Noncollapsed degeneration of Einstein 4-manifolds I
Tristan Ozuch


    
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. 


Localization of Compact Binary Sources with Second Generation Gravitational-wave Interferometer Networks
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    December 30, 2023
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§ 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 Λ, the equation

    (g)=Λ g,

where  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↦∫_M _g dvol_g, where  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↦∫_M |_g|^2dvol_g. Indeed, By Chern-Gauss-Bonnet formula, we have

    ∫_M |_g|^2dv_g = 8π^2 χ(M) +∫_M |^0_g|^2dv_g,

where χ is the Euler characteristic and where ∫_M |^0_g|^2dv_g⩾ 0 vanishes if and only if g is Einstein. Notice that in this case, the quantity ∫_M |_g|^2dv_g = 8π^2 χ(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 <cit.> 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. 

 Let (M^4_i,g_i)_i∈ℕ be a sequence of Einstein 4-manifolds satisfying,


  * the diameters of the (M^4_i,g_i) are uniformly bounded,

  * the volumes of the (M^4_i,g_i) are uniformly bounded from below,

  * the integrals ℰ_i:= ∫_M^4_i|(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 <ref>), (M^4_∞,g_∞).
 
 Moreover, for any singular point p_∞∈ M_∞, there exists a sequence (p_i)_i∈ℕ of points of M_i^4 with 1/t_i:= |(g_i)|(p_i)→∞ such that (M^4_i,g_i/t_i,p_i) converges in the pointed Gromov-Hausdorff sense to a Ricci-flat asymptotically locally Euclidean (ALE) manifold, see Definition <ref>.



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 <cit.>, is a topological condition by Chern-Gauss-Bonnet formula.


This theorem leaves several questions open, in particular the following one. 
[<cit.>]
    "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 <ref>]"

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 perturb 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 do not have any L^2-infinitesimal Einstein deformation) asymptotically hyperbolic Einstein manifolds with one singularity ℝ^4ℤ_2. This is realized by a particular gluing procedure of an Eguchi-Hanson metric to the singularity, see <cit.>. 

Strikingly, this particular desingularization is only possible if an obstruction is satisfied: denoting 𝐑 the Riemannian curvature seen as an endomorphism on 2-forms, the orbifold must satisfy 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 <cit.> holds under much less assumptions. The least satisfactory 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 <cit.> cannot be true in general if one drops any of the following assumption:

    
  * the orbifold is rigid (it does not admit infinitesimal Einstein deformations),
    
  * there is only one singular point,
    
  * the singularity is ℝ^4ℤ_2.

The reasons are the following:

    
  * the convergence speed towards the orbifold could be arbitrarily slow compared to the singularity formation,
    
  * different singularities may form at different speed,
    
  * 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 symmetric 2-tensors decaying in the neck regions. These spaces are necessarily different from <cit.> by the above reasons. The main result is the following theorem.


	Let D_0,v_0>0, l∈ℕ, then, there exists δ = δ(D_0,v_0,l) >0 such that if (M,g^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 ||≤ 3.
	
	and for which there exists an Einstein orbifold (M_o,g_o) with
	
    d_GH((M,g^ℰ),(M_o,g_o))⩽δ,

	then, (M,g^ℰ) is the result of a gluing-perturbation procedure detailed in <cit.>.

Notice that we are exactly in the context of Theorem <ref>, 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 <ref> actually implies that there is a satisfactory C^∞-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^∞-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, 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 <cit.> and the Kähler Ricci-flat ALE spaces which are quotients of gravitational instantons have been classified in <cit.>. 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.


	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 in the positive orientation.
	
	Then, at any singular point p of (M_o,g_o), we have
	
    𝐑_g_o(p) =0.



The condition on the singularity model can be reformulated as a topological assumption by <cit.> and we have the following example.
[<cit.>]
    Consider 𝕊^4ℤ_2 the Einstein orbifold with two ℝ^4ℤ_2 singularities obtained as the quotient of 𝕊^4 by {±} in a global geodesic chart. Then, there exists a differentiable manifold M^4, such that for any 1⩽ p<∞, we can construct a sequence of metrics (M,g_i) with both 
    (g_i)-3g_i_L^p(g_i)→ 0,  and (g_i)⩾ 3 g_i,

    while 
    (M,g_i) (𝕊^4ℤ_2,g_𝕊^4ℤ_2),

    but there does not exist any sequence of Einstein metrics with
    
    (g_i)=3g_i,
 and 
    (M,g_i) (𝕊^4ℤ_2,g_𝕊^4ℤ_2).



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


    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 
    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 <ref> 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 appropriate 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 ϵ-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 <cit.>.



 §.§ 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.



§ TREES OF SINGULARITIES AND NAÏVE DESINGULARIZATIONS


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



 §.§ Einstein orbifolds and Ricci-flat ALE orbifolds


The singular limit spaces in Theorem <ref> are Einstein orbifolds (with isolated singularities).


    We will say that a metric space (M_o,g_o) is an orbifold of dimension n∈ℕ if there exists ϵ_0>0 and a finite number of points (p_k)_k of M_o called singular such that we have the following properties:
    
        
  * the space (M_o\{p_k}_k,g_o) is a manifold of dimension n,
        
  * for each singular point p_k of M_o, there exists a neighborhood of p_k, U_k⊂ M_o, a finite subgroup acting freely on 𝕊^n-1, Γ_k⊂ SO(n), and a diffeomorphism Φ_k: B_e(0,ϵ_0)⊂ℝ^nΓ_k → U_k⊂ M_o for which, the pull-back of Φ_k^*g_o  on the covering ℝ^n is smooth.
    



    Orbifold metrics will most of the time be indexed by a little o for orbifold.


Theorem <ref> 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.


    An ALE orbifold of dimension n∈ℕ, (N,g_b) is a metric space for which there exists ϵ_0>0, singular points (p_k)_k and a compact K⊂ N for which we have:
    
        
  * (N,g_b) is a orbifold of dimension n,
        
  * there exists a compact subset K⊂ N and a diffeomorphism Ψ_∞: (ℝ^nΓ_∞)\ B_e(0,ϵ_0^-1) → N\ K such that we have 
    r_e^l|∇^l(Ψ_∞^* g_b - g_e)|_C^2(g_e)⩽ C_l r_e^-n.

    


    ALE metrics will most of the time be indexed by a little b for bubble.




 §.§ Naïve desingularizations of an orbifold by trees of singularities



By <cit.>, under the assumptions of Theorem <ref>, at each singular point, a tree of Ricci-flat orbifolds forms. We index them by b_j for j-th bubble.


    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 ℝ^4Γ_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↦ p_j∈ S_o∪⋃_k S_b_k, where the singularity at p_j is ℝ^4Γ_j. We will call D:= ((M_o,g_o,S_o),(N_j,g_b_j,S_b_j)_j, p) a tree of singularities or a desingularization pattern for (M_o,g_o) depending on the point of view.


Let us define metrics which will mimic the degeneration of Einstein manifolds of Theorem <ref>, 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 ϵ_0>0 of Definitions <ref> and <ref>.

    Let 0<2ϵ<ϵ_0, 0<t<ϵ^4, and (M_o,g_o) be an orbifold and Φ: B_e(0,ϵ_0)⊂ℝ^4Γ→ U a local chart around a singular point p∈ M_o which satisfies Φ^*g_o = g_e + 𝒪(r_e^2). Let also (N,g_b) be an ALE orbifold asymptotic to ℝ^4Γ, and Ψ_∞: (ℝ^4Γ_∞)\ B_e(0,ϵ_0^-1) → N\ K a chart at infinity in wich Ψ_∞^* g_b = g_e + 𝒪(r_e^-4). Consider finally χ a C^∞-function from ℝ^+ to ℝ^+ supported on [0,2] and equal to 1 on [0,1] and for all s>0, define ϕ_s: x∈ℝ^4Γ→ sx ∈ℝ^4Γ.
    
    We then define the orbifold M_o#N as N glued to M_o thanks to the diffeomorphism Φ∘ϕ_√(t)∘Ψ^-1 : Ψ(A_e(ϵ_0^-1,ϵ_0t^-1/2))→Φ(A_e(ϵ_0^-1√(t),ϵ_0)) which identifies their annuli Φ(A_e(ϵ_0^-1√(t),ϵ_0)) and Ψ(A_e(ϵ_0^-1,ϵ_0t^-1/2)).



    We define a naïve desingularization of an orbifold (M_o,g_o) at a singular point p by an ALE metric (N,g_b) at scale t, which we will denote (M_o#N,g_o#_p,tg_b) by putting g_o#_p,tg_b=g_o on M_o\ U, g_o#_p,tg_b=tg_b on K⊂ N, and 
    
    g_o#_p,tg_b = χ(t^-1/4r_e)Φ^*g_o + (1-χ(t^-1/4r_e)) ϕ_t^1/2,*Ψ_∞^*g_b

    on A(t,ϵ):=A_e(ϵ^-1√(t),bϵ).


More generally, it is possible to desingularize iteratively by trees of Ricci-flat ALE orbifolds. Let 0<2ϵ<ϵ_0, (M_o,g_o) be an Einstein orbifold and (N_j,g_b_j)_j be a tree of ALE Ricci-flat orbifolds such that M = M_o#_jN_j is the result of a desingularization pattern D, and let 0<t_j<ϵ^4 be relative gluing scales.

 

    The naïve desingularization metric g^D_t on M = M_o#_jN_j 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= ∅. If p_j∈ N_k and we replace (N_k,g_b_k,S_j) and (N_j,g_b_j,∅) by (N_k#N_j,g_b_k#_p_j,t_jg_b_j,S_k\{p_j}) and restrict p as l→ p_l for l≠ j in D and consider another deepest bubble, the same works if p_j∈ M_o.


    Moreover, if N_j is glued to p_j∈ N_j_1, and N_j_1 is glued to p_j_1∈ 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_1t_j_2...t_j_k. This way, on each N_j(2ϵ), the metric is T_jg_b_j.


    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 ℝ^4Γ_k. There is therefore a gluing gauge ϕ_k∈ Isom(ℝ^4Γ_k) at each point. These choices are equivalent to gluing different ALE spaces and they lead to different metrics, hence we will not worry too much about this degree of freedom here.




 §.§ Coordinates on a naïve desingularization


The above metric g^D_t on M of Definition <ref> 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


With the notations of definitions <ref> and <ref>, for 0<ϵ⩽ϵ_0, we will denote

    
  * M_o(ϵ):= M_o\(⋃_k Φ_k(B_e(0,ϵ)) )⊂ M_o,
    
  * N_j(ϵ):= N_j\(⋃_k Ψ_k(B_e(0,ϵ)) ∪Ψ_∞((ℝ^4Γ_∞)\ B_e(0,ϵ^-1)))⊂ N_j,
    
  * A_j(t,ϵ):=A_e(ϵ^-1√(T_j)√(t_j),ϵ√(T_j))⊂ℝ^4Γ_k.

