Noncollapsed degeneration of Einstein 4-manifolds II
Tristan Ozuch




    In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.
    
     The description of Einstein metrics as the result of a gluing-perturbation procedure sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we extend the obstruction to the desingularization of Einstein orbifolds found by Biquard, and prove that it holds for any desingularization by trees of quotients of gravitational instantons only assuming a mere Gromov-Hausdorff convergence instead of specific weighted Hölder spaces. This is conjecturally the general case, and can at least be ensured by topological assumptions such as a spin structure on the degenerating manifolds. We also identify an obstruction to desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE spaces.



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


An Einstein metric, g satisfies, for some real Λ, the equation

    (g)=Λ g.

In dimension 4, they are considered optimal for the homogeneity of their Ricci curvature, as critical points of the Einstein-Hilbert functional with fixed volume, g↦∫_M _g dvol_g, and more importantly as minimizers of the L^2-norm of Riemann curvature tensor, g↦∫_M |_g|^2dvol_g. 

From dimension 4, even under natural assumptions of bounded diameter (compactness) and lower bound on the volume (noncollapsing) Einstein metrics can develop singularities. One major goal for 4-dimensional geometry is therefore to understand the moduli space of Einstein metrics on a differentiable manifold M^4 defined as

    𝐄(M^4) := {(M^4,g) | ∃Λ∈ℝ, (g)=Λ g, (M^4,g)= 1}𝒟(M^4).

and to compactify it with a useful structure. This has been done in an L^2 and then Gromov-Hausdorff (GH) sense in <cit.>. More precisely, if we denote by 𝐄(M^4)_GH the compactification of the moduli space 𝐄(M^4) for the (pointed) Gromov-Hausdorff distance, d_GH, we have a decomposition

    𝐄(M^4)_GH = 𝐄(M^4)∪∂_o𝐄(M^4)∪∂_∞𝐄(M^4),

where ∂_∞𝐄(M^4) consists in limits with infinite diameter, and ∂_o𝐄(M^4) consists in singular limits with bounded diameter. 

We will focus on local questions and for simplicity assume most of the time that we work on spaces with uniformly bounded diameter and therefore study the Gromov-Hausdorff neighborhood of ∂_o𝐄(M^4). We therefore work on the d_GH-completion of 𝐄(M^4), which is 𝐄(M^4)∪∂_o𝐄(M^4). The metric spaces in ∂_o𝐄(M^4) and the associated singularity blow-ups in the Gromov-Hausdorff sense have been understood for a long time in <cit.>: they are respectively Einstein orbifolds and Ricci-flat ALE orbifolds. 

Anderson then asked the converse question for instance in <cit.>, namely, are all Einstein orbifolds limits of smooth Einstein ? To answer this question one has to understand if the reverse of the degeneration, the desingularization, of Einstein orbifolds in ∂_o𝐄(M^4) is possible. A natural way to desingularize an orbifold is by a gluing-perturbation technique. 

The goal of the present paper is to develop a gluing-perturbation procedure which attains any noncollapsed Einstein 4-manifold which is sufficiently close to an Einstein orbifold in the Gromov-Hausdorff sense. This in particular elucidates the d_GH-neighborhood of the boundary ∂_o𝐄(M^4) in 𝐄(M^4), and we will use this description in future works. In this paper, we will use it to prove that not all Einstein orbifolds can be desingularized by Einstein metrics in the Gromov-Hausdorff sense with an expected topology which partially answers the above question of Anderson.



 §.§ Desingularization of Einstein 4-orbifolds and obstructions


A natural technique to desingularize orbifolds is the following procedure: we glue Ricci-flat ALE manifolds to the singularities of the orbifold to obtain an approximate Einstein metrics, and then try to perturb it into an actual Einstein metric. We will call such gluings, naïve desingularizations of the orbifold and often denote them g^D_t (see Definition <ref>), where t is the set of gluing parameters which are small positive real numbers. The main result of <cit.> is that the Gromov-Hausdorff proximity of an Einstein metric to an Einstein orbifold implies the proximity to a naïve desingularization g^D_t in the sense of a weighted Hölder norm denoted C^2,α_β,*(g^D_t). This norm is bounded on symmetric 2-tensors decaying in the neck regions where the gluing takes place.

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


    Let g^D_t be a naïve desingularization of an Einstein orbifold (M_o,g_o) with small enough gluing parameters.
    
    Then, there exists a small C^2,α_β,*(g^D_t)-neighborhood of g^D_t in which there exists a unique metric ĝ_t which is Einstein modulo obstructions while satisfying some gauge conditions with respect to g^D_t.

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

These Einstein modulo obstructions metrics ĝ_t are not interesting for themselves when they aren't Einstein as they are not geometrically motivated. Their purpose is for instance different from the metrics of <cit.> which are critical for some geometric functionals obtained by perturbing a connected sum of Einstein metrics.


Let us note that the Ricci flow starting at Einstein modulo obstructions metrics however exhibits interesting behaviors with respect to the Ricci flow. Indeed, in <cit.>, an obstruction is identified to a particular desingularization of 𝕋^4ℤ_2 and an ancient solution to the Ricci flow smoothing out the orbifold 𝕋^4ℤ_2 is constructed thanks to it.



Our construction however produces every smooth Einstein desingularization in a Gromov-Hausdorff sense. Indeed, together with the convergence of <cit.>, as a direct consequence, we have the following complete description of the Einstein metrics in a Gromov-Hausdorff neighborhood of an Einstein 4-orbifold. 

    Let (M_o,g_o) be an Einstein 4-orbifold. Then, there exists δ >0 such that if (M,g^ℰ) is an Einstein manifold satisfying
	
    d_GH((M,g^ℰ),(M_o,g_o))⩽δ,

	then, (M,g^ℰ) is isometric to a result of the gluing-perturbation procedure of Theorem <ref>.





 §.§ A premoduli space in the neighborhood of a singular metric


Classically, studying a moduli space requires understanding its compactification with a useful structure. The compactification (<ref>) a priori does not carry a useful structure as it comes from the rough Gromov-Hausdorff distance. The moduli space 𝐄(M^4) however admits a real-analytic structure around smooth metrics.

    Let (M,g_0) be an Einstein manifold. Then, there exists a d_GH-neighborhood of g_0 in 𝐄(M) which is the quotient by the isometry group of g_0 of a real-analytic subvariety of a finite dimensional real-analytic submanifold, W, of the space of smooth metrics on M.

The finite dimensional real-analytic submanifold, W, consists in metrics which are Einstein modulo the cokernel of the linearization of the Einstein equation at g_0 as is usually obtained by Lyapunov-Schmidt reduction. The Einstein metrics are exactly the metrics for which these obstructions vanish. Our description extends this local description of 𝐄(M^4) to  the boundary ∂_o𝐄(M^4), and the set of Einstein modulo obstructions metrics ĝ_t of Theorem <ref> is the analogue of the ambient space W of Theorem <ref>.

Theorem <ref> is an important local result which implies for instance that 𝐄(M) is locally finite. Anderson asked in <cit.> whether this structure extends to 𝐄(M^4)∪∂_o𝐄(M^4). The new description of the neighborhood of ∂_o𝐄(M) in (𝐄(M)_GH,d_GH) of Corollary <ref> provides a promising setting in which one can tackle this question. In particular, in Section <ref>, we provide an adaptation to the singular setting of Koiso's premoduli space around metrics of ∂_o𝐄(M).




 §.§ Degeneration of Kähler-Einstein manifolds


Even if our purpose here is to study the real Einstein equation and not Kähler-Einstein metrics, our analysis in weighted Hölder spaces extends the analysis leading to the gluing-perturbation theorems of <cit.> in the Kähler setting. Indeed, it allows us to glue and perturb multiple trees of singularities with arbitrary scales and Einstein deformations. It would therefore be interesting to extend the constructions of <cit.> to remove the “generic” (<cit.>) or “non degenerate” (<cit.>) conditions which correspond to restricting the gluing scales depending on the size of the Einstein deformation. We should also be able to allow general degenerations forming trees of singularities.

For instance, in Section <ref>, we precise the construction of <cit.> in the case of Kronheimer's gravitational instantons and prove that any tree of Kähler Ricci-flat ALE spaces can be glued and perturbed to a single Kähler Ricci-flat ALE metric with uniform controls (in our weighted Hölder norms) only depending on the group at infinity.





 §.§ Obstructions to the Gromov-Hausdorff desingularization of Einstein orbifolds


Our main application in this paper is a nonexistence result: there exists Einstein orbifolds which cannot be approached by smooth Einstein metrics with specific topologies in the Gromov-Hausdorff sense. For this, it is enough to prove that the obstructions of Theorem <ref> do not vanish.

The hyperkähler ALE spaces which are called gravitational instantons have been classified in <cit.> and their Kähler quotients have been classified in <cit.>. It is a famous conjecture, <cit.>, that all Ricci-flat ALE spaces are Kähler.

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




	Let (M_i,g_i)_i be a sequence of Einstein manifolds converging in the Gromov-Hausdorff sense to an Einstein orbifold (M_o,g_o), and assume that there exists a subsequence (M_i,g_i)_i whose possible blow-up limits are Kähler Ricci-flat ALE orbifolds glued in the same orientation. 
	
	Then, at every singular point p of (M_o,g_o), we have
	
    𝐑_g_o(p) =0.


This answers positively a question from <cit.> and extends it to the case of several singularities and allows the formation of trees of singularities. It more precisely states that the obstruction of <cit.> holds for any known possible configuration of singularity models, and that it holds even assuming the weakest possible convergence instead of a convergence in particular weighted Hölder spaces.


    Note that this obstruction is very different in nature from the result of <cit.> which shows that most compact Kähler-Einstein 4-orbifolds with positive Ricci-curvature cannot be limits of Kähler-Einstein manifolds. Indeed, our obstruction 𝐑 =0 is always satisfied in this situation, and it remains unknown if these metrics can be desingularized by real Einstein metrics. It is also different from the obstruction found in <cit.> where the obstruction 𝐑=0 is also satisfied by the orbifold 𝕋^4ℤ_2.


Under topological assumptions, it is known that the singularity models appearing are Kähler and glued in the same orientation, in particular we have the following illustration of our obstruction.
[Corollary <ref>]
    Consider 𝕊^4⊂ℝ^5 and the quotient by ℤ_2 given by (x_1,x_2,x_3,x_4,x_5) ∼ (x_1,-x_2,-x_3,-x_4,-x_5). We will denote this space 𝕊^4ℤ_2 which is an Einstein orbifold with two ℝ^4ℤ_2 singularities. The minimal resolutions of the two singularities ℝ^4ℤ_2≈ℂ^2ℤ_2 has the topology M:= 𝕊^4ℤ_2#T^*𝕊^2#T^*𝕊^2, where # denotes the gluing of an ALE space to an orbifold along their asymptotic cone.

    
0.68
Then, for any 1⩽ p<∞, there exists a sequence of metrics (M,g_i)_i with both 
    (g_i)-3g_i_L^p(g_i)→ 0  and (g_i)⩾ 3g_i
 while 
    (M,g_i) (𝕊^4ℤ_2,g_𝕊^4ℤ_2),

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


    0.29

    < g r a p h i c s >





    In the same fashion, a conjecture of Anderson states that there is no sequence of asymptotically hyperbolic Einstein metrics on T^*𝕊^2 desingularizing the hyperbolic orbifold ℍ^4ℤ_2 obtained by antipodal identification in a global geodesic chart. It was proven in <cit.> assuming among other things a convergence speed in weighted spaces towards the orbifold depending on the maximum of the curvature. We can prove it assuming a pointed Gromov-Hausdorff convergence together with a suitable control in weighted spaces at infinity, this time independent on the maximum of the curvature. It is again possible to desingularize ℍ^4ℤ_2 with Ricci pinched in any L^p, space for 1⩽ p<∞ or with  bounded above or below by -3.




  §.§.§ Hitchin-Thorpe inequality and degeneration of Einstein manifolds


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

    2χ(M)⩾ 3|τ(M)|,

where χ is the Euler characteristic, and τ the signature. These topological invariants have definitions adapted to orbifolds and ALE spaces which we will denote χ̃ and τ̃, and any orbifold M_o admitting an Einstein metric satisfies

    2χ̃(M_o)⩾ 3|τ̃(M_o)|.

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


    Let (M_o,g_o) be an Einstein orbifold, and assume that (M,g_i)_i is a sequence of Einstein metrics converging to (M_o,g_o) in the Gromov-Hausdorff sense.
    
    Then, we have the following inequality,
    
    2χ(M)-3|τ(M)|⩾ 2χ̃(M_o)-3|τ̃(M_o)|.

    Moreover, there is equality if and only if M is a desingularization of M_o by gluing of trees of Kähler Ricci-flat ALE orbifolds in the same orientation (with the same sign for τ̃). 
    In this equality case, we have the condition 
    𝐑(g_o)=0
 at every singular point.




  §.§.§ Degeneration of Einstein metrics on spin manifolds


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


    Let (M_i,g_i)_i be a sequence of spin Einstein 4-manifolds  converging to an Einstein orbifold (M_o,g_o) in the Gromov-Hausdorff sense. Then, (M_o,g_o) is spin, and at any of its singular points whose group is in SU(2), we have the obstruction 
    𝐑_g_o = 0.






  
General obstructions for spherical and hyperbolic orbifolds.

Our Theorem <ref> holds for any singularity model which might be non-Kähler. We will use it lastly to identify an obstruction to desingularizing spherical or hyperbolic orbifolds by any Ricci-flat ALE manifold in Theorem <ref>. This provides an obstruction to any standard gluing-perturbation technique but will only imply an actual obstruction to the Gromov-Hausdorff desingularization by Ricci-flat ALE manifolds whose deformations are integrable (this is the case of all known examples).
	
	Spherical and hyperbolic orbifolds cannot be desingularized in the Gromov-Hausdorff sense by Ricci-flat ALE spaces which are integrable (see Definition <ref>).
	
