6.3.

With the above notations, if we denote
$g_{e}:=ds^{2}+s^{2}g^{\mathbb{S}^{n-1}}$, we have the following controls on
$\Phi^{*}g$: for all $l\in\mathbb{N}$, there exists $C(l,n,v_{0},D_{0})>0$
such that for all $s\in\big{[}\rho_{1},\frac{\rho_{2}}{2}\big{]}$, on the
annulus $[s,2s]$, we have

$\|u-1\|_{C^{l}(g_{e})}\leqslant C\eta(s),$

and

$\|h-g^{\mathbb{S}^{n-1}/\Gamma}\|_{C^{l}(g_{e})}\leqslant C\eta(s),$

and therefore, for all $s\in[\rho_{1},\frac{\rho_{2}}{2}]$, on the annulus of
radii $s$ and $2s$,

$\|\Phi^{*}g-g_{e}\|_{C^{l}(g_{e})}\leqslant C\eta(s).$

###### Proof.

All along the proof, $C$ will denote a positive constant that may vary from
line to line, but which only depends on the constants $n$, $v_{0}$, $D_{0}$
and the order of regularity $l\in\mathbb{N}$, $0<\alpha<1$.

Let us start by noting that the equation (18) implies that for all $s$,
$u_{s}$ satisfies the following equation

$\frac{n-1}{s^{2}}=\Big{(}\Delta^{s}+\big{|}A(s)\big{|}^{2}+\operatorname{\textup{Ric}}(s)\big{(}N(s),N(s)\big{)}\Big{)}u_{s}.$
(19)

Indeed, by following the normal perturbation
$u_{s}N^{\tilde{\Sigma}_{s}}=-\textup{grad}\big{(}\frac{1}{s}\big{)}$, the
variation of the mean curvature is exactly $-\frac{n-1}{s^{2}}$. In
particular, for all $s$, we have

$\Big{(}\Delta^{s}+\big{|}A(s)\big{|}^{2}+\operatorname{\textup{Ric}}\big{(}N(s),N(s)\big{)}\Big{)}(u_{s}-1)=\frac{n-1}{s^{2}}-\big{|}A(s)\big{|}^{2}-\operatorname{\textup{Ric}}(s)\big{(}N(s),N(s)\big{)},$
(20)

According to Proposition 3.2, while
$\|h_{s}-g^{\mathbb{S}^{n-1}/\Gamma}\|_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma})}$
is sufficiently small, there exists $C=C(l)>0$ such that for all $s$,

$\Big{\|}\frac{A(s)}{s}-h_{s}\Big{\|}_{C^{l,\alpha}(h_{s})}\leqslant
C\eta(s).$ (21)

Since
$s^{2}\big{|}\operatorname{\textup{Ric}}(s)\big{(}N(s),N(s)\big{)}\big{|}<\eta(s)$,
this implies that the right hand side of (20) satisfies: there exists
$C=C(\alpha,n)>0$ such that for all $s$,

$\Big{\|}n-1-s^{2}\big{|}A(s)\big{|}^{2}-s^{2}\operatorname{\textup{Ric}}(s)\big{(}N(s),N(s)\big{)}\Big{\|}_{C^{\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant
C\eta(s).$

Likewise, for the left hand side, for any function
$v:\mathbb{S}^{n-1}/\Gamma\to\mathbb{R}$ we have

$\displaystyle\Big{\|}s^{2}\Big{(}\Delta^{s}+\big{|}A(s)\big{|}^{2}+\operatorname{\textup{Ric}}(s)\big{(}N(s),N(s)\big{)}\Big{)}v-$
$\displaystyle\;\Big{(}\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1\Big{)}v\Big{\|}_{C^{\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}$
$\displaystyle\;\leqslant
C\eta(s)\|v\|_{C^{2,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}.$

Since $\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1$ is invertible for
$\Gamma\neq\mathrm{Id}$, we get by the inverse function theorem, Lemma 4.6,
the following control: there exists $C=C(n,\alpha)>0$ such that for all $s$

$\|u_{s}-1\|_{C^{2,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant C\eta(s).$
(22)

Let us now show that for all $s\in[\rho_{1},\rho_{2}]$,
$\|h_{s}-g^{\mathbb{S}^{n-1}/\Gamma}\|_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}$
remains small. We will actually show that there exists $C=C(n,\alpha)>0$ such
that
$\|h_{s}-g^{\mathbb{S}^{n-1}/\Gamma}\|_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant
C\eta(s)$. The expression (15) can be rewritten