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




    The sets M_o(ϵ) and N_j(ϵ) are the compact sets of the orbifolds minus their singular points which appear in <cit.>. They are exhaustive as ϵ tends to zero.



They moreover form a covering of M = M_o#_jN_j adapted to a degeneration at relative scales (t_j)_j:

    M = ⋃_j(N_j(ϵ)∪ A_j(t,ϵ)) ∪   M_o(ϵ).





 §.§ Coordinates on an Einstein metric


The goal will be to identify each of these regions with a region of an Einstein manifold (M,g) close enough to an Einstein orbifold (M_o,g_o) in the Gromov-Hausdorff sense.


Let ϵ>0, l∈ℕ and fix t = (t_j)_j with t_j<ϵ^4 for all j. We will say that a Riemannian manifold (M,g) is ϵ-approximated in  C^l-norm by a naïve desingularization (M,g^D_t) of (M_o,g_o), if there exists a diffeomorphism Φ: M→ M such that if we denote Φ(M_o(ϵ)) = ℳ_o(ϵ)⊂ M and Φ(N_j(ϵ)) = 𝒩_j(ϵ)⊂ M and 𝒜_k(t,ϵ) the region of M\ℳ_o(32ϵ)∪⋃_j 𝒩_j(32ϵ) with nonempty intersection with Ψ_∞( B_e(0,ϵ^-1))⊂𝒩_k(ϵ):
	
	
  * on ℳ_o(ϵ) in M, we have
		
    Φ^*g-g_o_C^l(g_o)=Φ^*g-g^D_C^l(g^D)⩽ϵ,

	
  * on the zone 𝒩_j(ϵ) in M, we have
	
    Φ^*g/T_j-g_b_j_C^l(g_b_j) = Φ^*g-g^D_C^l(g^D)⩽ϵ,

	
  * and on 𝒜_k(t,ϵ), we have
	
    ∫_𝒜_k(t,ϵ)||^2dv_g<ϵ^2.

	


Under the assumptions of Theorem <ref>, given a sequence (M_i,g_i)_i of Einstein manifolds converging in the Gromov-Hausdorff sense to an Einstein orbifold (M_o,g_o), there exist a subsequence still denoted (M_i,g_i)_i and naïve desingularizations (M_i,g^D_t_i) such that for all ϵ>0 and l∈ℕ, for i large enough, the manifold (M_i,g_i) is ϵ-approximated in C^l-norm by (M_i,g^D_t_i).


Let (M_i,g_i)  (M_o,g_o) be a sequence of manifolds satisfying the assumptions of Theorem <ref>. 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 |(g_i)|(p_i)→∞ such that: (M^4_i,g_i/T_i,p_i) 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 ϵ>0 and l∈ℕ, by considering the compact N_j(ϵ), there exists i_j∈ℕ such that for all i>i_j, there is a diffeomorphism on its image Φ_i: N(ϵ)→ M_i with Φ_i^*g_i/T_i-g_b_j_C^l≤ϵ and similarly, there exists a rank i_o from which we have the C^l-closeness on M_o(ϵ). 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 <cit.>.


The main objective of this paper will be to construct good coordinates in the neck regions 𝒜_k(t,ϵ). These regions 𝒜_k(t,ϵ) between N_k and N_j are included in metric annuli A(1/8T_k^1/2ϵ^-1,8T_j^1/2ϵ) centered at p_k, and in A(1/8t_k^1/2ϵ^-1,8ϵ) 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.


    For any 0<δ<1, there exists ϵ>0 such that if a manifold (M,g) is ϵ-approximated by a naïve desingularization g^D_t in C^0-norm for t<ϵ^4 with (g)⩾ -3g, then, if 𝒜_k(t,ϵ)⊂ A(1/8T_k^1/2ϵ^-1,8T_j^1/2ϵ) = A(ρ_1,ρ_2), we have
    
    |_g(B_g(δ^-1ρ_2))/_g(B_g(δρ_1))-(δ^-1ρ_2)^4/(δρ_1)^4|⩽δ.



    Let us consider the case of an annular region 𝒜_k(t,ϵ) between a Ricci-flat ALE (N,g_b) and an orbifold (M_o,g_o), which are asymptotic to ℝ^4Γ, at scale 0<t<ϵ^4 for ϵ>0 which we will choose small enough. By definition of an ϵ-approximated metric, the ball (B(δ^-1ρ_2),g) is arbitrarily close in the Gromov-Hausdorff sense to the ball of radius δ^-1ϵ of (M_o,g_o) for ϵ arbitrarily small, and by definition of an orbifold metric, as δ^-1ϵ→ 0, we have 
    _g_o(B_g_o(δ^-1ϵ))/(δ^-1ϵ)^4→ω_4/|Γ|,

    the volume of the unit ball of ℝ^4Γ. 
    
    Similarly, the ball (B_g/t(δϵ^-1),g/t) is arbitrarily close in the Gromov-Hausdorff sense to the ball of radius δϵ^-1 of (N,g_b) whose volume satisfies for δ^-1ϵ→ 0
    
    _g_b(B_g_b(δϵ^-1))/(δϵ^-1)^4→ω_4/|Γ|.

    As a consequence, by the continuity of volume for the Gromov-Hausdorff distance between manifolds with ⩾- 3g, <cit.>, we have the stated result for ϵ small enough.



The goal is now to construct a diffeomorphism from a flat cone of (ℝ^nΓ,g_e) to our intermediate annulus. Thanks to the controls on the curvature of <cit.>, 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.

	

§ FIRST COORDINATES

The main difficulty here in constructing our coordinates when compared to the context of <cit.> 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 local coordinates which is only partially satisfactory. The main goal of this section is the proof of the following proposition.


		For all δ >0, D_0>0, v_0>0, E>0 and l∈ℕ, there exists ϵ_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(ρ_1,ρ_2) with 4ρ_1<ρ_2 satisfying: 
	
    ∫_A(ρ_1,ρ_2)||^2⩽ϵ_2^2,

	and ∫_B(ρ_2)||^2⩽ E,
	
  * the volume growth in the annulus is almost Euclidean 
    |(B(ρ_2))/(B(ϵ_2ρ_1))-ρ_2^4/(ϵ_2ρ_1)^4|⩽ϵ_2.

	
	Then, for any ρ∈[2ρ_1,1/4ρ_2] there exists a region Â(ρ,2ρ) close to the annulus A(ρ,2ρ) in the following sense,
	
    A((1+δ)ρ,(2-δ)ρ) ⊂Â(ρ,2ρ)⊂ A((1-δ)ρ,(2+δ)ρ),
 
	such that there exists Γ a finite subgroup of SO(4) and a diffeomorphism 
    Φ_ρ : A_e(1,2)⊂ℝ^4Γ→Â(ρ,2ρ)⊂ M,

	for which we have :
	
    Φ_ρ^*g/ρ^2-g_e_C^l(A_e(1,2))⩽δ.

	



 §.§ ϵ-regularity in the neck regions



	The C^k,α-norms will be taken at the scale of the injectivity radius. Which means that we define the C^k,α-norm of a tensor s in the following way:
	
    s_C^k,α:= s_C^0+...+∇_g^ks_C^0 + sup_x[∇_g^ks]_C^α(g)(x),

	where, for a tensor u, a point x, α>0 and a metric g, if we denote exp_x the exponential map at x whose injectivity radius is _g(x), we define the Hölder seminorm of u on M as
    
    [u]_C^α(g)(x):= sup_{y∈ T_xM,|y|< _g(x)}| exp_x^*u(0)-exp_x^*u(y)/|y|^α|_exp_x^*g.



We will also sometimes use the harmonic radius which is defined at x∈ M, as the supremum of the r>0 for which there exists a diffeomorphism Φ: B(x,r)→ℝ^n, Φ(y)=(h_1(y),...,h_n(y)) such that for all i∈{1,...,n}, 
    Δ_g h_i =0,

and denoting g_ij the push forward of g by Φ,

    g_ij-δ_ij_C^0+ r^1+αg_ij_C^1+α⩽1/100.



	Under the assumptions of Proposition <ref>, there exists a constant r_0>0 only depending on D_0 and v_0, such that for all points of A(2ρ_1,1/2ρ_2) at distance 2ρ_1<ρ<1/2ρ_2 from the center, the injectivity radius and the harmonic radius are bounded below by r_0 ρ.


Let us consider a point x at distance ρ∈ [2ρ_1,1/2ρ_2] from the center of the annulus and the ball B(x,ρ/2). Thanks to Bishop-Gromov inequality, the volume of B(x,ρ/2) is bounded from below by C_1ρ^4 for a constant C_1>0 only depending on v_0 and D_0. To control the curvature, we use the following ϵ-regularity theorem:

    In an Einstein manifold (M,g) satisfying the assumptions of Proposition <ref>, for any C_0>0 there exists a constant ϵ=ϵ(C_0,v_0,D_0) such that if at a point x∈ M and for ρ>0 we have _g(B_g(x,ρ))>C_0 ρ^4 and

    ∫_B(x,2ρ)|(g)|^2 dv(g) < ϵ^2,
 
then, we have

    |(g)|⩽1/ρ^2
 on B(x,ρ).


Therefore for ρ∈ [4ρ_1, 1/4ρ_2], up to shrinking the constant ϵ_2 of the statement to be smaller than ϵ(16C_1)>0, the ball B(x,ρ/2) satisfies (<ref>). Hence, the curvature at distance ρ is smaller than 1/ρ^2 by (<ref>).

By <cit.>, these controls on the curvature and the volume at scale ρ imply that there exists a constant r_1>0 depending on the volume of the ball of radius ρ and on a bound on the curvature at this scale such that at any point at distance ρ from the center of the annulus, the injectivity radius is bounded below by r_1ρ. 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_0ρ.
	



 §.§ 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.
		 
		For all δ>0, there exists κ>0 such that for any r>0, if (M^n,g) satisfies, (g) ⩾ -(n-1)κ r^-2 and there exists a point p∈ M at which,
		
    (B(p,κ r))/v_-κ r^-2(κ r)-(B(p,2r))/v_-κ r^-2(2r)<κ,
 
		where v_-κ r^-2(s) is the volume of the ball of radius s in the simply connected space with sectional curvatures constant equal to -κ r^-2,
	    then, there exists a metric cone (C(X),d_C(X),0) such that diam (X)⩽π and,
		
    d_GH((B(p,r),p),(B_C(X)(0,r),0))≤δ r.

		

		We can refine this result in the case of 4-dimensional Einstein manifolds.
		
		For all δ>0, l∈ℕ, v_0>0, and E>0, there exists κ>0 such that for all r>0, if (M^4,g) satisfies (g) = Λ g with, |Λ|⩽ 3κ r^-2 and if there exists a point p∈ M for which, 
		
    (B(p,r)) ⩾ v_0 r^4,

		
    (B(p,κ r))/v_-κ r^-2(κ r)-(B(2r))/v_-κ r^-2(2r)<κ,

		and 
    ∫_B(p,2r) ||^2dv ⩽ E,

		then, there exists a finite subgroup Γ⊂ SO(4) (with order bounded in terms of v_0), and a diffeomorphism
		Φ:A_e(δ , (1-δ )) →Â(δ r, (1-δ )r), where,
		  
    A(2δ r, (1-2δ )r) ⊂Â(δ r, (1-δ )r) ⊂ A(1/2δ r, (1-1/2δ)r),

		 such that on the annulus A_e(δ r, (1-δ )r), we have:
		 
    Φ^*g/r^2-g_e_C^l(g_e)⩽δ .