	
	
	


 §.§ Outline of the paper

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

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

In Section 4, we prove that any naïve desingularization can be perturbed to a metric which is Einstein modulo some obstruction, that is, an approximate cokernel of the linearization of the gauged Einstein operator. The point is that every possible Einstein metric is produced this way, and that whenever the obstructions do not vanish, it is impossible to perturb the naïve desingularization to an Einstein metric. The proof again relies on a Lyapunov-Schmidt reduction in our weighted Hölder spaces. We then extend Koiso's definition of a premoduli space in the neighborhood of a singular metric.

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

In Section 6, we test the above obstructions on degenerations of Einstein manifolds forming trees of Kähler Ricci-flat ALE orbifolds. By developing our analysis on trees of singularities, we prove that the obstruction of <cit.> for the Eguchi-Hanson metric extends to any tree of quotients of gravitational instantons and holds under a mere Gromov-Hausdorff convergence. An important step is to prove that a gluing of gravitational instantons in the same orientation can be uniformly perturbed to an Einstein metric in our norms.

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

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




 §.§ Acknowledgements


I would like to thank my PhD advisor, Olivier Biquard, for his constant support as well as his mathematical and writing advice. I would also like to thank Aaron Naber for inviting me to Northwestern University in Spring 2017, and for pointing out and discussing the issue addressed in Section 8. I would also like to thank Gilles Carron and Michael Singer for their comments and suggestions when reviewing this work as a part of my PhD thesis. I would finally like to thank Aude Genevay for her help with the graphics.



§ ORBIFOLDS, ALE SPACES AND NAÏVE DESINGULARIZATIONS


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



 §.§ Einstein orbifolds and ALE spaces


For Γ a finite subgroup of SO(4) acting freely on 𝕊^3, let us denote (ℝ^4Γ,g_e) the flat orbifold obtained by the quotient by the action of Γ, and r_e:= d_e(.,0).


    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.
    


    Consequently, the analysis on an orbifold is exactly the same as the analysis on a manifold up to using finite local coverings at the singular points.



    We define r_o, a smooth function on M_o satisfying r_o:= (Φ_k)_* r_e on each U_k, and such that on M_o\ U_k, we have ϵ_0⩽ r_o<1 (the different choices will be equivalent for our applications).
    
    We will denote, for 0<ϵ⩽ϵ_0, 
    M_o(ϵ):= {r_o>ϵ} = M_o\(⋃_k Φ_k(B_e(0,ϵ)) ).



Let us now turn to ALE Ricci-flat metrics.


    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)|_g_e⩽ C_l r_e^-n.

    



We define r_b a smooth function on N satisfying r_b:= (Ψ_k)_* r_e on each U_k, and r_b:= (Ψ_∞)_* r_e on U_∞, and such that ϵ_0⩽  r_b⩽ϵ_0^-1 on the rest of N (the different choices are equivalent for our applications).

For 0<ϵ⩽ϵ_0, we will denote 
    N(ϵ):= {ϵ<r_b<ϵ^-1} = N\(⋃_k Ψ_k(B_e(0,ϵ)) ∪Ψ_∞((ℝ^4Γ_∞)\ B_e(0,ϵ^-1))).


Now, consider a subset S_o of the singular points of M_o (respectively S of N).

    We define the functional r_o,S_o (respectively r_b,S) exactly like in Definitions <ref> (respectively <ref>) by only considering the sets U_k containing points of S_o (respectively S).




 §.§ Naïve desingularizations


Let us now recall the definition of a naïve desingularization of an orbifold from <cit.>.



  
Gluing of ALE spaces to orbifold singularities.

    Let 0<2ϵ<ϵ_0 be a fixed constant, t>0, (M_o,g_o) an orbifold and Φ: B_e(0,ϵ_0)⊂ℝ^4Γ→ U a local chart of Definition <ref> around a singular point p∈ M_o. 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 of Definition <ref>.
    
    Define s>0, ϕ_s: x∈ℝ^4Γ→ sx ∈ℝ^4Γ. For t<ϵ_0^4, we define 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)).
 Consider moreover χ:ℝ^+→ℝ^+, a C^∞ cut-off function supported on [0,2] and equal to 1 on [0,1].
    

    We define a naïve gluing of (N,g_b) at scale 0<t<ϵ^4 to (M_o,g_o) at the singular point p, which we will denote (M_o#N,g_o#_p,tg_b) by putting g_o#_p,tg_b=g_o on M\ U, g_o#_p,tg_b=tg_b on K, and 
    
    g_o#_p,tg_b =  χ(t^-1/4r_e)Ψ_∞^*g_b + (1-χ(t^-1/4r_e))Φ^*g_o

    on 𝒜(t,ϵ):=A_e(ϵ^-1√(t),2ϵ).



    It is possible to compose ϕ_√(t) with any isometry of ℝ^4Γ. This is equivalent to gluing a different Ricci-flat ALE metric.


More generally, it is possible to desingularize iteratively by trees of Ricci-flat ALE orbifolds. Consider (M_o,g_o) an Einstein orbifold (the index o stands for orbifold), and S_o a subset of its singular points and (N_j,g_b_j)_j (the index b_j stands for j-th bubble) 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 finally assume that there is a one to one 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 desingularization pattern.


    Let 0<2ϵ<ϵ_0, D be a desingularization pattern for (M_o,g_o), and let 0<t_j<ϵ^4 be relative gluing scales. The metric g^D_t is then the result of the following finite iteration: 
    
        
  * start with a deepest bubble (N_j,g_b_j), that is, j such that S_j= ∅,
        
  * if p_j∈ N_k, replace (N_k,g_b_k,S_j) and (N_j,g_b_j,∅) in D 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.
        
  * choose another deepest bubble and do the same.
    
    
    For t = (t_j)_j, 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^16t, the metric is T_jg_b_j.


 Let (M_o,g_o) be an Einstein orbifold, and (M,g^D) a naïve desingularization of (M_o,g_o) by a tree of ALE Ricci-flat orbifolds (N_j,g_b_j) glued at scales T_j>0. 

Here, the manifold M is also covered as M = M_o^t∪⋃_jN_j^t, where

    M_o^t: = M_o\(⋃_k Φ_k(B_e(0,t_k^1/4)) ),

 where t_k>0 is the relative gluing scale of N_k at the singular point p_k∈ M_o, and where 
 
    N_j^t:= (N_j\Ψ_∞((ℝ^4Γ_∞)\ B_e(0, 2t_j^-1/4) )) \(⋃_k Ψ_k(B_e(0,t_k^1/4) ).

On M_o^16t⊂ M_o^t, we have g^D = g_o and on each N_j^16t⊂ N_j^t, we have g^D = T_j g_b_j. We also define t_max:= max_j t_j. By Definition <ref>, on the intersection N_j^t∩ M_o^t we then have √(T_j)r_b_j = r_o, and on the intersection N_j^t∩ N_k^t, we have √(T_j)r_b_j = √(T_k)r_b_k.


    On a naïve desingularization (M,g^D), we define a function r_D in the following way:
    
        
  * r_D = r_o on M_o^t,
        
  * r_D = √(T_j)r_b_j on each N_j^t.
    
    The function r_D is smooth on M.



    Let (N_k,g_b_k) be a Ricci-flat ALE orbifold of the above tree of singularities. We define 𝒜_k(t,ϵ) as the connected region with ϵ^-1√(T_k)<√(T_k)r_b_k=r_D<ϵ t_k^-1/2√(T_k) with a nonempty intersection with N_k^t.





    We define the following cut-off functions thanks to the cut-off function χ used in Definition <ref>.
    
        
  * χ_M_o^t, equal to 1 on M_o^16t and equal to 1-χ(t_k^-1/4r_o) on each annulus 𝒜_k(t,ϵ). It is supported on M_o^t.
        
  * χ_N_j^t, equal to 1 on N_j^16t and equal to 1-χ(t_k^-1/4r_b_j) on each annulus 𝒜_k(t,ϵ) at its singular points and χ(t_j^1/4r_b_j) in a neighborhood of infinity. It is supported on N_j^t.
        
  * χ_𝒜_k(t,ϵ) equal to 1 on 𝒜_k(t,1/2ϵ), and equal to χ(ϵ^-1t_k^1/2r_b_k) - χ(ϵ r_b_k). It is supported on 𝒜_k(t,ϵ).
        
  * χ_B(p_k,ϵ) for p_k∈ M_o equal to 1 on r_o<ϵ equal to χ(ϵ^-1r_o) around p_k. It is supported in  supported in r_o<2ϵ around p_k.
    
    Note that since t_k^1/2r_b_k = r_o on the gluing region, we have χ_B(p_k,ϵ)-χ_𝒜_k(t,ϵ) = χ(ϵ r_b_k) = χ(ϵ t_k^-1/2 r_o).
    
    The definition extends to deeper Ricci-flat ALE orbifolds thanks to the iteration of Definition <ref>.

This in particular yields a partition of unity,

    1 = χ_M_o^t + ∑_j χ_N_j^t.




§ WEIGHTED HÖLDER SPACES AND DECOUPLING NORMS


We now present the spaces in which the analysis of the rest of the article takes place.



 §.§ Weighted Hölder spaces


Let us construct weighted Hölder spaces adapted to our situation. Let (M,g^D_t) be a naïve desingularization of an orbifold (M_o,g_o) by Ricci-flat ALE orbifolds (N_j,g_b_j) at scales T_j>0. 



  §.§.§ Weighted Hölder spaces on orbifolds and ALE spaces


Let us first define weighted spaces on manifolds asymptotic to cones or with conical singularities. For a tensor s, a point x, 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 s on M as

    [s]_C^α(g)(x):= sup_{y∈ T_xM,|y|< _g(x)}| exp_x^*s(0)-exp_x^*s(y)/|y|^α|_exp_x^*g.


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

    Let β∈ℝ, k∈ℕ, 0<α<1 and (M_o,g_o) an orbifold. Then, for all tensor s on M_o, we define
    
    s _C^k,α_β(g_o)   := sup_M_or_o^-β(∑_i=0^k r_o^i|∇_g_o^i s|_g_o + r_o^k+α[∇_g_o^ks]_C^α(g_o)).



    The injectivity radius at a point x∈ M_o is equivalent to r_o.


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

For β∈ℝ, k∈ℕ and 0<α<1 on an orbifold ALE (N,g_b), we define
   
    s _C^k,α_β(g_b):= sup_N{max(r_b^β,r_b^-β)( ∑_i=0^kr_b^i|∇_g_b^i s|_g_b + r_b^k+α[∇_g_b^ks]_C^α(g_b))}.



    The injectivity radius at a point x∈ N is equivalent to r_b.



    We similarly define the norms for C^k,α_β(g_o,S_o) and C^k,α_β(g_b,S) by replacing r_o and r_b by r_o,S_o and r_b,S of Definition <ref>.




  §.§.§ Weighted Hölder spaces on trees of singularities


Let us assume that (M,g^D_t) is a naïve desingularization of (M_o,g_o) by a tree of singularities (N_j,g_b_j). For t_max:= max_j t_j<ϵ_0^4, for ϵ_0>0 the constant of Subsection <ref> only depending on g_o and the g_b_j, we define the global weighted norm in the following way.


		Let β∈ℝ and k∈ℕ, 0<α<1. We define for s∈ TM^⊗ l_+⊗ T^*M^⊗ l_- a tensor of type (l_+,l_-)∈ℕ^2, with l:= l_+-l_- the associated conformal weight.
		
    s_C^k,α_β(g^D):=χ_M_o^t s _C^k,α_β(g_o) + ∑_j T_j^l/2χ_N_j^ts_C^k,α_β(g_b_j).




    The factor T_j^l/2 in (<ref>) comes from the fact that on N_j^t, the metric g^D is close to T_j g_b_j. For a tensor s of conformal weight l, we have |s|_T_jg_b_j = T_j^l/2|s|_g_b_j, and therefore T_j^l/2χ_N_j^ts_C^k,α_0(g_b_j)=χ_N_j^ts_C^k,α_0(T_jg_b_j).



For a function, being bounded for this norm means being bounded in C^0_β-norm means being bounded everywhere and having a particular decay in the neck regions.




Its main advantage is that it is totally adapted to trees of singularities.





    If (M,g^D) is a partial desingularization at the singular points of S_o and S_j, we define the C^k,α_β(g^D)-norm similarly thanks to C^k,α_β(g_o,S_o) and C^k,α_β(g_b,S) of Remark <ref>.


Thanks to this norm, we can for example rewrite and extend the statement of <cit.>.


	Let D_0,v_0>0, l∈ℕ, and β= β(v_0,D_0)>0 obtained in <cit.>. Then, for all ϵ>0, there exists δ = δ(ϵ,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 above by D_0,
		
  * the Ricci curvature is bounded ||≤ 3.
	
	and for an Einstein orbifold (M_o,g_o),
	
    d_GH((M,g^ℰ),(M_o,g_o))⩽δ,

	then, there exists a naïve desingularization (M,g^D) of (M_o,g_o) by a tree of singularities and a diffeomorphism ϕ: M→ M such that 
    ϕ^*g^ℰ-g^D_C^l_β(g^D)⩽ϵ.



    Let l∈ℕ. Let us give a proof by contradiction and consider a sequence of counter examples, that is a sequence of Einstein manifolds (M_i,g_i)_i such that
    
        
  * (g_i)⩾ v_0>0,
        
  * diam(g_i)<D_0 and
        
  * |(g_i)|_g_i⩽ 3
    
    converging in the Gromov-Hausdorff sense to an Einstein orbifold (M_o,g_o), but such that there exists ϵ>0 for which, for all i∈ℕ and any naïve desingularization (M_i,g^D_i) of (M_o,g_o), and all diffeomorphism Φ_i:M_i → M_i, we have Φ_i^*g_i-g^D_i_C^l_β(g^D)>ϵ.
    