$\partial_{s}\big{(}h_{s}-g^{\mathbb{S}^{n-1}/\Gamma}\big{)}=\frac{2}{s}\Big{(}u\frac{A^{s}}{s}-h_{s}\Big{)},$

but by the control (22), and Proposition 3.2, there exists $C=C(n,\alpha)>0$
such that we have
$\Big{\|}u\frac{A^{s}}{s}-h_{s}\Big{\|}_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant
C\eta(s)$. Recall that

$\eta(s):=C\epsilon\Big{[}\Big{(}\frac{\rho_{1}}{s}\Big{)}^{\beta_{1}}+\Big{(}\frac{s}{\rho_{2}}\Big{)}^{\beta_{2}}\Big{]},$

to deduce that for $\bar{\rho}<s_{0}<\rho_{2}$, we have

$\displaystyle\big{\|}h_{s_{0}}-g^{\mathbb{S}^{n-1}/\Gamma}\big{\|}_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant$
$\displaystyle\;C\Big{(}\eta(\bar{\rho})+\int_{\bar{\rho}}^{s_{0}}\frac{2}{s}\eta(s)ds\Big{)}$
$\displaystyle=$
$\displaystyle\;C\eta(\bar{\rho})+\Big{(}\frac{2C\epsilon}{\rho_{2}^{\beta_{2}}}\int_{\bar{\rho}}^{s_{0}}s^{\beta_{2}-1}ds\Big{)}$
$\displaystyle=$
$\displaystyle\;C\epsilon\frac{\bar{\rho}^{\beta_{2}}}{\rho_{2}^{\beta_{2}}}+\Big{(}\frac{2C\epsilon}{\rho_{2}^{\beta_{2}}}(s_{0}^{\beta_{2}}-\bar{\rho}^{\beta_{2}})\Big{)}$
$\displaystyle\leqslant$ $\displaystyle\;C\eta(s_{0}).$

and similarly, for $\rho_{1}<s_{0}<\bar{\rho}$,

$\big{\|}h_{s_{0}}-g^{\mathbb{S}^{n-1}/\Gamma}\big{\|}_{C^{1,\alpha}(g^{\mathbb{S}^{n-1}/\Gamma)}}\leqslant
C\eta(s_{0}).$

To obtain controls on higher derivatives of $u$ and $h_{s}$, we use the other
equalities of [HP99, Theorem 3.2]: according to (16), and the control (22),
there exists $C=C(n,\alpha)$ such that for all $s$ we have

$\|\partial_{s}N(s)\|_{C^{1,\alpha}(h_{s})}\leqslant C\eta(s)$ (23)

according to (17) and the controls (22), (21) and (23), and since the $l$-th
derivatives of the curvature are bounded by $s^{-2-l}\eta(s)$, there exists
$C=C(n,\alpha)>0$ such that for all $s$, we have:

$\Big{\|}\partial_{s}\Big{(}\frac{A(s)}{s}\Big{)}\Big{\|}_{C^{\alpha}(h_{s})}\leqslant
C\eta(s).$ (24)

Differentiating the equality (18) with respect to $s$, and using the
inequalities $\|\partial_{s}h_{s}\|_{C^{1,\alpha}}\leqslant C\eta(s)$, and
(24), there exists $C=C(n,\alpha)>0$ such that for all $s$, we have:

$\Big{\|}\Big{(}\Delta^{\mathbb{S}^{n-1}/\Gamma}+n-1\Big{)}\partial_{s}u_{s}\Big{\|}_{C^{\alpha}(h_{s})}\leqslant
C\eta(s).$

We can therefore conclude that for all $s$,
$\|\partial_{s}u_{s}\|_{C^{2,\alpha}(h_{s})}\leqslant C\eta(s)$.

Iterating this for higher derivatives in all directions, we obtain the stated
controls. ∎

###  6.4 Proximity between the Einstein metric and the naïve desingularization
in weighted $C^{k}$-norm

Now, to construct coordinates on the whole manifold $M$, we just "glue" the
coordinates we have on each part together by using the local diffeomorphisms
with the common asymptotic cone of each part. We obtain the following control
for the metric.