		

    The assumptions on the Riemannian curvature and the volume are there to bound the order of Γ. They prevent our Einstein manifolds to be too close to ℝ^4Γ_i where the order of Γ_i goes to infinity. In such a situation, the sequence could converge to the cone ℝ^+. These hypotheses are moreover redundant by <cit.>, the hypotheses on the Ricci curvature and the volume imply a bound on the L^2-norm of the curvature.
		
		
		By rescaling we can restrict our attention to r =1 and assume towards a contradiction that there exist δ>0, E>0 and v_0>0, and a sequence of counterexamples (M^4_i,g_i,p_i) satisfying 
		    
  * (B_i(p_i,1))>v_0>0,
		    
  * ∫_B_i(p_i,2) |_i|^2v_i ⩽ E,
		    
  * there exists κ_i→ 0 and Λ_i, |Λ_i|⩽ 3κ_i for which (g_i) = Λ_i g_i and (B_i(p_i,κ_i ))/v_-κ_i (κ_i )-(B_i(p_i,2))/v_-κ_i (2)<κ_i,
		 
		but such that for any Γ finite subgroup of SO(4), there does not exist a diffeomorphism Φ:A_e(δ , (1-δ )) →Â(δ r, (1-δ )r) where Â(δ r, (1-δ )r) satisfies (<ref>), and such that we have the control (<ref>).
		
		Since (B_i(p_i,1))>v_0>0 and (g_i)⩾ -3g_i, 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 <ref>, up to taking a subsequence, for all i, we can assume that κ_i is small enough to have 
    d_GH((B_i(p,1),p_i),(B_C(X_i)(0,1),0))⩽1/i,
 for a particular metric cone (C(X_i),d_C(X_i),0) with diam (X_i)⩽π. By the triangular inequality, the sequence (B_C(X_i)(0,1),0) 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)((r,x),(s,y)):= √(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 <cit.>, (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 ℝ^4Γ_∞ for Γ_∞ a finite subgroup of SO(4). 
		
		Now, the convergence of Einstein metrics is smooth on compact subsets of the complement of the singularities by <cit.>, and there is only one here at the singular point of ℝ^4Γ_∞, hence, there exists a diffeomorphism Φ:A_e(δ , (1-δ )) →Â(δ r, (1-δ )r) satisfying (<ref>) and (<ref>), which is a contradiction.
		

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


	Let δ >0, D_0>0, v_0>0, E>0, Λ∈ [-3,3] and l∈ℕ, and a 4-dimensional manifold, (M,g) satisfiying :
	
	
  * (g) = Λ g,
	
  * the diameter is bounded by D_0>0,
	
  * the volume is bounded below by v_0>0,
	
  * there exists p∈ M, and ρ_2>0, such that we have : 
	∫_B(p,ρ_2)||^2⩽ E.
	
 Let us first remark that these imply a control on Λρ_2^2 by Bishop-Gromov inequality

    E   ⩾∫_B(ρ_2)||^2v
       ⩾ 6∫_B(ρ_2)||^2v
       ⩾ 24Λ^2(B(ρ_2))
       ⩾ 24v_0Λ^2_g_-|Λ|(B_g_-|Λ|(ρ_2))/_g_-|Λ|(B_g_-|Λ|(D_0))
       ⩾ 24v_0Λ^2_g_-3(B_g_-3(ρ_2))/_g_-3(B_g_-3(D_0))
       ⩾ C(D_0,v_0)Λ^2ρ_2^4,

hence Λρ^2_2⩽E^1/2/C(D_0,v_0)^1/2 =: C'(E,D_0,v_0). 

Let l∈ℕ, and denote 0<κ_0 <1/4 the constant of Lemma <ref> associated to our constants δ, l et E. We are looking for ϵ_2>0 small enough to satisfy the hypotheses of Lemma <ref> with constant κ_0>0. 

For any 0<4ρ_1≤ρ_2 for which we have :

    ∫_A(ρ_1,ρ_2)||^2⩽ϵ_2^2,

and

    |(B(ρ_2))/(B(ϵ_2ρ_1))-ρ_2^4/(ϵ_2ρ_1)^4|⩽ϵ_2,

we actually have, thanks to Bishop-Gromov inequality, for
C”(E,D_0,v_0)>0,

    |(B(ρ_2))/(B(ρ_1))-ρ_2^4/ρ_1^4|⩽ C”(E,D_0,v_0)ϵ_2,

and for any ρ and κ_0>0 for which ρ_1<κ_0ρ<2ρ<ρ_2,

    |(B(2ρ))/(B(κ_0ρ))-(2ρ)^4/(κ_0ρ)^4|⩽ C”(E,D_0,v_0)ϵ_2.

We can therefore assume that (A(ρ_1,ρ_2))⩾ c(E,D_0,v_0)ρ^4_2 for 4ρ_1<ρ_2 and c(E,D_0,v_0)>0 by choosing ϵ_2 small enough. Let us then look for a bound on Λ in order to verify the assumptions of Lemma <ref>. Since = Λ g, we have

    ϵ_2^2   ⩾∫_A(ρ_1,ρ_2)||^2v
       ⩾ 6∫_A(ρ_1,ρ_2)||^2v
       ⩾ 24Λ^2(A(ρ_1,ρ_2))
       ⩾ 24c(E,D_0,v_0)Λ^2ρ^4_2

which rewrites |Λ| ⩽ϵ_2(24c(E,D_0,v_0))^-1/2ρ_2^-2. Therefore, for ϵ_2 small enough, we verify the assumptions of Lemma <ref> for any ρ by choosing ϵ_2 for which we have 
    κ_0 = ϵ_2max((24c(E,D_0,v_0))^-1/2,1,C”(E,D_0,v_0)),
 because we control the volume growth between κ_0 ρ and 2ρ thanks to (<ref>). The conclusion of Proposition <ref> therefore holds for any ρ_1<ρ<1/4ρ_2 with δ(κ_0)>0.
	
	

	
	

 §.§ Consequences of the existence of the first coordinates.

	    The diffeomorphism of Proposition <ref> allows us to pull back our Einstein metric on a flat annulus A_e(ρ_1, ρ_2) ⊂ℝ^4Γ, where the Einstein metric g is C^∞-close to the flat metric g_e. This will let us construct hypersurfaces of almost constant second fundamental form for our Einstein metric.
		
	    
	    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^Σ(X,Y):= ⟨ X,∇_YN^Σ⟩,
 which ensures that spheres in the Euclidean space have positive second fundamental form.
	    
		
			
			For any l∈ℕ, there exists C>0 and δ_0>0 such that if a metric g on A_e(1/4,4) satisfies,
				
    g-g_e_C^l+1(A_e(1/4,4))⩽δ_0,

		we have, for all s∈[1/4,4], the sphere of radius s of ℝ^nΓ, 𝒮_s, has a second fundamental form for g, A_g^𝒮_s, which satisfies
		
    A_g^𝒮_s /s-g^𝒮_s/s^2_C^l(g^𝒮_s /s^2)⩽ Cg-g_e_C^l+1(A_e(1/4,4)).

		We moreover have
		
    g^𝒮_s /s^2-g^𝕊^n-1Γ_C^l+1(g^𝕊^n-1Γ)⩽ Cg-g_e_C^l+1(A_e(1/4,4)).

		
		
		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 𝒮_s have a second fundamental form constant equal to g^𝕊^n-1Γ/s^2, we conclude that we indeed control their second fundamental form 
    A_g^𝒮_s /s-g^𝒮_s/s^2_C^l(g^𝒮_s /s^2)⩽ Cg-g_e_C^l+1(A_e(1/4,4)).
 In a similar fashion, we get the control on the metric.
		
		
	Given a smooth and orientable hypersurface, Σ, we define for x∈Σ and s ∈ℝ, F^g(x,s):=γ^g_x(s), where γ^g_x is the geodesic for g such that γ^g_x(0) = x, and γ'_x^g(0) is the normal to Σ at x. We define the normal injectivity radius of Σ, denoted _g^⊥(Σ) as the supremum of the constants r>0 for which the map F^g: Σ× [-r,r]→ M is a diffeomorphism on its image.
	
		
		For all l∈ℕ and  ϵ>0, there exists C>0 and δ>0 such that if a metric g satisfies g-g_e_C^l+2(A_e(1/4,4))⩽δ,
		then for any hypersurface Σ⊂ A_e(1/2,2) satisfying A^Σ_g_e_C^l+1⩽ 10, we have the two following properties:
 
 		
  * the normal injectivity radii of Σ for g _g^⊥(Σ), and g_e,   _g_e^⊥(Σ) satisfy
 
    |_g^⊥(Σ)/_g_e^⊥(Σ) - 1| ⩽ϵ.

 
		
  * the two normal exponential maps are also close to each other on the set 
    Σ×[-(1-ϵ)_g_e^⊥(Σ),(1-ϵ)_g_e^⊥(Σ)]:
 there exists a vector field X on ℝ^nΓ, X_C^l+1(g_e)<ϵ,  for which we have,
			
    F^g∘  (F^g_e)^-1 = ϕ_X,

			where ϕ_X:p↦exp^g_e_p(X(p)).
			
	
	
		The curves F^g(x,.) = γ^g_x(.) and F^g_e(x,.) = γ^g_e_x(.) are respectively the solutions of the geodesic equation for g and g_e,
		
    {∇_γ_x'γ_x'     = 0,
    γ_x(0)     = x,
    γ_x'(0)     = N_x.
          .

where N^g_x and N^g_e_x are the outwards normal to Σ 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 ∇^g and ∇^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 Σ, for all ϵ>0, there exists δ>0 such that if g-g_e_C^l+2⩽δ, 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 γ^g_e_x such that 
    γ^g_x(s) = ϕ_X(γ^g_e_x(s)):= exp_γ^g_e_x(s)(X(γ^g_e_x(s))),

and X_C^l+1(g_e)<ϵ. Hence, the equality γ^g_x(s) = ϕ_X(γ^g_e_x(s)) rewrites:

    F^g = ϕ_X∘ F^g_e.

	
	
	In particular, we deduce that if a hypersurface Σ' is a small normal perturbation of Σ for the metric g, then, it is also a small normal perturbation of Σ for the metric g_e, and conversely.
	