    According to <cit.>, this implies that there exists a subsequence (M,g_i)_i with fixed topology, and a sequence (M,g^D_i)_i contradicting the assumption for i large enough by definition of the weighted norm.


On the annuli of low curvature 𝒜_k(t,ϵ_0) pulled back on flat annuli A_e(ρ_1,ρ_2)⊂ℝ^4Γ, the weighted norm on (M,g^D) is equivalent is equivalent to a particular norm which allows us to control independently of the radii the sum of tensors decaying at the center of the annulus and of tensors decaying at infinity.

	
Let 0<ρ_1<ρ_2, β∈ℝ, k∈ℕ, 0<α<1 and a tensor s on (A_e(ρ_1,ρ_2),g_e). We define

    η(r_e):=max((ρ_1/r_e)^β,(r_e/ρ_2)^β)⩽ 1,

 and the norm,

    s_C^k,α_β(A_e(ρ_1,ρ_2)):   = sup_A_e(ρ_1,ρ_2)[ η^-1(r_e)( ∑_i=0^k r_e^i|∇_g_e^is|_g_e + r_e^k+α[∇_g_e^ks]_C^α(g_e))].



In the rest of this article, we will often use spaces denoted fC^k,α_β for a positive function f. They will always be equipped with the following norm

    s_fC^k,α_β:= s/f_C^k,α_β.


    By definition of r_D, for all m, there exists a constant C>0 only depending on the cut-off functions of Definition <ref> such that

    1/Cs_r^m_DC^k,α_β(g^D)⩽χ_M_o^t s _r^m_oC^k,α_β(g_o) + ∑_j T_j^l-m/2χ_N_j^ts_r^m_b_jC^k,α_β(g_b_j)⩽ Cs_r^m_DC^k,α_β(g^D).




    The metric g^D is equal to g_o on {r_D>ϵ}∩ M_o^t, and to T_jg_b_j on {√(T_j)ϵ<r_D<2√(T_j)ϵ^-1}∩ N_j^t. Since on the 𝒜_k(t,ϵ) between N_k and N_j (resp. M_o) identified with A_e(ϵ^-1√(t_kT_j),ϵ√(T_j)) (resp. A_e(ϵ^-1√(t_k),ϵ)), g^D is arbitrarily close to g_e, we see that defining η̃ : M ↦ℝ^+ a function equal to 
    
        
  * 1 on {r_D>ϵ}∩ M_o^t and on {√(T_j)ϵ<r_D<2√(T_j)ϵ^-1}∩ N_j^t and
        
  * equal to the function η⩽ 1 defined in (<ref>) on the associated euclidean annulus A_e(ϵ^-1√(t_kT_j),ϵ√(T_j)) (or A_e(ϵ^-1√(t_k),ϵ)).
    
    Then, the norm r_D^mC^k,α_β(g^D) is equivalent (independently of t) to the norm which to a tensor s associates
    
    s=sup_Mr_D^-mη̃(r_D)^-1(∑_i=0^k r_D^i|∇_g^D^i s|_g^D + r_D^k+α[∇_g^D^ks]_C^α(g^D)).





    Let β⩽β', k+α⩽ k'+α', and m,m'∈ℤ. 

    For all the previously mentioned weighted Hölder spaces generically denoted r^m C^k,α_β, we have the following properties: for any tensors s and s'
    
        
  * s_r^mC^k,α_β⩽s_r^mC^k',α'_β',
        
  * ∇^ks_r^mC^k'-k,α_β⩽s_r^m+kC^k',α_β
        
  * if * is a composition, a product of a contraction of tensors, there exists C= C(*, k , α)>0 such that 
        
    s*s' _r^m+m'C^k,α_β+β'⩽ C s_r^mC^k,α_βs'_r^m'C^k,α_β'.

    
    Let us give an explanation for that last inequality (<ref>) for bilinear operations and assume first that k=0, α=0 and consider depending on the situation 
    
        
  * (w,w') = (r_o^-β,r_o^-β') on (M_o,g_o),
        
  * (w,w') = (max(r_b^β,r_b^-β),max(r_b^β',r_b^-β')) on (N,g_b)
        
  * or (w,w')=(η̃^-1,.η̃'.^-1) with the weights η̃ and η̃' used in (<ref>) above respectively associated to β and β' on (M,g^D).
    
    The goal is to bound 
    w.w'r^-m-m'|s*s'| uniformly and this is done using the definitions of the norms which yield
    w r^-m |s|⩽s_r^mC^0_β and
    w'r^-m'| s'|⩽s'_r^m'C^0_β' for any of the above spaces. The derivatives are treated thanks to the second above inequality and Leibniz rule.




  §.§.§ Weighted Schauder estimates


Weighted Schauder estimates hold in these norms for the operator P:= 1/2∇^*∇-. 


    For all β>0 and 0<α<1, there exists C>0 and ϵ>0  such that if h is a symmetric 2-tensor on (M,g^D), and g a metric on M satisfying 
    g-g^D_C^2,α_β(g^D)⩽ϵ,

    then, we have
    
    h_C^2,α_β(g^D)⩽ C (P_gh_r^-2_DC^α_β(g^D)+h_C^0_β(g^D)).



    Let g be a metric on M satisfying g-g^D_C^2,α_β(g^D)⩽ϵ, for ϵ>0 which we will choose small enough along the proof.
    On the compacts M_o(ϵ_0) of the orbifold and N_j(ϵ_0) of the ALE orbifolds minus their singular points, we have an elliptic estimate for the operators P_g_o and P_g_b_j: there exists C_1>0 such that for any symmetric 2-tensors h_o on M_o(ϵ_0) and h_j on N_j(ϵ_0), we have
    
    (h_o)_|M_o(2ϵ_0)_C^2,α(g_o)⩽ C_1 ((P_g_oh_o)_|M_o(ϵ_0)_C^α(g_o)+(h_o)_|M_o(ϵ_0)_C^0(g_o)),

    and
    
    (h_j)_|N_j(2ϵ_0)_C^2,α(g_b_j)⩽ C_1 ((P_g_b_jh_j)_|N_j(ϵ_0)_C^α(g_b_j)+(h_j)_|N_j(ϵ_0)_C^0(g_b_j)).

    
    By assumption, there exists C>0 only depending on g_o and g_b_j such that g-g_o_C^2,α(g_o)⩽ Cϵ on M_o(ϵ_0) and g/T_j-g_b_j_C^2,α(g_b_j)⩽ Cϵ on N_j(ϵ_0). We conclude that for ϵ small enough, the operators P_g and P_g/T_j, which are close to the operators P_g_o and P_g_b_j, satisfy for all h_o on M_o(ϵ_0) and h_j on N_j(ϵ_0),
    
    (h_o)_|M_o(2ϵ_0)_C^2,α(g_o)⩽ 2C_1 ((P_gh_o)_|M_o(ϵ_0)_C^α(g_o)+(h_o)_|M_o(ϵ_0)_C^0(g_o)),

    and
    
    (h_j)_|N_j(2ϵ_0)_C^2,α(g_b_j)⩽ 2C_1 ((P_g/T_jh_j)_|N_j(ϵ_0)_C^α(g_b_j)+(h_j)_|N_j(ϵ_0)_C^0(g_b_j)).

    
    On each almost flat annulus 𝒜_k(t,ϵ_0), let us denote 𝒜(ρ,ρ'):={ρ⩽ r_D ⩽ρ'}. There exists a diffeomorphism ϕ_ρ: A_e(1/2,4) →𝒜(ρ/2,4ρ) such that 
    
    ϕ_ρ^*g/ρ^2-g_e_C^1,α(A_e(1/2,4))⩽ Cη(ρ)ϵ.

    then, by ellipticity, for ϵ small enough, there exists C_2>0, such that for all symmetric 2-tensor h on A_e(1/2,4) we have,
    
    h_C^2,α(A_e(1,2))⩽  2C_2( h_C^0(A_e(1/2,4)) + P_ϕ_ρ^*g/ρ^2 h_C^α(A_e(1/2,4))).

    Coming back to (M,g), this implies that for ϵ small enough, we have for h a symmetric 2-tensor on 𝒜(ρ/2,4ρ),
    
    h_C^2,α(𝒜(ρ,2ρ),g/ρ^2)⩽  4C_2( h_C^0(𝒜(ρ/2,4ρ),g/ρ^2) + P_g/ρ^2 h_C^α(𝒜(ρ/2,4ρ),g/ρ^2)).

    The norm of a symmetric 2-tensor s behaves in the following way by rescaling, for t>0
    
    |s|_g/t = t|s|_g,

    and the operator P behaves in the following way by rescaling, for t>0:
    P_g/t = tP_g. Multiplying both sides of the equality by r_D^-2, we get
    
    h_C^2,α((ρ,2ρ),g)⩽  4C_2( h_C^0((ρ/2,4ρ),g) + r_D^2P_g h_C^α((ρ/2,4ρ),g)).

    Given the controls on the derivatives of r_D, we deduce the stated result by definition of the weighted norms by multiplying both sides of the inequality by the weight of the norm.


Analogous estimates also hold for the elliptic operator δδ^* with the same proof.


    For all β>0 and 0<α<1 there exists C>0 and ϵ>0 such that if X is a vector field on (M,g^D), and g a metric on M satisfying 
    g-g^D_C^2,α_β(g^D)⩽ϵ,

    then, we have
    
    X_r_DC^3,α_β(g^D)⩽ C (δ_gδ^*_gX_r^-1_DC^1,α_β(g^D)+X_r_DC^0_β(g^D)).





 §.§ Decoupling norms


We will see here that to expect good controls for the operators P and δδ^* in the annular regions of our manifold, we need to consider separately the influence of traceless constant 2-tensors for P and linear vector fields of the kernel of δ_eδ^*_e for δδ^*.



  §.§.§ Estimates on annuli A_e(ϵ,ϵ^-1) of (ℝ^4, g_e).

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


    Let 0<β<1, 0<α<1, and P = 1/2∇^*∇ -. There exists C_e>0, and ϵ_e>0 such that for any symmetric 2-tensor h on an annulus of radii 0<ϵ<ϵ_e and 1/ϵ, there exists a constant symmetric 2-tensor H_0 and a symmetric 2-tensor H_* satisfying 
         
    ∇^*_e∇_e H_*=0,
         
         
    H_*_C^2,α_1(A_e(2ϵ,(1/2)ϵ^-1))⩽ C_e h-H_0_C^0_β(A_e(ϵ,ϵ^-1)),

    (notice the norm C^2,α_1 for the left hand side) and,
    
    h-H_0-H_*_C^2,α_β(A_e(2ϵ,(1/2)ϵ^-1))⩽ C_eP_g_e h_r^-2_eC^α_β(A_e(ϵ,ϵ^-1)).

    This implies in particular the following control: for all x ∈ A_e(1/2,2),
    
    |h-H_0(x)|_g_e+|   ∇ h(x)|_g_e +|∇^2 h(x)|_g_e + [∇^2 h]_C^α(g)(x) 
    ⩽     C_e ( (2ϵ)^βP_g_e h_r^-2_eC^α_β(A_e(ϵ,ϵ^-1))+ 2ϵh-H_0_C^2,α_β(A_e(ϵ,ϵ^-1))).




    This is a strictly better estimate than the elliptic estimates of Proposition <ref> which would only have given
    
    |h-H_0(x)|_g_e+|   ∇ h(x)|_g_e +|∇^2 h(x)|_g_e + [∇^2 h]_C^α(g_e)(x) 
    ⩽     C_e ( (2ϵ)^βP_g_e h_r^-2_eC^α_β(A_e(ϵ,ϵ^-1))+ (2ϵ)^βh-H_0_C^2,α_β(A_e(ϵ,ϵ^-1))).

    The difference will be crucial in the proof of Proposition <ref>.


    Let us start by noting that (<ref>) is a consequence of (<ref>) and (<ref>). Indeed, h-H_0 = (h-H_0-H_*)+H_*, and we have therefore, denoting 
    s_C^2,α(1/2,2):= sup_x∈ A_e(1/2,2)|s(x)|_g_e+|∇ s(x)|_g_e +|∇^2 s(x)|_g_e + [∇^2 s]_C^α(g_e)(x),

    
    h-H_0_C^2,α(1/2,2)⩽    h-H_0-H_*_C^2,α(1/2,2) + H_*_C^2,α(1/2,2)
    ⩽    (2ϵ)^βh-H_0-H_*_C^2,α_β(A_e(ϵ,ϵ^-1)) + 2ϵH_*_C^2,α_1(A_e(ϵ,ϵ^-1))
    ⩽    C_e((2ϵ)^βP_g_e h_r^-2_eC^α_β(A_e(ϵ,ϵ^-1)) + 2ϵh-H_0_C^2,α_β(A_e(ϵ,ϵ^-1))),

    by definition of the weighted norms and assuming, for C_e>0, the inequalities (<ref>) and (<ref>).

    
    On ℝ^4\{0}, the harmonic symmetric 2-tensors are sum of homogeneous harmonic symmetric 2-tensors whose coefficients in the canonical basis of ℝ^4 are proportional to r^j for j∈ℤ\{-1}.  These harmonic symmetric 2-tensors are more precisely of the form r_e^k H_k or r^-2-k_eH_k for k∈ℕ, where H_k is a homogeneous symmetric 2-tensor with |H_k|_g_e∼ r^0 whose coefficients, once restricted to the unit sphere are eigenfunctions of the spherical Laplacian with eigenvalue -k(k+2).
    

    
    For any symmetric 2-tensor h on A_e(ϵ,ϵ^-1), let us define H̃ the solution on A_e(ϵ,ϵ^-1) of the following Dirichlet problem, denoting for r>0, S_e(r):= {r_e = r},
    
    {   ∇^*_e∇_e H̃ =0, 
       H̃ = h  on  S_e(ϵ)∪ S_e(ϵ^-1).
          .