######  Theorem 6.4.

Let $D_{0},v_{0}>0$ and $(M^{\mathcal{E}}_{i},g^{\mathcal{E}}_{i})$ a sequence
of Einstein manifolds satisfying

  * •

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

  * •

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

  * •

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

Then, there exists a subsequence with fixed topology $M=M_{i}$ and a sequence
of naïve desingularizations $(M,g^{D}_{\phi,t_{i}})$ of an Einstein orbifold
$(M_{o},g_{o})$ such that: for all $l\in\mathbb{N}$ and $i$ large enough,
there exists $C(l,v_{0},D_{0})>0$, $\beta_{1}(v_{0},D_{0})>0$ and
$\beta_{2}(v_{0},D_{0})>0$, $\epsilon_{i}>0$, $\epsilon_{i}\to 0$, and a
diffeomorphism $\Phi_{i}:M\to M$ satisfying

  1. 1.

on $\mathcal{M}_{o}(\epsilon_{i})$, we have

$\Big{\|}\Phi^{*}_{i}g^{\mathcal{E}}_{i}-g_{o}\Big{\|}_{C^{l}(g_{o})}\leqslant
C\epsilon_{i}.$

  2. 2.

at the different scales $T_{i,j}$ (associated to $t_{i}$), on
$\mathcal{N}_{j}(\epsilon_{i})$ we have

$\Big{\|}\frac{\Phi^{*}_{i}g^{\mathcal{E}}_{i}}{T_{i,j}}-g_{b_{j}}\Big{\|}_{C^{l}(g_{b_{j}})}\leqslant
C\epsilon_{i}.$

  3. 3.

in the intermediate regions, $\mathcal{A}_{k}(t_{i},\epsilon_{i})$ included in
a metric annulus of rarii
$\rho_{1}^{i,k}=\frac{1}{8}T_{i,k}^{\frac{1}{2}}\epsilon_{i}^{-1}$,
$\rho_{2}^{i,k}=8T_{i,j}^{\frac{1}{2}}\epsilon_{i}$, for all
$\rho_{1}^{i,k}\leqslant\rho\leqslant\frac{1}{2}\rho_{2}^{i,k}$, and denoting
$\Phi_{i,\rho}=\Phi_{i}\circ\phi_{\rho}$, where $\phi_{\rho}$ is the
homothetic transformation of ration $\rho$ on $\mathbb{R}^{4}/\Gamma_{k}$ with
its flat metric $g_{e}$, we have

$\Big{\|}\frac{\Phi_{i,\rho}^{*}g^{\mathcal{E}}_{i}}{\rho^{2}}-g_{e}\Big{\|}_{C^{l}(A_{e}(1,2))}\leqslant
C\epsilon_{i}\Big{[}\Big{(}\frac{\rho_{1}^{i,k}}{\rho}\Big{)}^{\beta_{1}}+\Big{(}\frac{\rho}{\rho_{2}^{i,k}}\Big{)}^{\beta_{2}}\Big{]}.$

######  Remark 6.5.

This is indeed much better than a $C^{\infty}$ convergence on compacts to a
flat cone as we have
$\Big{[}\Big{(}\frac{\rho_{1}^{i,k}}{\rho}\Big{)}^{\beta_{1}}+\Big{(}\frac{\rho}{\rho_{2}^{i,k}}\Big{)}^{\beta_{2}}\Big{]}\leqslant
2$, and a decay in the annulus.

##  Appendix A Optimal $C^{k,\alpha}$-proximity to a round sphere

In this appendix, we are interested in metrics with pinched positive sectional
curvatures and want to estimate how close to a round metric they have to be.
To start with, a consequence of Cheeger-Gromov compactness is the following
lemma.

######  Lemma A.1.

For all $n$, $0<\alpha<1$, $\delta>0$ and $v_{0}>0$, there exists $\epsilon>0$
such that if $(M^{n},g)$ satisfies

  1. 1.

$1-\epsilon<Sec(g)<1+\epsilon$,

  2. 2.

and $\operatorname{\textup{Vol}}(M)>v_{0}$,

then, there exists $\Gamma$ a finite subgroup of $SO(n+1)$ (whose order is
bounded by a function of $v_{0}$), and a diffeomorphism
$\Phi:\mathbb{S}^{n}/\Gamma\to M$ for which:

$\big{\|}\Phi^{*}g-g^{\mathbb{S}^{n}/\Gamma}\big{\|}_{C^{1,\alpha}}\leqslant\delta,$

where $g^{\mathbb{S}^{n}/\Gamma}$ is a round metric on
$\mathbb{S}^{n}/\Gamma$.