			
		For all ϵ_0 l∈ℕ, there exists C>0 and δ_0>0, such that for any metric g satisfying,
				
    g-g_e_C^l+2(A_e(1/4,4))⩽δ_0,

		and for any hypersurface Σ⊂ A_e(1/2,2) satisfying A^Σ_g_e_C^l+1⩽ 10,
			we have: for any function w:Σ→ℝ with w_C^l⩽ϵ_0, and denoting Σ(w):= {F(x,w(x)), x∈Σ},
			
			
  * there exists w':Σ→ℝ with w'_C^l⩽ 2ϵ_0 such that
			
    Σ_g_e(w) = Σ_g(w'),

			
  * there exists w”:Σ→ℝ with w”_C^l⩽ 2ϵ_0 such that
			
    Σ_g(w) = Σ_g_e(w”).

			
		
		
		According to Lemma <ref>, we have F^g = ϕ_X∘ F^g_e, where X is arbitrarily small in C^l+1-norm for δ_0 arbitrarily small. Adding the assumption A^Σ_g_e_C^l+1⩽ 10, ensures that for ϵ_0>0, the metrics g_e:=(F^g_e)^*g_e and g_p:=(g_e)_|Σ + ds^2 on Σ× [-2ϵ_0,2ϵ_0],  are arbitrarily close in C^l-norm for ϵ_0 arbitrarily small. This implies that
		
    [ (F^g_e)^-1∘ϕ_X∘ F^g_e] = ϕ^e_X^e,

		where X^e is the pull-back of X by F^g_e, and ϕ^e_X^e(p):= exp^g_e_p(X^e(p)). Hence, we have X^e_C^l+1(g_e)⩽ϵ_0 by the proximity of the metrics g_e and g_p, ϕ^e_X^e = (ϕ^Σ_X_Σ^p,ϕ^ℝ_X_ℝ^p), where X_Σ^p_C^l(g^Σ)+ X_ℝ^p_C^l(ds^2)⩽ 2ϵ_0.
		
	    Now, for all (x,s) ∈Σ× [-2ϵ_0,2ϵ_0], we have
			
    F^g (x,s)     = F^g_e∘[ (F^g_e)^-1∘ϕ_X∘ F^g_e](x,s),
        = F^g_e∘ (ϕ^Σ_X_Σ^p,ϕ^ℝ_X_ℝ^p)(x,s)
        = F^g_e(ϕ^Σ_X_Σ^p(x),ϕ^ℝ_X_ℝ^p(s)).

			In particular, for all w: Σ→ [-2ϵ_0,2ϵ_0], we have 
    F^g (x,w(x)) = (ϕ^Σ_X_Σ^p(x),w'( ϕ^Σ_X_Σ^p(x))),

			where
			w':=ϕ^ℝ_X_ℝ^p∘ w∘(ϕ^Σ_X_Σ^p)^-1. By the controls of X_ℝ^p and X_Σ^p which we can assume as small as wanted by choosing δ_0 and ϵ_0 small enough, we get the stated result.
		

		


§ 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 <cit.>.



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



For all k∈ℕ and 0<α<1, there exists ϵ>0 and C>0 for which, if Σ̃ is a hypersurface of a manifold (M^n,g) satisfying:


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

  * there exists Γ a finite subgroup of SO(n), and a diffeomorphism Φ: 𝕊^n-1Γ→Σ̃ such that we have
	
    Φ^*g^Σ̃ - g^𝕊^n-1Γ_C^1,α(𝕊^n-1Γ)⩽ϵ.

	
  * the ambient curvature is small on Σ̃,
	
    (g)_C^k(g)⩽ϵ,

	
	then:
	
		
  * the second fundamental form is almost constant at the scale of curvature,
		
    A^Σ̃ - g^Σ̃_C^k,α(g^Σ̃)⩽ C(g)_C^k(g),
	
	
  * there exists a diffeomorphism Ψ: 𝕊^n-1Γ→Σ̃ such that,
	
    Ψ^*g^Σ̃ - g^𝕊^n-1Γ_C^k+1,α(𝕊^n-1Γ)⩽ C(g)_C^k(g).

		


Let us precise the notations we will use for this proof:
	
		
  * ∇^Σ̃: the Levi-Civita connection of the hypersurface,
		
  * ∇: the Levi-Civita connection of the ambient space,
		
  * N^Σ̃: the outwards normal to the hypersurface,
		
  * A^Σ̃(X,Y):= ⟨ X, ∇_Y N^Σ̃⟩: the second fundamental form of the hypersurface which will be seen as a symmetric 2-tensor of the tangent space to Σ̃,
		
  * H^Σ̃: the mean curvature of the hypersurface,
		
  * for B a 1-form on Σ̃ taking values in T^*Σ̃, we define 
				
    (d^∇^Σ̃ B)(X,Y):= ∇^Σ̃_XB(Y,.)-∇^Σ̃_YB(X,.),

		
  * we will denote δ^∇^Σ̃ the formal adjoint of d^∇^Σ̃.
	
	
	We have the following classical formulas (see for instance <cit.>) for the second formula for any hypersurface Σ: for all vector fields X,Y,Z on Σ,
	
    d^∇^Σ A^Σ(X,Y,Z)= g((X,Y)N^Σ,Z),

	and
	
    δ^∇^Σ A^Σ(X) =(N^Σ,X) - dH(X).

	Denoting A_0^Σ̃ the trace-free part of A^Σ̃ for g^Σ̃, if Σ̃ is a constant mean curvature hypersurface, by the above formulas, we even have
	
    d^∇^Σ̃ A_0^Σ̃(X,Y,Z)= g((X,Y)N^Σ̃,Z),

	and
	
    δ^∇^Σ̃ A_0^Σ̃(X) =(N^Σ̃,X).

	The right hand sides being well controlled, we control A_0^Σ̃ by noticing that the operator d^∇^Σ̃+δ^∇^Σ̃ is elliptic.
	
		The operator d^∇^Σ̃+δ^∇^Σ̃ acting on symmetric 2-tensors is elliptic. 
		Moreover, for any metric g sufficiently C^0-close to g^𝕊^n-1, the operator d^∇^𝕊^n-1+δ^∇^𝕊^n-1 is injective on the symmetric 2-tensors which are traceless for g.
	
		
	    It is well-known that the operator d^∇+δ^∇ is elliptic. According to <cit.>, the associated Laplacian: -Δ̂:= δ^∇ d^∇+d^∇δ^∇ takes the following form for a symmetric 2-tensor h,
		
    -Δ̂^Σ̃h = (∇^Σ̃)^*∇^Σ̃h -^Σ̃h+ h∘^Σ̃,
 
		where  is the action of curvature on symmetric 2-tensors: for an orthonormal basis (e_i)_i,
		
    (h)(X,Y)= ∑_i h((e_i,X)Y,e_i),

		or (h)_ij:= _iklj h_mng^kmg^ln in local coordinates. Hence, on the unit sphere of dimension n-1, we have ^𝕊^n-1_iklj = g_ijg_kl-g_ilg_kj, and therefore ^𝕊^n-1(h)_ij=tr(h)g_ij - h_ij, and α∘^𝕊^n-1 = (n-2)α. This gives
		
    -Δ̂^𝕊^n-1h = (∇^𝕊^n-1)^*∇^𝕊^n-1h +(n-1)h - tr(h)Id.

		
		Let h_0 be a C^1,α(𝕊^n-1) symmetric 2-tensor whose trace for g^𝕊^n-1 vanishes and which is in the kernel of the operator. The trace of h_0 vanishes so we have,
		
    (∇^𝕊^n-1)^*∇^𝕊^n-1h_0 +(n-1)h_0 = 0.

		Integrating by parts against h_0 yields
		
    ∇ h_0_L^2^2+ (n-1)h_0_L^2^2 =0,

		and finally h_0 = 0. The operator d^∇^𝕊^n-1+δ^∇^𝕊^n-1 is therefore injective on symmetric 2-tensors whose trace for g^𝕊^n-1 vanishes.
		

		
		Let us now consider a metric g close to g^𝕊^n-1, that is, assume that there exists ϵ>0 which we will choose small enough such that
		
    g-g^𝕊^n-1_C^0(𝕊^n-1)≤ϵ.

		We want to show that the operator d^∇^𝕊^n-1+δ^∇^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 ϵ small enough, there exists C=C(n) for which
		
    |tr^𝕊^n-1 h|_𝕊^n-1⩽ Cϵ |h|_𝕊^n-1.

		We still have
		
    -Δ̂^𝕊^n-1h = (∇^𝕊^n-1)^*∇^𝕊^n-1h +(n-1)h - tr^𝕊^n-1(h),

		and thanks to (<ref>), if (d^∇^𝕊^n-1+δ^∇^𝕊^n-1) h = 0, then
		
    0 ⩾∇^𝕊^n-1h^2 +(n-1-Cϵ)h^2,

		which gives h =0 if ϵ is small enough to have n-1>Cϵ. We finally conclude that d^∇^𝕊^n-1+δ^∇^𝕊^n-1 is injective on symmetric 2-tensors whose trace for g vanishes.
	
	
	
        For any 1⩽ p<+∞ and l∈ℕ, there exists ϵ>0 such that if Σ is a hypersurface of a Riemannian manifold (M^n,g) for which there exists Γ a finite subgroup of SO(n), and a diffeomorphism Φ : 𝕊^n-1Γ→Σ̃ such that we have 
	    
    Φ^*g^Σ̃ - g^𝕊^n-1Γ_W^l+1,p(𝕊^n-1Γ)⩽ϵ,

	    then, there exists C>0 such that for any symmetric 2-tensor u on Σ̃ whose g^Σ̃-trace vanishes, we have,
	    
    u_W^l+1,p(𝕊^n-1Γ)⩽ C(δ^∇^Σ̃u_W^l,p(𝕊^n-1Γ)+d^∇^Σ̃u_W^l,p(𝕊^n-1Γ)).

	
	
	We have the following control: for all p⩾ 1, there exists a constant C>0 such that for all symmetric 2-tensor u on 𝕊^n-1 we have
	
    Φ^*(d^∇^Σ̃+δ^∇^Σ̃)u-(d^∇^𝕊^n-1+δ^∇^𝕊^n-1)u_L^p(𝕊^n-1Γ)⩽ Cϵu_W^1,p(𝕊^n-1Γ).

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


	

	Let 0<α<1, and p > n/1-α. Then, W^1,p(g^Σ̃) embeds continuously in C^α(g^Σ̃), which means that there exists C_1 =C_1(p,α,n)>0 such that for all symmetric 2-tensor u, we have:
	
    u_C^α(g^Σ̃)≤ C_1u_W^1,p(g^Σ̃).