    More precisely, H̃ = ∑_k⩾ 0(ϵ r_e)^k H̃_k^+    + (ϵ^-1 r_e)^-2-kH̃_k^- where the H̃_k^± are homogeneous with |H̃_k^+|_g_e∼ r_e^0 and which, once restricted to the sphere are eigenvectors associated to -k(k+2). If we decompose in spherical harmonics h_|S_e(ϵ) =: ∑_k H_k(ϵ) and h_|S_e(ϵ^-1) =: ∑_k H_k(ϵ^-1), we have the system
    
    {   H_k(ϵ^-1) = H̃_k^+ + ϵ^4+2kH̃_k^-, 
       H_k(ϵ) = ϵ^2kH̃_k^+ + H̃_k^-,
          .

and therefore,

    {   H̃_k^+ = 1/1-ϵ^4+4k(H_k(ϵ^-1) -ϵ^4+2kH_k(ϵ) ), 
       H̃_k^- = 1/1-ϵ^4+4k(H_k(ϵ) - ϵ^2k H_k(ϵ^-1)),
          .


Denote H̃_*:= H̃-H̃_0^+. Since ∇^*_e∇_e H̃_* = 0, by elliptic regularity on the annulus A(ρ/2,4ρ)⊂ A_e(ϵ,ϵ^-1), there exists a constant C>0 independent of h such that we have,

    H̃_*_C^0(A(ρ,2ρ))⩽C/ρ^2H̃_*_L^2(A_e(ρ/2,4ρ)),

so to control the norm C^0_1(A_e(ϵ,ϵ^-1)) of H̃_*, we just have to control the L^2-norm of H̃_* on the different annuli A_e(ρ,2ρ)⊂ A(ϵ,ϵ^-1). Since the harmonic decompositions are L^2(S_e(1))-orthogonal, we have for a constant C>0 which may change from line to line

    ∫_A_e(ρ,2ρ)|H̃_*|_g_e^2 dv_g_e =    ϵ^4∫_ρ^2ρ∫_S_e(1)|H̃_0^-|_g_e^2r^-4dv_S_e(1)r^3dr
       + ∑_k⩾ 1∫_ρ^2ρ∫_S_e(1) |ϵ^kr^kH̃_k^+
    +ϵ^2+kr^-2-kH̃_k^-|_g_e^2 dv_S_e(1)r^3dr
    ⩽     C ϵ^4 ∫_S_e(1)|H̃_0^-|_g_e^2 dv_S_e(1)
       +C∑_k⩾ 1ϵ^2k∫_ρ^2ρ∫_S_e(1)|H̃_k^+|_g_e^2dv_S_e(1)r^2k+3dr
       +ϵ^4+2k∫_ρ^2ρ∫_S_e(1)|H̃_k^-|_g_e^2dv_S_e(1)r^-1-2kdr 
    ⩽     C ϵ^4 ∫_S_e(1)|H̃_0^-|_g_e^2 dv_S_e(1)
       +C∑_k⩾ 1ϵ^2kρ^2k+4∫_S_e(1)|H̃_k^+|_g_e^2 dv_S_e(1)+ϵ^4+2kρ^-2k∫_S_e(1)|H̃_k^-|_g_e^2 dv_S_e(1)
    ⩽     C ϵ^4ϵ^-3∫_S_e(ϵ)|H̃_0^-|_g_e^2 dv_S_e(ϵ)
       +C∑_k⩾ 1ϵ^2kρ^2k+4ϵ^3∫_S_e(ϵ^-1)|H̃_k^+|_g_e^2 dv_S_e(ϵ^-1)
       +ϵ^4+2kρ^-2kϵ^-3∫_S_e(ϵ)|H̃_k^-|_g_e^2 dv_S_e(ϵ)

Now, the equalities (<ref>) and the fact that the decompositions in spherical harmonics are orthogonal imply that for a constant C>0 which may change from line to line we have 

    ∑_k⩾ 0∫_S_e(ϵ)|H̃_k^-|_g_e^2 dv_S_e(ϵ)   ⩽ C∑_k 
        ∫_S_e(ϵ)|H_k(ϵ)|_g_e^2 dv_S_e(ϵ) + Cϵ^6+2k∫_S_e(ϵ^-1)|H_k(ϵ^-1)|^2_g_e dv_S_e(ϵ^-1)
       ⩽ C 
        ∫_S_e(ϵ)|h-H̃_0^+|_g_e^2 dv_S_e(ϵ) + Cϵ^6∫_S_e(ϵ^-1)|h-H̃_0^+|_g_e^2 dv_S_e(ϵ^-1)
       ⩽ C ϵ^3(h-H̃_0^+)_|S_e(ϵ)^2_C^0(g_e)  + Cϵ^3(h-H̃_0^+)_|S_e(ϵ^-1)^2_C^0(g_e)
       ⩽ C ϵ^3 h-H̃_0^+^2_C^0_β(A_e(ϵ,ϵ^-1))

because |(h-H̃_0^+)_|S_e(ϵ)|_g_e⩽h-H̃_0^+_C^0_β(A_e(ϵ,ϵ^-1)) and |(h-H̃_0^+)_|S_e(ϵ^-1)|_g_e⩽h-H̃_0^+_C^0_β(A_e(ϵ,ϵ^-1)) by definition of the norm, and similarly

    ∑_k⩾ 1∫_S_e(ϵ^-1)|H̃_k^+|_g_e^2 dv_S_e(ϵ^-1)⩽ C ϵ^-3h-H̃_0^+^2_C^0_β(A_e(ϵ,ϵ^-1)).

Together with (<ref>), (<ref>) and (<ref>), and since on A_e(ϵ,1/ϵ) for any k⩾ 1, ϵ(ρ+ρ^-1)⩾ϵ^kρ^± k, this yields the following estimate for ϵ small enough and a constant C>0,

    H̃_*_L^2(A_e(ρ,2ρ))^2⩽     C ρ^4ϵ^2(ρ^1 +ρ^-1)^2h-H̃_0^+_C^0_β(A_e(ϵ,ϵ^-1))^2.

 Combining (<ref>) and (<ref>), we get

    H̃_*_C^0_1(A_e(ϵ,ϵ^-1))⩽ C h-H̃_0^+_C^0_β(A_e(ϵ,ϵ^-1)).


Let us fix x_0∈ S_e(1) and modify our symmetric 2-tensor H̃ to get a symmetric 2-tensor H such that h-H vanishes at x_0 and on S_e(ϵ) while being constant on S_e(ϵ^-1). The only possible choice with harmonic symmetric 2-tensors is

    H:= (H̃_0^+-c_0/1-ϵ^2)+ (H̃_*+ϵ^2/1-ϵ^2c_0/r_e^2),

where c_0 = (H̃-h)(x_0).
We will show that h-H satisfies the estimate (<ref>), but let us start by proving the control (<ref>) stated. For this, denote H_0:=H̃_0^+-c_0/1-ϵ^2 the constant part of H, and H_*:=H̃_*+1/1-ϵ^2c_0/(ϵ^-1r_e)^2 its varying part.

According to (<ref>), the part ∑_k⩾ 1(ϵ r_e)^k H̃_k^+ + (ϵ^-1 r_e)^-2-kH̃_k^- is well controlled in L^2-norm by the varying parts of h-H_0 on S_e(ϵ) and S_e(ϵ^-1): they are the same as the varying parts of h-H̃_0. There remains to control the part in r_e^-2, that is

    1/1-ϵ^2(1-ϵ^2)H̃_0^-+c_0/ϵ^-2r_e^2.

In order to control this part, let us look at the mean values of h-H_0 on S_e(ϵ) and S_e(ϵ^-1). On S_e(ϵ), we have (h-H)_|S_e(ϵ) = 0, and therefore 
    (h-H_0)_|S_e(ϵ) = (h-H) + H̃_* + c_0/1-ϵ^2ϵ^2/r_e^2
 and its mean value on S_e(ϵ) is then

    H̃_0^- + c_0/1-ϵ^2.

Similarly, since h-H̃ = 0 on S_e(ϵ^-1) and

    (h-H_0)_|S_e(ϵ^-1) = (h-H̃) + H̃_* + c_0/1-ϵ^2,

and its mean value is therefore

    ϵ^4H̃_0^-+c_0/1-ϵ^2.

By considering linear combinations of (<ref>) and (<ref>), we control both |c_0|_g_e and |H̃_0|_g_e thanks to the mean values of h-H_0 on S_e(ϵ) and S_e(ϵ^-1), and we consequently have for some constant C>0,

    |c_0|_g_e + |H̃_0^-|_g_e⩽ Ch-H_0_C^0_β(A_e(ϵ,ϵ^-1)).


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

    H_*_C^0_1⩽ C h-H_0_C^0_β,

and therefore the stated inequality (<ref>).



    Let us prove the estimate (<ref>), and assume towards a contradiction that there exists a sequence of positive numbers ϵ_i→ 0, and a sequence of symmetric 2-tensors h_i on annuli A_e(ϵ_i,ϵ_i^-1) satisfying, h_i-H_i_C^0_β(A_e(ϵ,ϵ^-1)) = 1, and
    P_e h_i_r^-2_eC^0_β(A_e(ϵ_i,ϵ_i^-1))⩽1/i.
    
        The failure of these properties will indeed yield the estimate (<ref>) since by elliptic regularity we will get higher order estimates on the smaller domain A_e(2ϵ,(2ϵ)^-1)).
    
    Let then (x_i)_i be a sequence of points of A_e(ϵ_i,ϵ_i^-1) where the C^0_β(A_e(ϵ_i,ϵ_i^-1))-norm of h̅_i:=h_i-H_i is reached. We can extract a subsequence with one of the following behaviors:
    
        
  * r_e(x_i)→ +∞, and ϵ_i r_e(x_i)→ 0,
        
  * r_e(x_i)→ +∞, and ϵ_i r_e(x_i)→ c>0,
        
  * r_e(x_i)→ 0 and ϵ_i^-1r_e(x_i)→ +∞,
        
  * r_e(x_i)→ 0 and ϵ_i^-1r_e(x_i)→ c>0,
        
  * r_e(x_i)→ c>0.
    
    In all cases, we rescale to fix r_e(x_i)=1 by defining, for all x,
    
    h_i'(x):= h̅_i(r_e(x_i)x)/ϵ_i^β(r_e(x_i)^β+ r_e(x_i)^-β),

    which satisfies
    
    (∇_e^*∇_eh_i')(x) = r_e(x_i)^2( ∇_e^*∇_eh̅_i)(r_e(x_i)x).

    Since we had by assumption the controls
    
    h̅_i(x)⩽ϵ_i^β(r_e(x)^β+ r_e(x)^-β),

    and
    
    |∇_e^*∇_eh̅_i|_g_e(x)⩽1/ir_e(x)^-2ϵ_i^β(r_e(x)^β+r_e(x)^-β),

    our new symmetric 2-tensor h'_i vanishes at x_0/r_e(x_i) and on S_e(ϵ_i r_e(x_i)^-1), and is constant on S_e(ϵ_i^-1 r_e(x_i)^-1). It moreover satisfies
    
    |h_i'|_g_e(x)⩽((r_e(x_i)r_e(x))^β+(r_e(x_i)r_e(x))^-β)/r_e(x_i)^β+ r_e(x_i)^-β

    with equality at x_i and
    
    |∇_e^*∇_eh_i'|_g_e(x)⩽1/ir_e(x)^-2((r_e(x_i)r_e(x))^β+(r_e(x_i)r_e(x))^-β)/r_e(x_i)^β+ r_e(x_i)^-β.

    
    In the different situations, up to extracting a subsequence, we finally get one of the following limits
        
        
  * on  ℝ^4 \{0}, a solution h'_∞ of P_e h'_∞ = 1/2∇_e^*∇_eh'_∞ =0, and sup r^-βh'_∞ = 1, but there does not exist such a solution because the harmonic symmetric 2-tensors decay at least as 𝒪(r) at 0 if they vanish at 0 and must therefore grow at this rate at infinity. This is a contradiction.
        
  * on  B_e(1/c) \{0}, a solution h'_∞ of P_e h'_∞ =0, and sup r^-βh'_∞ = 1, and such that (h_∞')_|S_e(1/c) is constant. The unique solution to the Dirichlet problem with the zero condition at 0 and a constant condition on S_e(1/c) is h_∞'=0. This is a contradiction.
        
  * on  ℝ^4 \{0}, a solution h'_∞ of P_e h'_∞ =0, and sup r^βh'_∞ = 1, but there does not exist such a solution because the harmonic symmetric 2-tensors decaying at infinity decay at least like 𝒪(r^-2), and therefore blow up at least at this rate at 0, and finally, h_∞'=0. This is a contradiction.
        
  * on  ℝ^4 \ B_e(1/c), a solution h'_∞ of P_e h'_∞ =0, and sup r^βh'_∞ = 1 and (h'_∞)_|S_e(1/c)=0. The unique solution to the Dirichlet problem on  ℝ^4 \ B_e(1/c), decaying at infinity and vanishing on S_e(1/c) being zero, we have h_∞ =0. This is a contradiction.
        
  * on  ℝ^4 \{0}, a solution h'_∞ of P_e h'_∞ =0, and sup (r^β+r^-β)h'_∞ = 1 satisfying h'_∞(x_0/c)=0. The conditions P_e h'_∞ =0, and sup (r^β+r^-β)h'_∞ =1 imply that h'_∞ is constant, since h'_∞ vanishes at x_0/c, we have h'_∞ =0. This is a contradiction.
    
    We therefore deduce that there exists ϵ_e>0 and C_e>0 such that for all 0<ϵ<ϵ_e and all symmetric 2-tensor h on the annulus A_e(ϵ,ϵ^-1), we have
    
    h-H_0_C^0_β(A_e(ϵ,ϵ^-1))⩽ C_eP_g h_r^-2_eC^α_β(A_e(ϵ,ϵ^-1)).