In this appendix, we will prove the following proposition.

######  Proposition A.2.

Let $(M,g_{0})$ be a Riemannian manifold of dimension $n$ with sectional
curvatures equal to $1$, and $k\in\mathbb{N}$. Then, there exists $\delta>0$
and $C>0$, such that if a metric $g$ on $M$ satisfies:

$\|g-g_{0}\|_{C^{k+1,\alpha}}\leqslant\delta,$

then, there exists a diffeomorphism $\Phi:M\to M$, for which

$\|\Phi^{*}g-g_{0}\|_{C^{k+1,\alpha}}\leqslant
C\big{\|}\operatorname{\textup{Ric}}(g)-(n-1)g\big{\|}_{C^{k}}.$

As an immediate consequence of Lemma A.1 and Proposition A.2, we get the
following

######  Corollary A.3.

For all $n$, $0<\alpha<1$ and $v_{0}>0$, there exists $\epsilon_{0}>0$ and
$C>0$ such that for any $0<\epsilon<\epsilon_{0}$, if $(M^{n},g)$ satisfies:

  1. 1.

$1-\epsilon<Sec(g)<1+\epsilon$,

  2. 2.

and $\operatorname{\textup{Vol}}(M,g)>v_{0}$,

then, there exists $\Gamma$ a finite subgroup of $SO(n+1)$, and a
diffeomorphism $\Phi:\mathbb{S}^{n}/\Gamma\to M$ for which

$\|\Phi^{*}g-g_{0}\|_{C^{1,\alpha}}\leqslant C\epsilon.$

######  Remark A.4.

The crucial part for us is that the $\epsilon$ in the final estimate is the
same as in the assumption $1-\epsilon<Sec(g)<1+\epsilon$.

###### Proof of Proposition A.2.

Let $(M,g_{0})$ be a Riemannian manifold whose sectional curvatures are
constant equal to $1$. For $\lambda:=\frac{(n-2)(n-1)}{2}$, define the
operator

$E_{g_{0}}(g):=\operatorname{\textup{Ric}}(g)-\frac{\operatorname{\textup{R}}(g)}{2}g+\lambda
g+\delta^{*}_{g}\delta_{g_{0}}g.$

Then, following [And10, Section 5], by [Ebi70] and by the Bianchi identity,
there exists $\delta>0$ such that $E_{g_{0}}^{-1}(\\{0\\})\cap
B_{C^{1,\alpha}}(\delta)$ is exactly the set of Einstein metrics with constant
$\Lambda$ which are in divergence-free gauge with respect to $g_{0}$ and have
same volume.

Let $g$ be a metric on $M$ such that
$\|g-g_{0}\|_{C^{k+1,\alpha}}\leqslant\delta$ for $\delta>0$ a constant which
we will choose small enough in the rest of the proof. According to [Ebi70],
for $\delta$ small enough, there exists a diffeomorphism $\Phi:M\to M$ such
that

$\delta_{g_{0}}\Phi^{*}g=0.$

Hence, we have

$\|E_{g_{0}}(\Phi^{*}g)\|_{C^{k}}\leqslant\big{\|}\operatorname{\textup{Ric}}(g)-(n-1)g\big{\|}_{C^{k}}.$
(25)

Now, by [Bes87, Corollary 12.72 (Bourguignon, unpublished)], the linearization
of $E$ at $g_{0}$ is invertible (see again [And10, Section 5] for the
adaptation to the operator $E$) and by the inverse function theorem Lemma 4.6
we deduce that for $\delta$ small enough, and for all $1\leqslant p<+\infty$
and $k\in\mathbb{N}$ there exists $C>0$, such that we have

$\|\Phi^{*}g-g_{0}\|_{W^{k+2,p}}\leqslant
C\|E_{g_{0}}(\Phi^{*}g)\big{\|}_{W^{k,p}},$

and finally, by (25), we have

$\|\Phi^{*}g-g_{0}\|_{C^{k+1,\alpha}}\leqslant
C\|\operatorname{\textup{Ric}}(g)-(n-1)g\big{\|}_{C^{k}},$

because on the sphere, for $p$ large enough, $W^{k+2,p}$ embeds continuously
in $C^{k+1,\alpha}$, and so does $C^{k}$ in $W^{k,p}$. ∎

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