	
	Define on 𝕊^n-1 the operators, P:=δ^∇^𝕊^n-1+d^∇^𝕊^n-1 and P':= Φ^*( δ^∇^Σ̃+d^∇^Σ̃) between the Banach spaces W^1,p(𝕊^n-1) and L^p(𝕊^n-1). According to Lemma <ref>, P is injective on traceless symmetric 2-tensors. Thanks to the expressions (<ref>) and (<ref>) of the operator δ^∇+d^∇ applied to A^Σ̃_0, by Corollary <ref>, we have,	
	
    A_0^Σ̃_W^1,p(g^Σ̃)    ⩽ C(δ^∇^Σ̃A_0^Σ̃_L^p(g^Σ̃)+d^∇^Σ̃A_0^Σ̃_L^p(g^Σ̃)) 
        = C((.,.)N^Σ̃_L^p(g^Σ̃) + (N^Σ̃) _L^p(g^Σ̃))

	Since the hypersurface has bounded volume (actually close to the volume of 𝕊^n-1Γ), there exists C_2 = C_2(p,n)>0 such that we have
	
    (N^Σ̃) _L^p(g^Σ̃)⩽ C_2 (.,.)N^Σ̃_C^0(g^Σ̃) ,

	and
	
    (.,.)N^Σ̃_L^p(g^Σ̃)⩽ C_2 (.,.)N^Σ̃_C^0(g^Σ̃).

	Hence, by continuous embedding of W^1,p(g^Σ̃) in C^α(g^Σ̃), we have
	
    A_0^Σ̃_C^α(g^Σ̃)⩽ 2C_1C_2C (.,.)N^Σ̃_C^0(g^Σ̃),

	which is the stated result for k=0.
	

	
	Let us finally explain how to control the higher levels of regularity by a bootstrap argument once we have found optimal coordinates on our hypersurface Σ̃. Assume that we have the following controls for l∈ℕ, on the hypersurface Σ̃: there exists C>0 such that,
	
	
  * (g)_C^l+1(g)⩽ϵ,
	
  * A^Σ̃ - g^Σ̃_C^l,α(g^Σ̃)⩽ C(g)_C^l(g),
	
  * there exists a diffeomorphism Ψ:𝕊^n-1Γ→Σ̃ with 
    Ψ^*g^Σ̃ - g^𝕊^n-1Γ_C^l+1,α(𝕊^n-1Γ)⩽ C(g)_C^l(g).

	
	Notice that at this point, we have proven the estimates for l=0. Gauss equation gives:
	
    g(^Σ̃(X,Y)Z,W) =       g((X,Y)Z,W)  
        +  A^Σ̃(X,Z)A^Σ̃(Y,W)-A^Σ̃(Y,Z)A^Σ̃(X,W).

	And controls 1 and 2 above imply therefore that there exists C=C(α,l,n)>0 which gives
	
    (g^Σ̃)-(n-1)g^Σ̃_C^l,α(g^Σ̃)⩽ C (g)_C^l+1(g),

	and we can use Proposition <ref> proven in the appendix.
	 
Let (M,g_0) be an n-dimensional manifold with sectional curvatures equal to 1, and k∈ℕ.
    Then, there exists δ>0 and C>0, such that if a metric g on M satisfies:

    g-g_0_C^1,α(g_0)⩽δ

    then, there exists a diffeomorphism Φ: M→ M, such that we have,
    
    Φ^*g-g_0_C^k+1,α(g_0)⩽ C(g)-(n-1)g_C^k-1,α(g_0).

    
	Hence, in our case, there exists a diffeomorphism (independent of l) Ψ̃_Σ̃: 𝕊^n-1Γ→Σ̃ with
	
    Ψ̃_Σ̃^*g^Σ̃ - g^𝕊^n-1Γ_C^l+2,α(𝕊^n-1Γ)⩽ C(g)_C^l+1(g).

	This diffeomorphism combined to the controls in C^l+1-norm on the curvature imply by Corollary <ref> (just like in the proof with k=0) that there exists a constant C'=C'(l+1,α,n)>0 such that 
    A^Σ̃ - g^Σ̃_C^l+1,α(g^Σ̃)⩽ C'(g)_C^l+1(g).

	We therefore have the property for l+1 and the control on the metric Ψ̃_Σ̃^*g^Σ̃ improves thanks to Proposition <ref> and we can iterate this argument for all l⩽ k-1.





§ 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 perturb a hypersurface with almost constant second fundamental form to a constant mean curvature hypersurface.

For all n∈ℕ and 0<r_0< 1/6, there exists C>0 and 0<ϵ<r_0/C such that for any hypersurface Σ diffeomorphic to 𝕊^n-1Γ, for Γ≠{Id}, of a manifold (M,g) satisfying:
	
		
  * the normal injectivity radius of Σ and the injectivity radius of the points of Σ in M are bounded below by 3r_0,
		
  * A^Σ - g^Σ_C^α(g^Σ)⩽ϵ,
		
  * on the annulus {x,d^M_g(x,Σ)<3r_0}, we have (g)_C^2(g)<ϵ,
	
	
	Then, there exists a unique function w satisfying w_C^2,α(g^Σ)⩽ Cϵ and such that we have 
    H^Σ(w)≡ n-1,
 where we denoted Σ(w):= {γ_x(w(x)), x∈Σ}, where γ_x is the geodesic with γ_x(0) =x, and γ'_x(0) is the unit outwards normal to Σ. We moreover have the following control,
	
    w_C^2,α(g^Σ)⩽ C H^Σ-(n-1)_C^α(g^Σ).




    By adapting the arguments of Section 3, we know that a hypersurface satisfying the assumptions of Proposition <ref> is arbitrarily close (up to a diffeomorphism) in the C^1,α-topology to the round 𝕊^n-1Γ.



	Recall that we defined our Hölder norms at the scale of the injectivity radius, see Definition <ref>. In our applications, the controls on the injectivity radius will be consequences of Lemma <ref>, 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 <cit.>) because it is achieved by constructing a right-inverse for the operator Δ_𝕊^n-1+(n-1), where Δ_𝕊^n-1 is the Laplace-Beltrami operator (with nonpositive eigenvalues) which is not invertible. In our case, our hypersurfaces are not close to 𝕊^n-1, but to 𝕊^n-1Γ for Γ≠{Id}. The crucial difference it makes is that -(n-1) is not an eigenvalue anymore (see <cit.> 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 nonzero linear functions of ℝ^n to the sphere which cannot be Γ-invariant.
	
	
	    Geometrically, this comes from the fact that given 𝕊^n-1⊂ℝ^n, the translations provide constant mean curvature perturbations of the spheres. In the case of 𝕊^n-1Γ⊂ℝ^nΓ, the center cannot move as it is the fixed point of the group action.
	
	
	
	     

  
Normal geodesic coordinates.	
        Let us first give some controls on the geometry of the hypersurfaces equidistant to a given hypersurface Σ̃_1.
    
		For all n∈ℕ and 0<r_0<1/6, there exists C>0 and 0<ϵ<r_0/C such that if a hypersurface Σ̃_1 of a manifold (M,g) satisfies:
	
		
  * the normal injectivity radius of Σ̃_1 and the injectivity radius of points of Σ̃_1 in M are bounded below by 3r_0,
		
  * H^Σ̃_1≡ n-1, and A^Σ̃_1-g^Σ̃_1_C^α(g^Σ̃_1)<ϵ,
		
  * on the annulus {x,d^M_g(x,Σ)<3r_0}, we have _C^2(g)⩽ϵ,
	
	then,
		
    A^Σ_s/s-g^Σ_s/s^2_C^α(g^Σ_s/s^2)    ⩽ C(A^Σ̃_1-g^Σ̃_1_C^αg^Σ̃_1+ |s-1|_C^α(g) + |s-1|^2_C^2(g))
        ⩽ C(1 + r_0^2)ϵ,

and
	
    H^Σ_s/s-n-1/s^2_C^α(g^Σ_s/s^2)⩽ C(|s-1|_C^α(g) + |s-1|^2 _C^2(g))⩽ C( r_0 + r_0^2)ϵ.

	
	
	Given a hypersurface Σ_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,Σ)<3r_0} by setting N(x), for x∈Σ_s, as the outwards normal vector of the hypersurface Σ_s. 
	
	Denoting, S(X):= ∇_X N, for X a vector field on M, we have
	
    (∇_NS)(X) = - (S^2(X)+(X,N)N).

    Assuming S_|Σ_1-Id_C^α(g^Σ_1)<ϵ for ϵ>0 small enough, we can follow S along the normal geodesics, and integrating the previous Riccati equation (<ref>), we find that there exits C>0 such that for |s-1| ⩽ r_0 ⩽1/6,  
    S_|Σ_s/s - Id/s^2_C^α(g^Σ_s)⩽S_|Σ_1-Id_C^α(g^Σ_1) + C|s-1|_g_C^α(g) + C|s-1|^2 _g_C^2(g).
 Denoting κ_i(s) the eigenvalues of the second fundamental form of Σ_s, and (e_i(s)) an associated orthonormal basis of eigenvectors, if we project (<ref>) on e_i(s), we get:
	
    κ_i'(s) =  - (κ_i(s)^2 + g((N^s,e_i(s)) e_i(s),N^s))

    and finally, by summing on i, H'(s) = - (|A|^2(s)+ (N^s,N^s)). With the initial condition H(1) ≡ n-1, we finally have the second stated estimate.
	
    
    
    
    
    
    
    Let us now construct coordinates in the neighborhood of a hypersurface satisfying the assumptions of Proposition <ref>.
    
        Let Σ be a hypersurface satisfying the assumptions of Proposition <ref>.
        Then, there exists a diffeomorphism Φ : A_e(1-2r_0,1+2r_0)→{x,d(x,Σ)<2r_0} such that we have for C>0,
        
    Φ^*g-g_e_C^1,α(g_e)⩽ Cϵ,

        where ϵ>0 is the constant of Proposition <ref>.
    
    
        Let us consider the diffeomorphism F_g : Σ× [-2r_0,2r_0] obtained from the outwards normal geodesic of Σ.
        
        We then use our controls on the second fundamental form and the ambient curvature and integrate the Riccati equation satisfied by the Jacobi fields generated by the infinitesimal change of starting point on the hypersurface Σ. We therefore obtain like in Lemma <ref> the following control for C= C(n,v_0)>0, by noting s the parameter of the geodesics vanishing on Σ :
        
    F_g^*g-(ds^2+s^2g^Σ)_C^1,α⩽ Cϵ.

        By the assumptions of Proposition <ref>, up to choosing ϵ arbitrarily small, the second fundamental form is arbitrarily close to being constant and the ambient curvature arbitrarily small. In particular, according to Gauss equation (<ref>) together with Lemma <ref> and Proposition <ref>, this implies that there exists Γ⊂ SO(4) such that, up to reparametrizing Σ, we have for some C=C(n,v_0)>0, 
        
    g^Σ-g^𝕊^n-1Γ_C^1,α⩽ Cϵ.

        Combining this with the control (<ref>), we get the stated result.
    
	
	

  
Normal perturbations of the hypersurface.
	
	Consider normal perturbations of a hypersurface Σ of the form Σ(w):= {γ_x(w(x)), x∈Σ}. The computations of <cit.> adapted to our situation yield the following development of the mean curvature.
	