    
    In order to prove the estimate (<ref>) and go from a C^0_β(A_e(ϵ,ϵ^-1))-controls to C^2,α_β(A_e(2ϵ,(2ϵ)^-1))-controls, we use elliptic estimates which are satisfied on the flat annuli according to the end of the proof of Proposition <ref>.


With a completely analogous proof using  the harmonic decomposition of 1-forms on a cone of <cit.> (see also Section <ref> for particular case of flat cones), we have the same result for vector fields, but this time, we treat the linear kernel of δδ^* on ℝ^4Γ separately. On ℝ^4, the elliptic operator

    δ_eδ_e^* = ∇_e^* ∇_e-1/2 d_e^*d = dd_e^*+1/2d_e^*d,

has its kernel equal to the linear vector fields of the kernel of δ_eδ^*_e among the vector fields of order 𝒪(r_e^1-β+r_e^1+β) for 0<β<1, see Lemma <ref> for a proof of this and Section <ref> for a description of the kernel.


Let 0<β<1. There exists C_e>0, and ϵ_e>0 such that for any vector field X on an annulus of radii 0<ϵ<ϵ_e and 1/ϵ, there exists Y_0, a linear vector field of the kernel of δ_eδ^*_e, and an element Y_* of the kernel of δ_eδ_e^* satisfying 

    Y_*_r_eC^3,α_1(A_e(2ϵ,(2ϵ)^-1))⩽ C_e X -Y_0_r_eC^3,α_β(A_e(ϵ,ϵ^-1)),
 
    
    X-Y_0-Y_*_r_eC^3,α_β(A_e(2ϵ,(2ϵ)^-1))⩽ C_e δ_g_eδ^*_g_e X_r^-1_eC^1,α_β(A_e(ϵ,ϵ^-1)).

    
    In particular, this implies the following control, for all x ∈ A_e(1/2,2),
    
    |(X-Y_0)(x)|_g_e+|   ∇ (X-Y_0)(x)|_g_e +|∇^2 (X-Y_0)(x)|_g_e + [∇^2 (X-Y_0)]_C^α(g)(x) 
    ⩽    C_e ( (2ϵ)^βδ_eδ_e^*X_r^-1_eC^1,α_β(A_e(ϵ,ϵ^-1))+ 2ϵX-Y_0_r_eC^3,α_β(A_e(ϵ,ϵ^-1))).







  §.§.§ Approximate kernels


Let (M,g^D) be a naïve desingularization of an Einstein orbifold. For each annulus 𝒜_k(t,ϵ) (see Definition <ref>) between N_k and N_j or N_k and M_o, by construction there exists a diffeomorphism

    Φ_k: A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j))⊂ℝ^4Γ_k→𝒜_k(t,ϵ)⊂ M,

such that there exists C>0 for which, for all 0<β<1,

    Φ^*_kg^D-g_e_C^2,α_β(A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j)))⩽ C ϵ^2-β.


Because of the above constant symmetric 2-tensors and the linear vector fields, we cannot expect estimates independent of the gluing scales in the definition of (M,g^D) of the type h_C^2,α_β(g^D)⩽ CP_g^Dh_r_D^-2C^α_β(g^D) which are needed to apply an inverse function theorem. Indeed, we have the following estimates according to Proposition <ref> (which is proven below). Recall that the cut-off functions are in Definition <ref>.


    In most of the rest of this article, we will often abusively forget the diffeomorphism Φ_k to simplify the notations. For instance, a symmetric 2-tensor Φ_k^*(χ_𝒜_k(t,ϵ)H) will be denoted χ_𝒜_k(t,ϵ)H on M.



    On a naïve desingularization (M,g^D_t), for all 0<β<1, there exists C>0 such that for H_k a constant symmetric 2-tensor, and χ_𝒜_k(t,ϵ) the cut-off function defined in Definition <ref>,
    
    P_g^D(χ_𝒜_k(t,ϵ)H_k)_r_D^-2C^α_β(g^D)⩽ C |H_k|_g_e,

    but 
    χ_𝒜_k(t,ϵ)H_k_C^0_β(g^D)⩾1/2 t_max^-β/4|H_k|_g_e.



    

Linear vector fields in the kernel of δδ^* also rule out the existence of estimates independent of t for the operator δδ^* according to Proposition <ref> proven below.


    On a naïve desingularization (M,g^D_t), for all 0<β<1, there exists C>0 such that for X_k a linear vector field in the kernel of δ_eδ^*_e, 
    
    δ_g^Dδ_g^D^*(χ_𝒜_k(t,ϵ)X_k)_r_D^-1C^1,α_β(g^D)⩽ C X_k_r_eC^0_0(g_e),

    but 
    χ_𝒜_k(t,ϵ)X_k_C^0_β(g^D)⩾1/2 t_max^-β/4X_k_r_eC^0_0(g_e).



    

    


  
Weighted decoupling norms.
Propositions <ref> and <ref> actually show that we can control the inverses of our operators once we solve our equations modulo constant symmetric 2-tensors and the linear vector fields of the kernel of δ_eδ^*_e on ℝ^4 and Propositions <ref> and <ref> show that we cannot expect better. We therefore introduce new norms to reflect this. They are similar to the norms introduced in <cit.> for similar reasons.


    Let h be a symmetric 2-tensor on (M,g^D), (respectively (M_o,g_o) or (N,g_b)). We define its C^k,α_β,*-norm by
    
    h_C^k,α_β,*:= inf_h_*,H_kh_*_C^k,α_β + ∑_k |H_k|_g_e,

    where the infimum is taken among the couples (h_*,H_k) satisfying h= h_*+∑_k χ_𝒜_k(t,ϵ)H_k (respectively h= h_*+∑_k χ_B_o(ϵ)H_k or h= h_*+∑_k χ_B_b(ϵ)H_k), for H_k a constant traceless symmetric 2-tensor on ℝ^4Γ_k.



    Let X a vector field on (M,g^D) (respectively (M_o,g_o) or (N,g_b)). We define its rC^k,α_β,*-norm, where r is the function r_D (respectively r_o or r_b) by
    
    X_rC^k,α_β,*:= inf_X_*,X_kX_*_rC^k,α_β + ∑_k X_k_rC^0_0(g_e),

    where the infimum is taken among the couples (X_*,X_k) satisfying X = X_* + ∑_kχ_𝒜_k(t,ϵ)X_k (respectively X = X_* + ∑_kχ_B_o(ϵ)X_k or X = X_* + χ_B_b(ϵ)X_k).



    By definition of the weighted norms, on an orbifold or orbifold ALE, the decompositions h = h_*+ ∑_k χ_B(ϵ)H_k and X = X_* + ∑_kχ_B(ϵ)X_k are unique and determined respectively by the limits of h and of X/r when r→ 0 (where r = r_o or r=r_b). Indeed, in other cases, the expression we minimize is infinite.



    By definition, we have 
    
    ._C^k,α_β,*⩽._C^k,α_β, and  ._rC^k,α_β,*⩽._rC^k,α_β,

    and the spaces (C^k,α_β,*,._C^k,α_β,*) and (rC^k,α_β,*,._rC^k,α_β,*) are clearly Banach spaces.





  §.§.§ Estimates in the decoupling norms

Let us show that it is possible to control thanks to the r^mC^k,α_β-norm the images by the operators P and δδ^* of elements of r^m+2C^k+2,α_β,*.
  Let 0<β<1, and (M_∙,g_∙) one of the spaces (M_o,g_o), (N_j,g_b_j) or (M,g^D), g a metric, h a symmetric 2-tensor, and X a vector field on M_∙. We then have, the following controls:
    
    P_g_∙h_r^-2_∙ C^α_β(g_∙)⩽ Ch_C^2,α_β,*(g_∙),

    
    δ_g_∙δ^*_g_∙ X _r^-1_∙ C^1,α_β(g_∙)⩽ CX_r_∙ C^3,α_β,*(g_∙),

    
    P_g(h)-P_g_∙(h)_r^-2_∙ C^α_β(g_∙)⩽ Cg-g_∙_C^2,α_β,*(g_∙)h_C^2,α_β,*(g_∙),

    and
    
    δ_gδ^*_g (X) - δ_g_∙δ^*_g_∙ (X) _r^-1_∙ C^1,α_β(g_∙)⩽ Cg-g_∙_C^2,α_β,*(g_∙)X_r_∙ C^3,α_β,*(g_∙).



    Let us show the result for g^D, the proof for other spaces is very similar. For the two first inequalities, consider h a symmetric 2-tensor and X a vector field on M, and some decompositions h = h_*+∑_kχ_𝒜_k(t,ϵ)H_k and X = X_*+∑_kχ_𝒜_k(t,ϵ)X_k. Remark <ref> implies that we have the following controls for h_* and X_*,
    
    P_g^Dh_*_r^-2_D C^α_β(g^D)⩽ Ch_*_C^2,α_β(g^D),

    and
    
    δ_g^Dδ^*_g^D X_* _r^-1_D C^1,α_β(g^D)⩽ CX_*_r_D C^3,α_β(g^D).

    On ℝ^4Γ, we have P_eH_k = 0 and δ_eδ_e^*X_k=0, hence, since for all l∈ℕ, we have 
    
    |∇^lχ_k|_g^D⩽ C_lr_D^-l,

    and thanks to the control (<ref>), we have 
    P_g^D(χ_𝒜_k(t,ϵ)H_k)_C^α_β(g^D)⩽ C |H_k|_g_e,

    and
    
    δ_g^Dδ_g^D^*(χ_𝒜_k(t,ϵ)X_k)_r_D^-1C^1,α_β(g^D)⩽ C X_k_r_eC^0_0(g_e),

    where we pulled-back thanks to the diffeomorphism 
    Φ_k: A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j))⊂ℝ^4Γ_k→𝒜_k(t,ϵ)⊂ M.
 Summing the controls (<ref>) and (<ref>) on the one hand, and the controls (<ref>) and (<ref>) on the other hand, yields the two fist inequalities stated.
    

    
    Let us now focus on the two last inequalities, which are more difficult to obtain. The control we want is local, let us therefore write down the expressions of our operators in local coordinates in an orthonormal basis (e_i). For a symmetric 2-tensor h, denoting h_ij = h(e_i,e_j) and R_ijkl the Riemannian curvature in coordinates, we have 
    
    P_g(h)_ij= 1/2(∇_g^*∇_g h)_ij-g^kpg^lq R_ikjl h_pq,

    where ∇_i is the covariant derivative for g in the direction e_i.
    We directly see thanks to the estimates of Remark <ref> that we have the controls:
    
    P_g(h)-P_g^D(h)_r^-2_DC^α_β(g^D)⩽ Cg-g^D_C^2,α_β(g^D)h_C^2,α_β(g^D).

    
    Let us now consider ∑_k χ_𝒜_k(t,ϵ)H_k and ∑_k χ_𝒜_k(t,ϵ)H_k'. These tensors being all supported in the annuli 𝒜_k(t,ϵ), we just need to restrict our attention to them. The crucial remark is that in (<ref>), every term involves at least a derivative of h or of g-g^D. Hence, we have a more precise control on h a symmetric 2-tensor supported in 𝒜_k(t,ϵ_0)
    
    P_g(h)-P_g^D(h)_r^-2_DC^α_β(g^D)⩽    C(g-g^D_C^2,α_0(g^D)∇^2 h_r^-2_DC^α_β(g^D)
       +∇(g-g^D)_r^-1_DC^1,α_β(g^D)∇ h_r^-1_DC^1,α_β(g^D)
       +∇^2(g-g^D)_r^-2_DC^α_β(g^D) h_C^2,α_0(g^D)),

    (notice the norms C^2,α_0(g^D) in which we have χ_𝒜_k(t,ϵ)H_k_C^2,α_0(g^D)⩽ C|H_k|_g_e and χ_𝒜_k(t,ϵ)H'_k_C^2,α_0(g^D)⩽ C|H_k'|_g_e). There remains to control the derivatives of the tensors χ_𝒜_k(t,ϵ)H_k and χ_𝒜_k(t,ϵ)H_k'. Since the H_k and H'_k are constant on ℝ^4, and since the cut off functions are bounded in C^2_0(g^D) by (<ref>), for i ∈{1,2}, we have
    
    ∇^i(χ_𝒜_k(t,ϵ)H_k)_r^-i_DC^2-i,α_β(g^D)⩽ C |H_k|_g_e,

    and
    
    ∇^i(χ_𝒜_k(t,ϵ)H_k')_r^-i_DC^2-i,α_β(g^D)⩽ C |H_k'|_g_e,
 
    which together with (<ref>) let us conclude that the third estimate holds.
    

    
    For the vector fields, we have the following rewriting for X a vector field supported in 𝒜_k(t,ϵ_0)
    
    δ_gδ^*_g (X) - δ_g^Dδ^*_g^D (X) _r^-1_DC^1,α_β(g^D)⩽     C((δ_g-δ_g^D)(ℒ_Xg) _r^-1_DC^1,α_β(g^D)
       +δ_g^D(ℒ_X(g^D-g)) _r^-1_DC^1,α_β(g^D)).

    We moreover know that for X_k a linear vector field in the kernel of δ_eδ_e^*, then the symmetric 2-tensor ℒ_X_kg_e is constant, and more generally, for H_k a constant symmetric 2-tensor, we have δ_g_e(ℒ_X_kH_k)=0 on ℝ^4. Using these two facts and the controls of the cut-off functions, we conclude that the last estimate of the statement holds by an argument similar to the above one for 2-tensors and P.




  §.§.§ Elliptic estimates for the decoupling norms

Some elliptic estimates are still satisfied in these norms.


    Let 0<β<1, g a metric, h a symmetric 2-tensor and X a vector field  on M_o (respectively N_j, or M). Then, there exists ϵ_*=ϵ_*(g_o,g_b_j,g^D, β)>0 and C>0 such that if we have g-g_∙_C^2,α_β,*(g_∙)⩽ϵ_*, where g_∙ is one of the norms g_o, g_b_j or g^D, then,
    
    h_C^2,α_β,*(g_∙)⩽ C (P_gh_r^-2_∙ C^α_β(g_∙)+h_C^0_β,*(g_∙)),

    and
    
    X_r_∙ C^3,α_β,*(g_∙)⩽ C (δ_gδ^*_gX_r^-1∙ C^1,α_β(g_∙)+X_r_∙ C^0_β,*(g_∙)).



    Let g_∙ be one of the metrics g_o, g_b_j or g^D, and for all k, H_k a traceless constant symmetric 2-tensor on ℝ^4Γ_k, and X_k a Killing vector field on ℝ^4Γ_k. Let moreover h_* be a symmetric 2-tensor of C^2,α_β(g_∙) and X_* be a vector field of r_∙ C^3,α_β(g_∙), and define h = h_*+ ∑_k χ_∙ X_k and X= X_*+∑_k χ_∙ X_k, where χ_∙ is χ_𝒜_k(t,ϵ) or χ_B_g_∙(ϵ) (of Definition <ref>) depending on the metric.
    
    We then have the following controls:
    
    P_g_∙χ_∙ H_k_r^-2_∙ C^α_β(g_∙)⩽ C |H_k|_g_e,

    and
    
    δ_g_∙δ_g_∙^*χ_∙ X_k_r^-1_∙ C^1,α_β(g_∙)⩽ C X_k_r_e C^0_0(g_e).

    Hence, for h_*, we have
    
    P_g_∙h_*_r^-2_∙ C^α_β⩽ C(P_g_∙h_r^-2_∙ C^α_β(g_∙) + ∑_k|H_k|_g_e),

    and the expected estimate for g = g_∙ is then a consequence of the elliptic estimates in the weighted spaces of Lemma <ref> which give 
    
    h_*_C^2,α_β(g_∙)⩽ C(P_g_∙h_*_r^-2_∙ C^α_β(g_∙)+h_*_C^0_β(g_∙)),

    and imply therefore that
    
    h_C^2,α_β,*(g_∙)⩽ 2C^2 (P_gh_r^-2_∙ C^α_β(g_∙)+h_C^0_β,*(g_∙)).

    The same argument works for the operator δδ^* on the vector fields thanks to the elliptic estimates of Lemma <ref>.

    Proposition <ref> finally lets us go from the metric g_∙ to a metric g satisfying g-g_∙_C^2,α_β,*(g_∙)⩽ϵ_*. 






§ REDUCED DIVERGENCE-FREE GAUGE


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

    δδ^*X = - δ h,

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

In our degenerating situation, we want to obtain estimates in our weighted norms which are independent of the gluing scales. A difficulty is that our limit orbifold might have more symmetries than the Ricci-flat ALE spaces (for example, 𝕊^4ℤ_2 desingularized by Eguchi-Hanson metrics). The associated Killing vector fields would give an approximate kernel for δδ^* which would not be an actual kernel or cokernel. We will need to define a reduced divergence-free gauge to obtain uniform estimates as the gluing scales go to zero.



    All along this section, if nothing is precised, an Einstein orbifold (M_o,g_o) will be either compact or ALE.




 §.§ Kernel of the linearization

Let us focus on the operator δδ^*  on a flat cone (ℝ^4Γ,g_e), on an orbifold (M_o,g_o), and on Ricci-flat ALE orbifolds (N_j,g_b_j).



  
On a flat cone.
On the flat cone (ℝ^4Γ,g_e)= (ℝ^+×𝕊^3Γ, dr^2+r^2 g_𝕊^3Γ), according to <cit.>, any 1-form on ℝ^4Γ is a countable sum of 1-forms of one of the following types which are preserved by δδ^*:

    
  * p(r)ψ, where δ_𝕊^3Γψ = 0, and ψ is eigenvector of the Hodge Laplacian of 𝕊^3Γ,
    
  * r^-1l(r)ϕ dr + u(r)r d_𝕊^3Γϕ, and ϕ is eigenfunction of the Hodge Laplacian of 𝕊^3Γ,

where p,l,u: ℝ^+→ℝ and ϕ: 𝕊^3Γ→ℝ are functions, and where ψ is a 1-form on 𝕊^3Γ.

According to <cit.>, thanks to the computation of the eigenvalues of the Laplacian and of the Hodge Laplacian on the 1-forms of the sphere <cit.>, the solutions to δ_eδ_e^* ω = 0 are countable sums of 1-forms of the following types

    
  * r^a^±_jψ with a^±_j:= ± (1+j), j∈ℕ^*, where ψ is an eigenvector the Hodge Laplacian,
    
  * r^b^±_jd_𝕊^3Γϕ +  b^±_j r^b^±_j-1ϕ dr, or 2r^b^±_j+2d_𝕊^3Γϕ +  b^∓_j r^b^±_j+1ϕ dr, with b_j^± = -1 ± (1+j), j∈ℕ and where ϕ is an eigenfunction the Hodge Laplacian.


Since we are interested in solving an equation 

    δδ^*X = - δ h,

where X is a vector field, and h a symmetric 2-tensor is in C^2,α_β,*, we are naturally looking for X in r_DC^3,α_β,*. The exceptional values of δ_e δ_e^* are the values γ∈ℝ such that there exists a homogeneous 1-form whose norm is proportional to r_e^γ in the kernel of δ_e δ_e^*. We are interested in the exceptional values around the exceptional value 1 associated to the linear vector fields of the kernel of δ_eδ^*_e.


    On (ℝ^4Γ)\{0} for Γ≠{e}, 1 is the only exceptional value between -3 and 2.


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

For b^±_j+1, the values 0 and -1 are a priori possible, and for b^±_j-1, -1 and -2 are a priori possible. However, these values cannot appear on a flat cone ℝ^4Γ for Γ≠{e}. Indeed, the values b^±_j-1 =0 and b^±+1 = -2 only appear if -3 is an eigenvalue of the Laplacian on the link of the cone, but this is not the case for 𝕊^3Γ because there does not exist any non zero Γ-invariant linear function on ℝ^4. 

For the values b^±_j-1=-1 and b^±_j+1=-1, we use the form of the solutions. In the first case, b^±_j=0 gives

    r^b^±_jd_𝕊^3Γϕ +  b^±_j r^b^±_j-1ϕ dr = d_𝕊^3Γϕ,

for Δ_𝕊^3Γϕ=0, therefore ϕ is constant and finally d_𝕊^3Γϕ =0. In the second case, the equality b^±_j=-2, that is b^∓_j=0, gives

    2r^b^±_j+2d_𝕊^3Γϕ +  b^∓_j r^b^±_j+1ϕ dr = d_𝕊^3Γϕ,

for Δ_𝕊^3Γϕ=0, therefore ϕ is constant and finally d_𝕊^3Γϕ =0.


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

    
  * r^2ψ, where ψ is the dual of a Killing vector field of 𝕊^3,
    
  * r dr,
    
  * 2rϕ d r + r^2d_𝕊^3Γϕ.




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


Let (N_j,g_b_j) be a Ricci-flat ALE orbifold. For 0<β<1, the operator 
    δ_g_b_jδ_g_b_j^*: r_b_jC^3,α_β,*→ r_b_j^-1C^1,α_β

is bijective.

Let (M_o,g_o) be a compact Einstein orbifold. For 0<β<1, the operator 
    δ_g_oδ_g_o^*: r_oC^3,α_β,*→ r_o^-1C^1,α_β
 is Fredholm and both its kernel and its cokernel are equal to 𝐊_o, the set of Killing vector fields of (M_o,g_o).

As a consequence, there exist C_o>0 and ϵ_o>0 depending on g_o 
    such that if g-g_o_C^2,α_β,*(M_o)⩽ϵ_o, then we have for any vector field X ∈𝐊_o^⊥ on M_o
    
    X_r_oC^3,α_β,*(g_o)⩽ C_oδ_gδ_g^*X_r_o^-1C^1,α_β(g_o).

    
    There also exists C_j>0 and ϵ_j>0 depending on g_b_j
    such that if g-g_b_j_C^2,α_β,*(N_j)⩽ϵ_j then we have for any vector field X on N_j,
    
    X_r_b_jC^3,α_β,*(g_b_j)⩽ C_jδ_gδ_g^*X_r_b_j^-1C^1,α_β(g_b_j).



    For orbifold singularities, we will first authorize our tensors to behave like r^1-β for 0<β<1 at the singularities, instead of being in rC^3,α_β,* to use the theory of elliptic operators in weighted Hölder spaces, see for instance <cit.> of <cit.>. Let us start by considering an Einstein orbifold (M_o,g_o) and the operator δ_g_oδ_g_o^*: r_oC^3,α_-β→ r_o^-1C^1,α_-β (notice the -β). Its kernel is composed of Killing vector fields of g_o. Indeed, if for X∈ r_oC^3,α_-β we have  δ_g_oδ_g_o^* X=0, integrating by parts yields,
    
    0=    ∫_M_o⟨δ_g_oδ_g_o^*X,X⟩ dv_o
       =∫_M_o |δ_g_o^*X|^2_g_odv_o + lim_r→ 0∫_{r_o = r}δ^*_g_oX(n,X)
       =∫_M_o |δ_g_o^*X|^2_g_odv_o,

    where n=∇ r_o/|∇ r_o|, because the boundary term which is schematically lim_r→ 0(𝒪(r^-β+1-β+3)) vanishes. Similarly, its cokernel is equal to the kernel of δ_g_oδ_g_o^* on r_o^-3C^1,α_β which is also reduced to 𝐊_o because there is no exceptional value between -3 and 1. 
    
    On an ALE orbifold (N,g_b), let us assume that a vector field X∈ r_b^1-βC^3,α_0 satisfies δ_g_bδ_g_b^*X = 0. Since there is no exceptional value between -3 and 1, we actually have X = 𝒪(r_b) when r_b→ 0 at the singular points of (N,g_b), and X = 𝒪(r_b^-3) at infinity. Let us then consider the following integration by parts,
    
    0=    ∫_N ⟨δ_g_bδ_g_b^*X,X⟩ dv_b
       =∫_N |δ_g_b^*X|^2_g_bdv_b - lim_ρ→∞∫_{r_b = ρ}δ^*_g_bX(n,X)+ lim_r→ 0∫_{r_b = r}δ^*_g_bX(n,X)
       =∫_N |δ_g_b^*X|^2_g_bdv_b,

    where the boundary term vanishes because it is the sum of the limit for r_b→∞ of a 𝒪(|X|_g_b|∇ X|_g_br_b^3) = 𝒪(r_b^-4) and of the limit when r_b→ 0 of a 𝒪(|X|_g_b|∇ X|_g_br_b^3) = 𝒪(r_b^4).    Hence, we have δ_g_b^*X = 0, and since g_b is Ricci-flat, (δ_b+1/2d _b)δ_g_b^*X = ∇_b^*∇_b X =0, which implies that ∇_b X=0 by integration by parts against X, and finally, that X is parallel on N. Since X tends to 0 at infinity, we have X=0. The operator δ_g_bδ_g_b^*: r_b^1-βC^3,α_0→ r_b^-1-βC^1,α_0 is therefore injective. 
    
    The cokernel of the self adjoint operator δ_g_bδ_g_b^*: r_b^1-βC^3,α_0→ r_b^-1-βC^1,α_0 is equal to the kernel of δ_g_bδ_g_b^* on r_b^-3+βC^1,α_0 which is also reduced to {0} because there is no exceptional values between -3+β and 1-β. The operator δ_g_bδ_g_b^*: r_b^1-βC^3,α_0→ r_b^-1-βC^1,α_0 is therefore bijective.
    

    
    Let us finally work in the norms we are interested in and study the operators δ_g_oδ_g_o^*: r_oC^3,α_β,*→ r_o^-1C^1,α_β and δ_g_bδ_g_b^*: r_bC^3,α_β,*→ r_b^-1C^1,α_β.
    Since the spaces r_bC^3,α_β,* and r_oC^3,α_β,* are respectively only the direct sum of r_bC^3,α_β and r_oC^3,α_β with a space of finite dimension composed of cut-off of linear vector fields, the image remains closed and of finite codimension. We can be more precise by noticing that
    
    δ_g_oδ_g_o^* (r_oC^3,α_β,*) = δ_g_oδ_g_o^* (r_oC^3,α_-β)∩ r_o^-1C^1,α_β.

    Indeed, we have δ_g_oδ_g_o^* (r_oC^3,α_β,*) ⊂δ_g_oδ_g_o^* (r_oC^3,α_-β)∩ r_o^-1C^1,α_β because r_oC^3,α_β,*⊂ r_oC^3,α_-β and thanks to Proposition <ref>. Conversely, if for X∈ r_oC^3,α_-β we have δ_g_oδ_g_o^*X∈ r_o^-1C^1,α_β, then, since the only exceptional value between 1-β and 1+β is 1 and corresponds to the linear kernel of δ_eδ_e^*, we have X∈ r_oC^3,α_β,*.
    Similarly, we conclude that
    
    δ_g_bδ_g_b^* (r_bC^3,α_β,*) = δ_g_bδ_g_b^* (r_b^1-βC^3,α_0)∩ r_b^-1C^1,α_β,

    and finally, 
    δ_g_oδ_g_o^*: r_oC^3,α_β,*→ r_o^-1C^1,α_β is Fredholm with 𝐊_o as kernel and cokernel, and δ_g_bδ_g_b^*: r_bC^3,α_β,*→ r_b^-1C^1,α_β is bijective. We finally conclude by the open mapping theorem between Banach spaces which is stable by small perturbation of the operator.




 §.§ Controls on the inverse of the linearization


These controls will help us treat the case of trees of singularities with small enough gluing parameters.