	
		For any 0<r_0<1/6 and any smooth hypersurface Σ⊂ M whose injectivity radius and normal injectivity radius are larger than 3r_0, there exists ϵ>0 such that for all w: Σ→ℝ such that w_C^2,α(g^Σ)⩽ϵ, the mean curvature of Σ(w) has the development:
		
    H(Σ(w)) = H(Σ) - J_Σ,M w + Q_Σ,M(w),

		where 
    J_Σ,M:= Δ_Σ + |A^Σ|^2 + Ric^M(N,N)
 is the Jacobi operator of the hypersurface, and where Q_Σ,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_Σ,M(w) - Q_Σ,M(w')_C^α(g^Σ)⩽  C  w-w' _C^2,α(g^Σ)( w_C^2,α(g^Σ)+ w' _C^2,α(g^Σ)).

	
	
	    
	

Proposition <ref> is then a consequence of the following Lemma applied to Σ.
	
	
	For any r_0>0, there exists ϵ>0 such that if a metric g on the annulus A_e(1-r_0,1+r_0)⊂ℝ^nΓ for Γ⊂ SO(n) with Γ≠{} satisfies
	
	    
  * g-g_e_C^1,α(g_e)⩽ϵ,
	    
  * on S the unit sphere centered at zero in S⊂ℝ^nΓ, we have 
    A_g^S-g_|S_C^α(S)⩽ϵ ,

	    
  * on A_e(1-r_0,1+r_0), we have
	    
    _g_C^4(g_e)⩽ϵ,

	
	Then, there exists ϵ_1=ϵ_1(r_0,ϵ,n,α)>0 such that there exists a unique solution w satisfiying w_C^2,α(S)⩽ϵ_1 to the equation
		
    H_g(S(w)) ≡ n-1,

		where H_g is the mean curvature for g and where S(w)= {γ_x(w(x)),x∈ S} for γ_x the geodesic starting x and outwards normal to S.
		
		We moreover have the control:
		
    w_C^2,α(S)⩽ C H_g(S) - (n-1)_C^α(S).

	
	
	    This is a consequence of the following quantitative version of the inverse function theorem proven by Banach fixed point theorem.
		
				Let Φ: E→ F, be a smooth map between Banach spaces and let Q:= Φ - Φ(0)- d_0Φ.
				
				Assume that there exist q>0, r_0>0 and c>0 such that
				
				
  * for all x and y in B(0,r_0), we have the following control on the nonlinear terms
				
    Q(x)-Q(y)⩽ q (x+y)x-y.

				
  * the linearization d_0Φ is an isomorphism, and more precisely, we have
				
    (d_0Φ)^-1≤ c.

				
				
				If r⩽min(r_0,1/2qc) and Φ(0)⩽r/2c, then, the equation Φ(x) =0 admits a unique solution in B(0,r).
		
		We will equip S with the Hölder norms given by the restriction of g_e on S.	Let us consider the operator Φ : C^2,α(S) → C^0,α(S) defined by
		
    Φ :w∈ C^2,α(S) ↦ H_g(S(w))-(n-1)∈ C^0,α(S),

		and show that it satisfies the assumptions of Lemma <ref>, with Φ(0) = H_g(S)-(n-1), d_0Φ = J_S,g.
		
		    In this proof, we will precise the metric instead of the manifold for the Jacobi operator and the nonlinear terms of the mean curvature variations.
		
		
			
  * The operator J_S,g is invertible and has a bounded inverse for ϵ>0 small enough. Indeed, -(n-1) is not an eigenvalue of the Laplacian because Γ≠{}, therefore
			J_S,g_e is invertible with a bounded inverse. Moreover, by proximity of the metrics and the control of the C^α(S)-norm of the second fundamental form and the C^4-norm of the ambient curvature, there exists C_1>0 such that, 
			
    J_S,g_e - J_S,g⩽ C_1ϵ,
 for the operator norm from C^2,α(S) to C^α(S). The operator J_S,g is therefore invertible with a bounded inverse for ϵ small enough since this is an open condition.
			
			
  * The nonlinear terms are controlled in a quadratic way. There exists a constant C>0 depending on r_0, α and ϵ  :
			
    Q_S,g(w) - Q_S,g(w')_C^α⩽  C  w-w' _C^2,α(S)(w_C^2,α+w'_C^2,α(S)).

			
			
  * Since the second fundamental form is almost constant, we have:
			
    H_g(S) - (n-1)_C^α(S)⩽ϵ.

		
		Therefore, there exists a unique solution w to the equation Φ(w)=0 such that w_C^2,α(S)⩽ϵ_1 := min(1/2CC_1, 2C_1ϵ). This solution moreover satisfies for C'>0, 
    w_C^2,α(S)⩽ C' H_g(S)-(n-1)_C^α(S).

		


    According to Lemma <ref>, under the assumptions of Proposition <ref>, then, there exists a diffeomorphism Φ : A_e(1-2r_0,1+2r_0)→{x,d(x,Σ)<2r_0} such that we have for C>0,
        
            
  * the metrics are close, 
    Φ^*g-g_e_C^1,α(g_e)⩽ Cϵ,

            
  * for the unit-sphere centered at zero denoted S⊂ℝ^nΓ, we have 
    A_Φ^*g^S-(Φ^*g)_|S_C^α⩽ϵ ,

	    
  * for the curvature on A_e(1-r_0,1+r_0), we have
	    
    _Φ^*g_C^4⩽ϵ,

        
        where ϵ>0 is the constant in the assumptions of Proposition <ref>, for which the metric Φ^*g satisfies the assumptions of Lemma <ref> with S = Φ^*Σ the unit-sphere of ℝ^nΓ. Applying Lemma <ref>, we obtain the statement of Proposition <ref>.


	For Σ satisfying the assumptions of Proposition <ref>, let us define Σ̃ := Σ(w), where w
	is the unique solution given by the proof of Proposition <ref>. 



§ CONSTRUCTION OF A FOLIATION OF THE NECK REGIONS BY CONSTANT MEAN CURVATURE HYPERSURFACES

Let us now foliate our neck regions by constant mean curvature hypersurfaces.


 §.§ Local foliation by constant mean curvature hypersurfaces

We start by constructing our foliation locally.


    For all n∈ℕ and 0<r_0<1/6, there exists C>0 and 0<ϵ<r_0/C such that if a hypersurface Σ̃_1 diffeomorphic to 𝕊^n-1Γ, for Γ≠{Id}, of a manifold (M,g) satisfies:
	
		
  * the normal injectivity radius of Σ̃_1 and the injectivity radius of the points Σ̃_1 in M are bounded below by 3r_0,
		
  * H^Σ̃_1≡ n-1, and A^Σ̃_1-g^Σ̃_1_C^α(g^Σ̃_1)<ϵ,
		
  * on the annulus {x,d^M_g(x,Σ)<3r_0}, we have _g_C^2(g)⩽ϵ.
	
	
	Then, there exists a family of hypersurfaces Σ̃_s, s∈[1/1+2r_0, 1+2r_0] with constant mean curvature n-1/s foliating a region of the annulus Ã(1/1+2r_0, 1+2r_0) satisfying:	
	
    {x, d^M_g(x,Σ)<r_0}⊂Ã(1/1+2r_0, 1+2r_0)  ⊂{x, d^M_g(x,Σ)<3r_0}.


    The construction we propose consists in perturbing normally the level sets of the distance function to the hypersurface Σ̃_1 in constant mean curvature hypersurfaces by Proposition <ref>.
    

    
	Let us define the equidistant hypersurfaces Σ_s:= {γ_x(s-1), x∈Σ̃_1 } for s>0 small enough where γ_x is the outwards normal geodesic to the hypersurface Σ̃ starting at x. These hypersurfaces foliate the annulus {x,d^M_g(x,Σ)<3r_0} by definition of the normal injectivity and their geometry is well controlled around Σ̃_1.	
	
		
	Thanks to the controls (<ref>), and (<ref>), we can use Lemma <ref> to perturb normally each hypersurface Σ_s to a constant mean curvature hypersurface Σ̃_s. 
	Hence, there exists a family of hypersurfaces (Σ̃_s)_s∈[1/1+2r_0, 1+2r_0] with mean curvature constant equal to n-1/s for each s. These hypersurfaces are defined from a unique function w_s from Σ̃_1 to ℝ by Proposition <ref>,
	
    Σ̃_s = Σ(s(1+w_s)):= {γ_x(s(1+w_s(x))), x∈Σ̃_1},

	we moreover have the control
	
    w_s_C^2,α(g^Σ̃_1)⩽ Cϵ(|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'∈ [1/1+2r_0, 1+2r_0], close enough and such that s<s', Σ̃_s' is a normal perturbation of Σ̃_s by a strictly positive function w_s^s': Σ̃_s→ℝ. They are indeed normal perturbation of each other because they are small (in C^2,α(g^Σ̃_1)-norm) normal perturbations of the equidistant hypersurfaces from Σ̃_1.
	
	By Lemma <ref>, for s and s' close enough, the function w_s^s' satisfies w_s^s'_C^2,α((g^Σ̃_s))⩽ 2 |s'-s| and is a solution to 
    -J_Σ̃_s,M(w_s^s') =  n-1/s'-n-1/s + Q_Σ̃_s,M(w_s^s'),

 where there exists C=C(ϵ,n,α,r_0)>0 independent of s such that for any functions w, and w', we have Q_Σ̃_s,M(w) - Q_Σ̃_s,M(w')_C^α(g^Σ̃_s)⩽ C  w-w' _C^2,α(w_C^2,α(g^Σ̃_s)+w'_C^2,α(g^Σ̃_s)).
	
	We therefore get Q_Σ̃_s,M(w_s^s')_C^α(g^Σ̃_s)⩽ 4C |s'-s|^2, and since sΣ̃_s is close to 𝕊^n-1Γ, for |s'-s| small enough, J_Σ̃_s,M is close to Δ^𝕊^n-1Γ  + n-1, and we have:
	
    -(Δ^𝕊^n-1Γ  + n-1) (w_s^s'-(s'-s)) _C^α(g^Σ̃_s)⩽ C ((s'-s)^2 + ϵ (s'-s)) .

	Since -Δ^𝕊^n-1Γ+n-1 is invertible, there exists C=C(α, n)>0 such that 
    w_s^s'-(s'-s)_C^2,α(g^Σ̃_s)⩽ C ((s'-s)^2 + ϵ (s'-s)).
 Finally, we have w_s^s' = (s'-s) + 𝒪((s'-s)^2 + ϵ (s'-s)), and in particular, for s'-s>0 and ϵ small enough (depending on n and α), we have w_s^s' >0.
	
	This implies that for all s,s'∈ [1/1+2r_0, 1+2r_0], the hypersurfaces Σ̃_s and Σ̃_s' are all disjoint and foliate the annulus. Indeed, we have just seen that for s<s' and |s-s'| small enough, the hypersurface Σ̃_s' is strictly included in one side of Σ̃_s. Recalling that for all s, Σ̃_s = Σ̃_1(s(1+w_s)), this means that for all x, the function s↦ (s(1+w_s(x))) is strictly increasing, and therefore that we indeed have a foliation of the annulus.