For this, we approximate the kernel 𝐊_o on our naïve desingularization (M,g^D) in the following way. Note that 𝐊_o=0 for an ALE Ricci-flat orbifold (M_o,g_o) by Proposition <ref>. For all 𝐗_o∈𝐊_o, according to Remark <ref>, on an orbifold, there exists a unique decomposition

    𝐗_o = 𝐗_o,* + ∑_kχ_B(p_k,ϵ_0)𝐗_o,k,

such that 𝐗_o_r_oC^3,α_β,* = 𝐗_o,*_r_oC^3,α_β + ∑_k𝐗_o,k_r_eC^0_0 (other decompositions make the value infinite). We then define 𝐊̃_o as the space of the following vector fields on M

    𝐗̃_o,t:= χ_M_o^t𝐗_o,* + ∑_kχ_𝒜_k(t,ϵ_ 0)𝐗_o,k,
 for 𝐗_o∈𝐊_o. Note that we therefore have 𝐗̃_o,t = 𝐗_o on M_o^16t.


    By elliptic regularity on M_o, the norms L^2(g_o), r_DC^3,α_β,*(g_o) and r_D^-1C^1,α_β(g_o) are equivalent on the finite-dimensional space 𝐊_o. Since the C^4_0-norms of the cut-off functions are bounded, we conclude that for ϵ and t_max small enough, the norms L^2(g^D), r_DC^3,α_β,*(g^D) and r_D^-1C^1,α_β(g^D) are equivalent on 𝐊̃_o.



    We define the reduced divergence operator, δ̃_g:=π_𝐊̃_o^⊥δ_g, where π_𝐊̃_o^⊥ is the  L^2(g^D)-orthogonal projection on 𝐊̃_o^⊥. We will say that a metric g_1 is in reduced divergence-free gauge with respect to a metric g_2 if δ̃_g_2g_1 = 0.

 Let us start by noticing that the operator δ̃_g^D is actually very close to δ_g^D for a naïve desingularization g^D with small enough gluing parameters.


    There exists C>0 such that for any symmetric 2-tensor h∈ C^2,α_β,*(g^D), we have,
    
    (δ̃_g^D-δ_g^D) h _r_D^-1C^1,α_β(g^D)⩽ C t_maxh_C^2,α_β,*(g^D).



    If (M_o,g_o) is ALE, then, one has 𝐊_o = {0} and therefore δ̃_g^D=δ_g^D. Let us focus on the case when M_o is compact.
    
    Thanks to the equivalence of the different norms, see Remark <ref>, it is enough to show that the L^2(g^D)-projection on 𝐊̃_o of δ_g^D h is small to show the result. We naturally proceed by integration by parts. Let 𝐗̃_o,t∈𝐊̃_o for 𝐗_o∈𝐊_o be an approximate Killing vector field as above. We have,
    
    |∫_M(δ_g^Dh)(𝐗̃_o,t) dv_g^D| =    |∫_M⟨ h, δ_g^D^*(𝐗̃_o,t) ⟩_g^D dv_g^D|,

    and,
    
    δ_g^D^*(𝐗̃_o,t) =    δ^*_g_o𝐗_o +δ^*_g_o((χ_M_o^t-1)𝐗_o,*) + δ^*_g_o((χ_𝒜_k(t,ϵ_0)- χ_B_o(p_k,ϵ_0))∑_k𝐗_o,k)
       + (δ_g_o^*-δ_g^D^*)(χ_𝒜_k(t,ϵ_0)∑_k𝐗_o,k),

    where by definition δ^*_g_o𝐗_o =0.
    Thanks to Definition <ref>, on M_o^t, χ_M^t_o𝐗_o,* is equal to 𝐗_o,* except on the annuli of radii t_k^1/4 and 2t_k^1/4, and χ_𝒜_k(t,ϵ_0)- χ_B_o(p_k,ϵ_0) is supported in ϵ_0^-1√(t)_k<r_D<2ϵ_0^-1√(t)_k and well-defined on M. For any l∈ℕ, the cut off functions are moreover uniformly bounded in C^l_0. If we denote 1_A the indicator function of A,
    for the vector fields, we therefore have
    
    r_D^l(|∇^l𝐗_o,*|_g_o+|∇^l∑_k𝐗_o,k|_g_o)⩽ C_l 𝐗_o_r_oC^l_β,* r_D,

    and
    
    r_D^l|∇^l(g^D-g_o)|_g_o⩽ C_l 1_{r_D<t_k^1/4}(r_D^2+t_k^2r_D^-4).

    
    As a consequence, because of the properties of the norms detailed in Remark <ref>, we have
    
    |δ^*_g_o((χ_M_o^t-1)𝐗_o,*)|_g_o⩽  C1_{t_k^1/4<r_D<2t_k^1/4}𝐗_o_r_oC^1_0(g_o),

    on the annulus of radii ϵ_0^-1√(t)_k<r_D<2ϵ_0^-1√(t)_k, we have
    
    |δ^*_g_o((χ_𝒜_k(t,ϵ_0)- χ_B_o(p_k,ϵ_0))∑_k𝐗_o,k)|⩽ C1_{ϵ_0^-1√(t)_k<r_D<2ϵ_0^-1√(t)_k}𝐗_o_r_oC^1_0(g_o),

    and
    
    |(δ_g_o^*-δ_g^D^*)(χ_𝒜_k(t,ϵ_0)∑_k𝐗_o,k)|⩽ C 1_{ϵ_0^-1√(t)_k<r_D<t_k^1/4}(r_D^2+t_k^2r_D^-4)𝐗_o_r_oC^1_0(g_o).

    Finally, since we have (A_e(t_k^1/4,2t_k^1/4))≈ t_k, (A_e(ϵ_0^-1√(t_k),2ϵ_0^-1√(t_k)))≈ t_k^2, and also ∫_{ϵ_0^-1√(t_k)<r_D<t_k^1/4}(r_D^2+t_k^2r_D^-4) dv_g^D≈ t_k^3/2+t_k^2|log t_k|, we have
    
    |∫_M⟨δ_g^Dh, 𝐗̃_o,t⟩_g^D dv_g^D|⩽ C t_maxh_C^0_0(g^D)𝐗_o_r_oC^1_0(g_o).

    Finally, let us denote 𝐘̃_o,t = π_𝐊_oδ_g^Dh, we have δ̃_g^Dh = π_𝐊̃_o^⊥δ_g^Dh= δ_g^Dh - 𝐘̃_o,t with 
    𝐘̃_o,t_r_D^-1C^1,α_β(g^D)⩽ C t_maxh_C^0_0(g^D),

    by the equivalence of the norms of Remark <ref>.



    Let 0<β<1, 0<α<1 and (M,g^D) a naïve desingularization of a compact or ALE Einstein orbifold by a tree of singularities. Then, there exists τ_D >0 and ϵ_D>0 and C_D>0, only depending on β and the constants of Proposition <ref>, such that for t_max<τ_D, and any metric g satisfying g-g^D_t_C^2,α_β,*(g^D)⩽ϵ_D, the operator 
    δ̃_gδ_g^*: 𝐊̃_o^⊥∩ r_DC^3,α_β,*(g^D) →𝐊̃_o^⊥∩ r^-1_DC^1,α_β(g^D)
 is invertible and we have for any vector field X⊥𝐊̃_o on M,
    
    X_r_DC^3,α_β,*(g^D)⩽ C_Dδ̃_gδ_g^*X_r^-1_DC^1,α_β(g^D).



    Let 0<ϵ<ϵ_D^1/2-β<ϵ_0 for ϵ_D and ϵ which we will choose small enough along the proof, and assume that t_max< ϵ^4. Therefore, by construction, on each annulus 𝒜_k:=𝒜_k(t,ϵ) between N_k and N_j or N_k and M_o (in which case, we will fix T_o=1), we have a diffeomorphism 
    Φ_k: A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j))⊂ℝ^4Γ_k→𝒜_k(t,ϵ)⊂ M,

    such that for all 0<β<1, there exists C>0, for which we have
    
    Φ^*_kg^D-g_e_C^2,α_β(A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j)))⩽ C ϵ^2-β<Cϵ_D.

   Until the end of the proof, we will use the notation
    
    A_k:= A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j)).

    
    According to the estimate (<ref>), for t_max small enough, it is enough to have
    
    X_r_DC^3,α_β,*(g^D)⩽C_D/2δ_g^Dδ_g^D^*X_r^-1_DC^1,α_β(g^D)

    to obtain the stated result.
    
    The diffeomorphisms Φ_k: A_k→𝒜_k allow us to pull the situation back on ℝ^4, where the ratio of the annuli A_k is ϵ^2t_k^-1/2 which is arbitrarily large for t_max arbitrarily small. According to the estimate (<ref>) of Proposition <ref> and thanks to the controls of Proposition <ref>, for t_max and ϵ_D small enough, then, there exist linear vector fields X_k of the kernel of δ_eδ^*_e such that the vector fields χ_𝒜_kX_k in the annuli 𝒜_k satisfy
    
    Φ_k^*X-X_k_r_eC^3,α_0(A(T_j^1/2t_k^1/4))⩽     C_e T_j^1/2t_k^1/4(ϵ^-βt_k^β/4δ_g_eδ_g_e^*Φ_k^*X_r^-1_eC^1,α_β(A_k)
       + 2 ϵ^-1t_k^1/4Φ_k^*X-X_k_r_eC^3,α_β(A_k))
    ⩽     2 C_eT_j^1/2t_k^1/4(ϵ^-βt_k^β/4(δ_g^Dδ_g^D^*X)_|𝒜_k_r^-1_DC^1,α_β(g^D)
       + 2 ϵ^-1t_k^1/4(X-χ_𝒜_kX_k)_|𝒜_k_r_DC^3,α_β(g^D))

     on A(√(T_j)t_k^1/4):= A_e((1/2)√(T_j)t_k^1/4,4√(T_j)t_k^1/4). Let us then consider the decomposition 
    
    X = X_*+∑_kχ_𝒜_kX_k

    with the above linear vector fields X_k in the kernel of δ_eδ_e^* for the rest of the proof.
    
    The objective is now to show that there exists a constant C_D>0 such that 
    X_*_r_DC^3,α_β(g^D)+ ∑_k X_k_r_eC^0_0(g_e)⩽ C_D δ_g^Dδ_g^D^*X_r^-1_DC^1,α_β(g^D).

    In order to do this, we will reduce our situation to M_o and to the N_j where such controls have been shown in Proposition <ref>. 
    

    
    On M_o^t/16, g^D-g_o is supported in M_o^t/16\ M_o^16t, that is where 1/2 t_k^1/4⩽ r_D< 2 t_k^1/4 on each annulus 𝒜_k, and for all l∈ℕ, there exists C_l>0 such that in these regions, we have
    
    t_k^l/4|∇^l (g^D-g_o)|_g_o⩽ C_l t_k^1/2.

    
    Consider the cut-off function χ_M_o^t/16 of Definition <ref> supported in M_o^t/16 such that χ_M_o^t/16≡ 1 on M_o^t and such that for all l∈ℕ, there exists C_l>0 for which in each 𝒜_k, 
    
    t_k^l/4|∇^lχ_M_o^t/16|_g^D⩽ C_l.

    We then define a vector field X_o on M_o by
    
    X_o:= χ_M_o^t/16X_* + ∑_k∈ K_oχ_B_o(p_k,ϵ)X_k,

    where K_o is the set of k such that the annulus 𝒜_k has a nonempty intersection with M_o^t. 
    
    By construction, X_*=X_o,* on M_o^t and we therefore have the following obvious control. Denoting X_o,*:=  χ_M_o^t/16X_*, we have
    
    X_o,*_r_oC^3,α_β(g_o)⩾X_o,*_r_oC^0_β(g_o)⩾(X_*)_|M_o^t_r_DC^0_β(g^D).

    
    On M_o^t/16, we have,
    
    δ_g^Dδ_g^D^*X= δ_g_oδ_g_o^*X_o+δ_g_oδ_g_o^*(X-X_o)+(δ_g^Dδ_g^D^*-δ_g_oδ_g_o^*)X.

    Since the cut off functions are bounded in norm C^3,α_0(g^D) and C^3,α_0(g_o) by (<ref>), and since their derivatives are supported in M_o^t/16\ M_o^t, we have the following bound on the last two terms of (<ref>): for C>0 depending on the cut off function, we have
    
        
  * δ_g_oδ_g_o^*(X-X_o)_r^-1_DC^1,α_β(g^D)⩽ C(X_*)_|M_o^t/16\ M_o^t_r_DC^3,α_β(g^D) since the difference between X and X_o on M_o^16t only comes from the cut-off on X_*, and
        
  * ((δ_g^Dδ_g^D^*-δ_g_oδ_g_o^*)X)_|M_o^t/16_r^-1_DC^1,α_β(g^D)⩽ C∑_k∈ K_o t_k^1/2X_r_DC^3,α_β,* thanks to (<ref>).
    
    Consequently, by (<ref>), and using (<ref>), for C>0 depending on the above constants, we have
    
    (δ_g^Dδ_g^D^*X)_|M_o^t/16_r^-1_DC^1,α_β(g^D)⩾    δ_g_oδ_g_o^*X_o_r^-1_oC^1,α_β(g_o)
       - C(X_*)_|M_o^t/16\ M_o^t_r_DC^3,α_β(g^D)
       -C∑_k∈ K_o t_k^1/2X_r_DC^3,α_β,*
    ⩾    δ_g_oδ_g_o^*X_o_r^-1_oC^1,α_β(g_o)
       - 2C( (δ_g^Dδ_g^D^*X)_|𝒜_k_r^-1_DC^1,α_β(g^D)
       +∑_k∈ K_oϵ^β-1t_k^1-β/4(X-χ_𝒜_k(t,ϵ)X_k)_𝒜_k_r_DC^3,α_β(g^D))
       -C∑_k∈ K_o t_k^1/2X_r_DC^3,α_β,*.