	
	
	    It was not directly possible to ensure that two hypersurfaces Σ̃_s and Σ̃_s' do not intersect for s,s'∈ [1/1+2r_0, 1+2r_0] from our previous controls. Indeed, the control w_s_C^2,α(g^Σ̃_1)⩽ C(ϵ|s-1| +ϵ |s-1|^2) is not enough to rule out w_s(x) = w_s'(x') when s-1 is much larger than s-s'.
	



 §.§ 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 <ref>, this manifold is ϵ-approximated by a tree of Ricci-flat ALE orbifolds desingularizing the orbifold (M_o,g_o). We have satisfactory coordinates on the regions ℳ_o(ϵ) and 𝒩_k(ϵ) and want to find good coordinates in the annuli 𝒜_k(t,ϵ). They are included in metric annuli A(ρ_1,ρ_2) for ρ_1:= 1/8ϵ^-1 T_k^1/2 and ρ_2:= 8 ϵ T_j^1/2 on which we have the following properties:

	
  * the curvature is controlled by <cit.>: for all l∈ℕ, there exists C>0 and β_1,β_2>0 depending on g_o and the g_b_j such that at distance ρ∈[ρ_1/2,2ρ_2] from p, we have
	
    ρ^2+l|∇^l |⩽η(ρ):= C ϵ[(ρ_1/ρ)^β_1+(ρ/ρ_2)^β_2],

	
	    In <cit.>, the result is only stated for l=0, but the regularity of Einstein metrics, and in particular the fact that the curvature is harmonic (for the Hodge Laplacian) yields the estimate for higher derivatives by local elliptic estimates.
	
	
  * by Proposition <ref>, there exists δ(ϵ) such that δ(ϵ)→ 0 as ϵ→ 0 and for any 2ρ_1<ρ<1/4ρ_2, there exists a region Â(ρ,2ρ) satisfiying
	
    A((1+δ)ρ,(2-δ)ρ) ⊂Â(ρ,2ρ)⊂ A((1-δ)ρ,(2+δ)ρ),

	Γ a finite subgroup of SO(4) and a diffeomorphism 
    Φ_ρ : A_e(1,2)⊂ℝ^4Γ→Â(ρ,2ρ)⊂ M,

	for which we have,
	
    Φ_ρ^*g/ρ^2-g_e_C^l+2(A_e(1,2))⩽δ.

	
	
  * by Lemma <ref>, there exists a hypersurface Σ_ρ_1 whose second fundamental form satisfies A_g^Σ_ρ_1/ρ_1-g/ρ_1^2_C^l+1(g/ρ_1^2)⩽δ,

	
  * by Lemma <ref> and Lemma <ref> 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 Σ_ρ_1 by Lemma <ref> is bounded below by r_0ρ_1.
	
	
The control (<ref>) means that in the middle of the annulus, we have a much better control than initially expected since the classical ϵ-regularity theorems <cit.> would only give ρ^2||⩽ C _L^2(A(ρ_1,ρ_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 <cit.>.


    Analogous estimates hold in any dimension n by assuming some much less natural L^n/2-controls on the Riemann curvature.

	
	The main difficulty in constructing good coordinates here is that the interior radius, ρ_1, is neglectible when compared to the radius at which our controls are optimal, ρ̅:= (ρ_1^β_1ρ_2^β_2)^1/β_1+β_2 which minimizes η. Constant mean curvature hypersurfaces are convenient in this context as they can be constructed in the entire zone, and can be controlled by Proposition <ref>.


For all n∈ℕ, there exists ϵ_0>0, and C>0 such that for any 0<ϵ<ϵ_0, we have the following property. Let 𝒜_k(t,ϵ) be an intermediate annulus of an Einstein metric which is ϵ-approximated by a naïve desingularization, and assume that it is included in a metric annulus A(4ρ_1,1/4ρ_2) with ∫_A(ρ_1,ρ_2)||^2dv< ϵ^2. Then, there exists a foliation of a region Ã(2ρ_1,1/2ρ_2) bounded by two hypersurfaces of mean curvature respectively constant equal to  n-1/2ρ_1 and n-1/1/2ρ_2, with 	
	
    𝒜_k(t,ϵ)⊂ A((2+Cϵ)ρ_1,(1/2-Cϵ)ρ_2)⊂Ã(2ρ_1,1/2ρ_2) ⊂ A((2-Cϵ)ρ_1,(1/2+Cϵ)ρ_2),

	by hypersurfaces denoted Σ̃_s whose mean curvature equals n-1/s.



Considering Φ_2ρ_1, the diffeomorphism of Proposition <ref> at scale 2ρ_1, we define the hypersurface Σ_2ρ_1:=Φ_2ρ_1,*S⊂ M, where S is the unit-sphere centered at zero of ℝ^4Γ. According to Lemmata <ref> and <ref>, and Corollary <ref>, we satisfy the assumptions of Lemma <ref>. We can therefore perturb Σ_2ρ_1 to a hypersurface Σ̃_2ρ_1 whose mean curvature is constant equal to n-1/2ρ_1. 

According to Proposition <ref>, there exists a foliation by hypersurfaces Σ̃_s whose mean curvature is constant equal to n-1/s for all 2ρ_1<s<(1+2r_0)2ρ_1. Moreover, for any s, there exists a constant C=C(n,α)>0 such that if we recall the notation

    η(s) := C ϵ[(ρ_1/s)^β_1+(s/ρ_2)^β_2],
 
we have the following a priori controls on the Σ̃_s thanks to Proposition <ref>:
	
		
  * the normal injectivity radius of Σ̃_s and the injectivity radius of the points of Σ̃_s in M are bounded below by 3sr_0 according to Lemmata <ref> and <ref>,
		
  * H^Σ̃_1≡n-1/s, and A^Σ̃_s/s-g^Σ̃_s/s^2_C^α<Cη(s),
		
  * on the annulus {x,d^M_g(x,Σ̃_1)<3sr_0}, we have _g_C^4(g)⩽Cη(s)/s^2.
	
	We can therefore apply Proposition <ref> from the hypersurface Σ̃_2(1+2r_0)ρ_1 in order to obtain a foliation by hypersurfaces with constant mean curvature equal to n-1/s up to s = 2(1+2r_0)^2ρ_1. Once again, the a priori controls of Proposition <ref> of order Cη(s) and the injectivity and normal injectivity controls of Lemmata <ref> and <ref> let us iterate this process in order to construct hypersurfaces Σ̃_s with mean curvature constant equal to n-1/s up to s= 2(1+2r_0)^kρ_1 for any k∈ℕ with 2(1+2r_0)^kρ_1<1/2ρ_2 which is the stated result.






§ 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:= n-1/H(Σ̃_s) whose level sets are the hypersurfaces Σ̃_s. The main result is the following Proposition:


	Under the same assumptions as Proposition <ref>, there exists β_1>0, β_2>0, Γ≠{Id} a finite subgroup of SO(4) such that for all l∈ℕ, there exists C>0, and a diffeomorphism: 	
	
    Φ: A_e(2ρ_1,1/2ρ_2) →Ã(2ρ_1,1/2ρ_2) ,

	such that for all ρ∈ [2ρ_1,1/4ρ_2], denoting Φ_ρ:= Φ∘ϕ_ρ: A_e(1,2)→Ã(2ρ_1,1/2ρ_2) , where ϕ_ρ is the homothetic transformation of ratio ρ on ℝ^4Γ, we have:	
	
    Φ^*_ρ g/ρ^2-g_e_C^l(A_e(1,2))⩽ C ϵ((ρ_1/ρ)^β_1+(ρ/ρ_2)^β_2).
 



    Our proofs again work in any dimension n∈ℕ for metrics with a region close to a flat cone ℝ^nΓ (for Γ≠{Id}) where the curvature is small in L^n/2.




 §.§ Coordinates based on our constant mean curvature foliation

In the foliation constructed in Proposition <ref>, one particular hypersurface is better controlled than the others at its scale. It is Σ̃_ρ̅ where η(ρ̅)= min_ρ_1<ρ<ρ_2η(ρ), which satisfies the following properties by (<ref>) and Proposition <ref>:

    
  * there exists a diffeomorphism ϕ: 𝕊^n-1Γ→Σ̃_ρ̅ such that: for all k∈ℕ and 0<α<1, there exists C=C(k,α,n,r_0)>0 for which,
    
    ϕ^*g^Σ̃_ρ̅/ρ̅^2-g^𝕊^nΓ_C^k,α(g^𝕊^nΓ)< C η(ρ̅),

    
  * for all k∈ℕ and 0<α<1, there exists C=C(k,α,n)>0 such that
    
    A^Σ̃_ρ̅/ρ̅ - g^Σ̃_ρ̅/ρ̅^2_C^k,α(g^Σ̃_ρ̅/ρ̅^2)⩽ Cη(ρ̅).

    
  * for all k∈ℕ, there exists C=C(k,n)>0 such that on the annulus [1/2ρ̅,2ρ̅] we have,
    
    ρ̅^2+k|∇^k |_g ⩽ C η(ρ̅).



We then define a diffeomorphism Φ: [ρ_1,ρ_2]×𝕊^n-1Γ→Ã(ρ_1,ρ_2) in the following way: at ρ̅,

    Φ(ρ̅,x):= ϕ(x),

where ϕ is the diffeomorphism of the above point 1, and for all s∈ [ρ_1,ρ_2],

    ∂_sΦ(s,x) = u_s(x) N^Σ̃_s(Φ(s,x)) =  -∇ s/|∇ s|^2,

where u_s is such that Φ(s,𝕊^n-1Γ) = Σ̃_s. This is a diffeomorphism on its image since two curves following the vector field -∇ s/|∇ s|^2, which never vanishes, cannot intersect if they are not equal.




 §.§ Control of the metric in these coordinates


In the coordinates given by Φ, the metric has the following form,

    Φ^*g(s,x) = u_s(x)^2ds^2 + s^2 h_s(x),

where h_s is a metric on 𝕊^n-1Γ.



Let us take the following notations for the rest of the section in which we will always work on ℝ^nΓ:

	
  * ϕ_s(x) = Φ(s,x), which implies that s^2h_s = ϕ_s^*g^Σ̃_s.
	
  * H(s) = ϕ_s^*H^Σ̃_s≡n-1/s,
	
  * A(s) = ϕ_s^*A^Σ̃_s,
	
  * N(s) = ϕ_s^*N^Σ̃_s
	
  * f_i(s) = d_(s,.)Φ(0,e_i), where the e_i form an orthonormal basis of T_x𝕊^n-1Γ,
	
  * ∇^s = ϕ_s^*∇^Σ̃_s,
	
  * Δ^s = ϕ_s^*Δ^Σ̃_s, where Δ^Σ̃_s is the Laplace-Beltrami operator associated to the metric g^Σ̃_s,
	
  * ^s is the gradient for the metric s^2h_s,
	
  * K(s)_i^j = g(_g(N(s),f_i(s))f_j(s),N(s)).

We have the following variations for the different geometric quantities.



	By <cit.>, given a variation ∂_sΦ(s,x) = u_s N^Σ̃_s, we have the following variation formulas:
	
    ∂_s (s^2h_s) = 2 u_s A(s),

	
    ∂_s N(s) = ^s u_s,

	
    ∂_s A(s)_ij = - ∇^s_i∇^s_j u_s + (∑_kA(s)_ikA(s)_j^k+ K(s)_i^j),

	
    ∂_s H(s) = - (Δ^s+|A(s)|^2+(s)(N(s),N(s)))u_s.

	

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


 With the above notations, if we denote g_e:=ds^2+s^2g^𝕊^n-1, we have the following controls on Φ^*g: for all l∈ℕ, there exists C(l,n,v_0,D_0)>0 such that for all s∈[ρ_1,ρ_2/2], on the annulus [s,2s], we have
 
    u-1_C^l(g_e)⩽ C η(s),

 and
 
    h-g^𝕊^n-1Γ_C^l(g_e)⩽ Cη(s),

 and therefore, for all s∈[ρ_1,ρ_2/2], on the annulus of radii s and 2s,
 
    Φ^*g-g_e_C^l(g_e)⩽ Cη(s).



    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∈ℕ, 0<α<1.

    Let us start by noting that the equation (<ref>) implies that for all s, u_s satisfies the following equation
	
    n-1/s^2 = (Δ^s+|A(s)|^2+(s)(N(s),N(s)))u_s.

	Indeed, by following the normal perturbation u_s N^Σ̃_s =-∇ s/|∇ s|^2, the variation of the mean curvature is exactly -n-1/s^2. In particular, for all s, we have
		
    (Δ^s+|A(s)|^2+(s)(N(s),N(s)))(u_s-1) = n-1/s^2-|A(s)|^2-(s)(N(s),N(s)),

	
	According to Proposition <ref>, while h_s-g^𝕊^n-1Γ_C^1,α(g^𝕊^n-1Γ) is sufficiently small, there exists C = C(l)>0 such that for all s,

    A(s)/s-h_s_C^l,α(h_s)⩽ C η(s).
		
	Since s^2|(s)(N(s),N(s))|<η(s), this implies that the right hand side of (<ref>) satisfies: there exists C=C(α,n)>0 such that for all s,
	
    n-1-s^2|A(s)|^2-s^2(s)(N(s),N(s))_C^α(g^𝕊^n-1Γ)⩽ Cη(s).
	
	Likewise, for the left hand side, for any function v: 𝕊^n-1Γ→ℝ we have
	
    s^2(Δ^s+|A(s)|^2+(s)(N(s),N(s)))v -     (Δ^𝕊^n-1Γ+n-1) v _C^α(g^𝕊^n-1Γ)
        ⩽ Cη(s) v_C^2,α(g^𝕊^n-1Γ).

	Since Δ^𝕊^n-1Γ+n-1 is invertible for Γ≠{Id}, we get by the inverse function theorem, Lemma <ref>, the following control: there exists C=C(n,α)>0 such that for all s
	 
    u_s-1_C^2,α(g^𝕊^n-1Γ)⩽ Cη(s).

	 
	 Let us now show that for all s∈ [ρ_1,ρ_2], h_s-g^𝕊^n-1Γ_C^1,α(g^𝕊^n-1Γ) remains small. We will actually show that there exists C=C(n,α)>0 such that h_s-g^𝕊^n-1Γ_C^1,α(g^𝕊^n-1Γ)⩽ C η(s).  
	The expression (<ref>) can be rewritten 
    ∂_s (h_s-g^𝕊^n-1Γ) =2/s(uA^s/s-h_s),

	but by the control (<ref>), and Proposition <ref>, there exists C=C(n,α)>0 such that we have uA^s/s-h_s_C^1,α(g^𝕊^n-1Γ)⩽ C η(s). Recall that 
    η(s):= Cϵ[(ρ_1/s)^β_1+(s/ρ_2)^β_2],
 to deduce that for ρ̅<s_0<ρ_2, we have
	
    h_s_0-g^𝕊^n-1Γ_C^1,α(g^𝕊^n-1Γ)⩽     C (η(ρ̅)+ ∫_ρ̅^s_02/sη(s) ds)
    
    	=     Cη(ρ̅)+(2Cϵ/ρ_2^β_2∫_ρ̅^s_0 s^β_2-1 ds)
    
    	=     Cϵρ̅^β_2/ρ_2^β_2+(2Cϵ/ρ_2^β_2(s_0^β_2-ρ̅^β_2) )
    ⩽     C η(s_0).

	and similarly, for ρ_1<s_0<ρ̅,
	
    h_s_0-g^𝕊^n-1Γ_C^1,α(g^𝕊^n-1Γ)⩽ C η(s_0).

	
	To obtain controls on higher derivatives of u and h_s, we use the other equalities of <cit.>: according to (<ref>), and the control (<ref>), there exists C =C(n,α) such that for all s we have

    ∂_s N(s)_C^1,α(h_s)⩽ Cη(s)
	
according to (<ref>) and the controls (<ref>), (<ref>) and (<ref>), and since the l-th derivatives of the curvature are bounded by s^-2-lη(s), there exists C=C(n,α)>0 such that for all s, we have:
	
    ∂_s (A(s)/s)_C^α(h_s)⩽ Cη(s).

	
	Differentiating the equality (<ref>) with respect to s, and using the inequalities ∂_s h_s_C^1,α(h_s)⩽ Cη(s), and (<ref>), 
	there exists C=C(n,α)>0 such that for all s, we have:
	
    (Δ^𝕊^n-1Γ+n-1)∂_su_s_C^α(h_s)⩽ C η(s).

	We can therefore conclude that for all s, ∂_su_s _C^2,α(h_s)⩽ Cη(s).
	
	Iterating this for higher derivatives in all directions, we obtain the stated controls.





 §.§ 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.

	Let D_0,v_0>0 and (M^ℰ_i,g^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 ||≤ 3.
	
	
	Then, there exists a subsequence with fixed topology M=M_i^ℰ and a sequence of naïve desingularizations (M,g^D_t_i) of an Einstein orbifold (M_o,g_o) such that: for all l∈ℕ and i large enough, there exists C(l, v_0, D_0)>0, β_1(v_0, D_0)>0 and β_2(v_0, D_0)>0, ϵ_i>0, ϵ_i→ 0, and a diffeomorphism Φ_i: M→ M satisfying
	
	
  * on ℳ_o(ϵ_i), we have
	
    Φ^*_ig^ℰ_i-g_o_C^l(g_o)⩽ Cϵ_i.


	
  * at the different scales T_i,j (associated to t_i), on 𝒩_j(ϵ_i) we have
	
    Φ^*_ig^ℰ_i/T_i,j-g_b_j_C^l(g_b_j)⩽ Cϵ_i.

	
	
  * in the intermediate regions, 𝒜_k(t_i,ϵ_i) included in a metric annulus of radii ρ_1^i,k= 1/8T_i,k^1/2ϵ_i^-1, ρ_2^i,k = 8 T_i,j^1/2ϵ_i, for all ρ_1^i,k⩽ρ⩽1/2ρ_2^i,k, and denoting Φ_i,ρ = Φ_i ∘ϕ_ρ, where ϕ_ρ is the homothetic transformation of ration ρ on ℝ^4Γ_k with its flat metric g_e, we have
	
    Φ_i,ρ^*g^ℰ_i/ρ^2-g_e_C^l(A_e(1,2))⩽ Cϵ_i[(ρ_1^i,k/ρ)^β_1+(ρ/ρ_2^i,k)^β_2].

	



    This is indeed much better than a C^∞ convergence on compacts to a flat cone as we have [(ρ_1^i,k/ρ)^β_1+(ρ/ρ_2^i,k)^β_2]⩽ 2, and a decay in the annulus.






	
	


§ OPTIMAL C^K,Α-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, see <cit.> for instance.


	For all n, 0<α<1, δ>0 and v_0>0, there exists ϵ>0 such that if (M^n,g) satisfies
	
	
  * 1-ϵ<Sec(g)<1+ϵ,
	
  * and (M)>v_0,
	 
	then, there exists Γ a finite subgroup of SO(n+1) (whose order is bounded by a function of v_0), and a diffeomorphism Φ: 𝕊^nΓ→ M for which:
	
    Φ^*g-g^𝕊^nΓ_C^1,α(g^𝕊^nΓ)⩽δ,

	where g^𝕊^nΓ is a round metric on 𝕊^nΓ.


In this appendix, we will prove the following proposition.


Let (M,g_0) be a Riemannian manifold of dimension n with sectional curvatures equal to 1, and k∈ℕ.
    Then, there exists δ>0 and C>0, such that if a metric g on M satisfies:
   
    g-g_0_C^1,α(g_0)⩽δ,

    then, there exists a diffeomorphism Φ: M→ M, for which 
    Φ^*g-g_0_C^k+1,α(g_0)⩽ C(g)-(n-1)g_C^k(g_0).

    
As an immediate consequence of Lemma <ref> and Proposition <ref>, we get the following statement.

For all n, 0<α<1 and v_0>0, there exists ϵ_0>0 and C>0 such that for any 0<ϵ<ϵ_0, if (M^n,g) satisfies:
	
	
  * 1-ϵ<Sec(g)<1+ϵ,
	
  * and (M,g)>v_0,
	 
		then, there exists Γ a finite subgroup of SO(n+1), and a diffeomorphism Φ: 𝕊^nΓ→ M for which
		
    Φ^*g-g_0_C^1,α(g_0)⩽ C ϵ.




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



    Let (M^n,g_0) be a Riemannian manifold whose sectional curvatures are constant equal to 1. For λ:= (n-2)(n-1)/2, define the operator
    
    E_g_0 (g):= (g)-(g)/2g+λ g + δ^*_gδ_g_0 g.

    Then, following <cit.>, by <cit.> and by the Bianchi identity, there exists δ>0 such that E_g_0^-1({0})∩ B_C^1,α(δ) is exactly the set of Einstein metrics with constant Λ 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,α(g_0)⩽δ for δ>0 a constant which we will choose small enough in the rest of the proof. According to <cit.>, for δ small enough, there exists a diffeomorphism Φ: M → M such that  
    δ_g_0Φ^*g=0.
    Hence, we have 
    
    E_g_0(Φ^*g)_C^k(g_0)⩽(g)-(n-1)g_C^k(g_0).

    Now, by <cit.>, the linearization of E at g_0 is invertible (see again <cit.> for the adaptation to the operator E) and by the inverse function theorem Lemma <ref> we deduce that for δ small enough, and for all 1⩽ p<+∞ and k∈ℕ there exists C>0, such that we have
    
    Φ^*g-g_0_W^k+2,p(g_0)⩽ CE_g_0(Φ^*g)_W^k,p(g_0),

    and finally, by (<ref>), we have
    
    Φ^*g-g_0_C^k+1,α(g_0)⩽ C(g)-(n-1)g_C^k(g_0),

    because on the sphere, for p large enough, W^k+2,p(g_0) embeds continuously in C^k+1,α(g_0), and so does C^k(g_0) in W^k,p(g_0).

	




























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