    
    Now, when t_max→ 0, we have 
    π_𝐊_o^⊥X_o_r_oC^3,α_β,*(g_o)/X_o_r_oC^3,α_β,*(g_o)→ 0

    because X⊥𝐊̃_o. Proposition <ref> therefore yields, for t_max small enough,
    
    X_o_r_oC^3,α_β,*(g_o)⩽ 2C_oδ_g_oδ_g_o^*X_o_r^-1_oC^1,α_β(g_o).

    Therefore, thanks to (<ref>) and (<ref>), for t_max small enough, and denoting by C'>0 a constant that may change from line to line but only depending on the previous ones of this proof, and therefore only on g_o and the g_b_j and γ(t_max):= ∑_kt_max^1/2 + t_max^1-β/4, we have
    
    (X_*)_|M_o^t_r_DC^0_β(g^D) + ∑_k∈ K_o   X_k_r_eC^0_0(g_e) -C' γ(t_max)X_r_DC^0_β,*(g^D)
    ⩽   X_o_r_oC^3,α_β,*(g_o)- C'γ(t_max)X_r_DC^0_β,*(g^D)
    ⩽    2C_oδ_g_oδ_g_o^*X_o_r^-1_oC^1,α_β(g_o)- C'γ(t_max)X_r_DC^0_β,*(g^D)
    ⩽    C'δ_g^Dδ_g^D^*X_r^-1_DC^1,α_β(g^D),

    where we successively used (<ref>), (<ref>) and (<ref>).
    Indeed, on an orbifold (M_o,g_o), the vector fields X_k of the decomposition (<ref>) reaching the infimum of the definition of the norm ._r_oC^3,α_β,* are determined by the limit of r_o^-1X_o at each singular point according to Remark <ref>. Here, the infimum is therefore reached with the X_k of the decomposition (<ref>).
    

    
    We next consider the vector field X_1:= X- ∑_k∈ K_oχ_𝒜_kX_k which satisfies for a constant C>0,
    
    δ_g^Dδ_g^D^*X_1 _r_D^-1C^1,α_β(g^D)⩽     C( δ_g^Dδ_g^D^*X _r_D^-1C^1,α_β(g^D)+γ(t_max)X_r_DC^0_β,*(g^D))

    thanks to the control (<ref>) of ∑_k∈ K_oX_k_r_eC^0_0(g_e).
    
    Given j∈ K_o, the Ricci-flat ALE orbifold (N_j,g_b_j) is glued to M_o and we can extend the vector field X_1 = X_* + ∑_k ∉ K_oχ_𝒜_lX_l to N_j by 
    
    X_j:= χ_N_j^t/16X_* + ∑_l∈ K_jχ_B_j(p_l,ϵ)X_l,

    where K_j is the set of k≠ j such that 𝒜_k has a nonempty intersection with N_j^t. 
    
    
        By considering X_1 instead of X, we do not have a linear vector field of the kernel of δ_eδ^*_e to extend at at infinity of N_j. The vector field X_j is therefore well controlled in r_b_jC^3,α_β,*(g_b_j).
    
    
    The difference g^D/T_j-g_b_j is supported in N_j^t/16\ N_j^16t and there exists for all l∈ℕ, C_l>0 such that we have the following controls. Around the singular points where 
    1/2√(T_j)t_k^1/4<r_D = √(T_j)r_b_j<2√(T_j)t_k^1/4,
 
    we have
    
    t_k^l/4|∇^l (g^D/T_j-g_b_j)|_g_b_j⩽ C_l t_k^1/2.

    and at infinity, where 
    1/2√(T_j)t_j^-1/4<r_D=√(T_j)r_b_j<2√(T_j)t_j^-1/4
 we have
    
    t_j^-l/4|∇^l (g^D/T_j-g_b_j)|_g_b_j⩽ C_l t_j^1/2.

    
    Denoting X_j,*:= χ_N_j^t/16X_*, we have
    
    X_j,*_r_b_jC^3,α_β(g_b_j)⩾X_j,*_r_b_jC^0_β(g_b_j)⩾(X_*)_N^t_j_r_DC^0_β(g^D),

    and thanks to (<ref>) and the inequalities (<ref>) and (<ref>), we have
    
    δ_g^Dδ_g^D^*X_1:= δ_g_b_jδ_g_b_j^*X_j+δ_g_b_jδ_g_b_j^*(X_1-X_j)+(δ_g^Dδ_g^D^*-δ_g_b_jδ_g_b_j^*)X_1,

    analogously to (<ref>), we find for C>0 depending on the above constants such that
    
    (δ_g^Dδ_g^D^*X_1)_|N_j^t/16_r^-1_DC^1,α_β(g^D)⩾    δ_g_b_jδ_g_b_j^*X_j_r^-1_b_jC^1,α_β(g_b_j)
       - C( (δ_g^Dδ_g^D^*X_1)_|𝒜_k_r^-1_DC^1,α_β(g^D)
       + ∑_k∈ K_jϵ^β-1t_k^1-β/4(X_*)_𝒜_k_r_DC^3,α_β(g^D))
       -C t_k^1/2X_1_r_DC^3,α_β,*- C t_j^1/2X_1_r_DC^3,α_β,*,

    where we remark that X_1_r_DC^3,α_β,*⩽X_*_r_DC^3,α_β + ∑_k∈ K_jX_k_r_eC^0_0(g_e).
    
    Proposition <ref> then yields
    
    X_j_r_jC^3,α_β,*(g_b_j)⩽ C_jδ_g_b_jδ_g_b_j^*X_j_r^-1_jC^1,α_β(g_b_j),

    and thanks to the control (<ref>), we then have for a constant C>0 only depending on the constants of Propositions <ref> and  <ref>, the control
    
    (X_*)_|N_j^t_r_DC^0_β(g^D) + ∑_k∈ K_jX_k_r_eC^0_0(g_e) - Cγ   (t_max)X_r_DC^0_β,*(g^D)
    ⩽     Cδ_g^Dδ_g^D^*X_r^-1_DC^1,α_β(g^D),

    similar to (<ref>).
    
    Iterating this to the other Ricci-flat ALE orbifolds of the tree of singularities, we get controls similar to (<ref>) on all the N_j which, with (<ref>) on M_o, give the following control on the whole manifold only depending on g_o and the g_b_j 
    
    X_*_r_DC^0_β(g^D) + ∑_k   X_k_r_eC^0_0(g_e) - Cγ(t_max) X_r_DC^0_β,*(g^D)
       ⩽  Cδ_g^Dδ_g^D^*X_r^-1_DC^1,α_β(g^D)

    and for t_max small enough. Together with the elliptic estimates of Proposition <ref>, this shows the stated result for g=g^D because M = M_0^t∪⋃_jN_j^t.
    

    
    To get the estimate for another metric g close to g^D, we just use Proposition <ref> to ensure that for g-g^D_C^2,α_β,*(g^D) arbitrarily small, δ̃_gδ^*_g is arbitrarily close to δ_g^Dδ^*_g^D for the operator norm on r_DC^3,α_β,*(g^D).
    
    Finally, notice that (δ̃_g^D)^* = δ^*_g^Dπ_𝐊̃_o^⊥, and therefore that δ̃_g^Dδ^*_g^D is self adjoint on 𝐊̃_o^⊥. Its injectivity implies its surjectivity by integration by parts on the compact manifold (M,g^D).  


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


    Let 0<β<1/2, and (M,g^D)=(M,g^D_t) be a naïve desingularization of a compact Einstein orbifold, (M_o,g_o). Then, there exist ϵ_D,τ_D,C_D>0 which only depends on the constants of Lemma <ref> such that for t_max⩽τ_D and for any metric g satisfying g-g^D_C^2,α_β,*(g^D)⩽ϵ_D, there exists a unique vector field X⊥𝐊̃_o on M for which, 
    
    δ̃_g^D(exp_X^*g)=0,

    where exp_X is the diffeomorphism exp_X: x∈ M ↦exp_x^g^D(X(x)).
    
    We moreover have, X_r_DC^3,α_β,*(g^D)⩽ C_D δ̃_g^D(g-g^D)_r^-1_DC^1,α_β(g^D), and therefore, there exists η: ℝ^+→ℝ^+ with lim_0η=0 such that we have 
    
    exp_X^*g-g_C^2,α_β,*(g^D)⩽η(g-g^D_C^2,α_β,*(g^D)).



    Let us fix g a metric on M, such that g-g^D_C^2,α_β,*⩽ϵ for ϵ>0 which we will choose small enough along the proof and define the operator F_g: r_DC^3,α_β,*(g^D)→ r^-1_DC^1,α_β(g^D) which to a vector field X associates 
    F_g(X):=δ̃_(exp_g^DX)_*g^Dg.

    
    The objective is therefore to find X such that F_g(X)=0, which will imply that δ̃_g^D(exp_g^DX)^*g = 0 because for any diffeomorphism ϕ, ϕ_*(δ_g^Dϕ^* g) = δ_ϕ_*g^Dg (by applying ϕ_* to g^D which is C^∞, we do not loose regularity). The map g↦ F_g is linear, and the linearization of the operator F_g^D around zero is δ̃_g^Dδ^*_g^D which is invertible between the orthogonals of 𝐊̃_o according to Lemma <ref>. 
    
    There remains to control the nonlinear terms in our norms. Let us denote them Q = F_g^D-δ̃_g^Dδ^*_g^D. We schematically have that
    Q(X) is a converging sum of terms of the form (∇_g^D^2X*X)*X*...*X, (∇_g^DX*∇_g^DX)*X*...*X, ((g^D)* X*X)*...*X which are at least quadratic in X, where * denotes various multilinear operations or contractions. 
    
    Using (<ref>) of Remark <ref>, since on a compact manifold r_D_C^3,α_0 is bounded, there exists C>0 such that 
    
    ._r_D^-1C^1,α_β(g^D)⩽ C._C^1,α_β(g^D),

    and for any vector field X, assuming β<1/2, we have
    
    ∇_g^D X_r_D^-1/2C^2,α_0(g^D)⩽∇_g^D X_r_D^-1/2C^2,α_β(g^D)⩽ X_r_D^1/2C^3,α_β(g^D)⩽ CX_r_DC^3,α_β,*(g^D).

    We therefore find
    
    Q(X)-    Q(X')_r_D^-1C^1,α_β(g^D)
    ⩽     C(X-X'_r_DC^1,α_0(g^D)(∇^2 X_r^-1_DC^1,α_β(g^D)+∇^2 X'_r^-1_DC^1,α_β(g^D))
       +(X_r_DC^1,α_0(g^D)+X'_r_DC^1,α_0(g^D))∇^2 (X-X')_r^-1_DC^1,α_β(g^D)
       +(X_r_DC^1,α_0(g^D)+X'_r_DC^1,α_0(g^D))(X-X'_r_DC^1,α_0(g^D))(g^D)_r^-2_DC^1,α_β(g^D)
       +∇ (X-X')_r_D^-1/2C^2,α_0(g^D)(∇ X_r_D^-1/2C^2,α_β(g^D)+∇ X'_r_D^-1/2C^2,α_β(g^D))
    ⩽     3C (X_r_DC^3,α_β,*(g^D)+X'_r_DC^3,α_β,*(g^D))X-X'_r_DC^3,α_β,*(g^D),

    notice the different norms with β and 0 for the weight power. We controlled the
    C^1,α_β(g^D)-norm (which is larger than the r_D^-1C^1,α_β(g^D)-norm by (<ref>)) in the first three lines and the r_D^-1C^1,α_β(g^D)-norm in the last one (notice the r_D^-1/2-norms controlled by (<ref>)).
    
        Using r_D^-1/2-norms was necessary because the first derivatives of the linear element of the kernel of δ_g_eδ^*_g_e do not decay in the neck regions, that is, ∇_g^Dχ_𝒜_k(t,ϵ)X_k∉ C^2,α_β(g^D).
    
    
    The crucial reason for such a control of the nonlinear terms, already noted in <cit.>, is that our norm is equivalent to a norm C^k,α(g^D) weighted by a function uniformly bounded below by 1 independently on t, see (<ref>). We can therefore finally put our metrics in gauge with respect to each other thanks to a fixed point theorem with explicit constant below, Lemma <ref>.

				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).

    
    Let us finally remark that for a linear vector field X_k in the kernel of δ_eδ_e^*, the symmetric 2-tensor δ_e^* X_k is constant. This lets us define for any metric g on M, a continuous map ψ_g : r_DC^3,α_β,*→ C^2,α_β,* by
    
    X↦ψ_g(X):=exp^*_X g.

    It is indeed continuous since for any diffeomorphism ϕ : M → M, we have
    
    (ϕ^* g)(x) _kl = g(ϕ(x))_ij∂_iϕ/∂ x^k∂_jϕ/∂ x^l

    in local coordinates and therefore, for any vector field X_*∈ r_DC^3,α_β, the symmetric 2-tensor exp^*_X g-g is arbitrarily small for the C^2,α_β,*-norm. For the constant part, it is enough to note that for a linear vector field X_k in the kernel of δ_eδ^*_e, and for a constant symmetric 2-tensor H_k, the symmetric 2-tensor exp^*_X_kH_k is also constant and controlled |exp^*_X_kH_k|_g_e⩽ CX_k_r_eC^0|H_k|_g_e.










 §.§ Einstein metrics in gauge


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

    E(g):= (g)-(g)/2g+n-2/2nR(g)g

on a compact manifold M, where R:= 1/(M,g)∫_M (g)dv_g. Notice that δ_gE(g)=0, again by the Bianchi identity.

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

It turns out that we can characterize the zeros of E in reduced divergence-free gauge as the zeros of a single operator Φ_g^D defined by

    Φ_g^D(g):=E(g) + δ_g^*δ̃_g^Dg.

Indeed, if we have E(g)=0 and δ̃_g^D g=0, then we have Φ_g^D(g) = 0. And conversely, if Φ_g^D(g) = 0, then since E(g) is divergence-free (for g) by the Bianchi identity, by taking the reduced divergence of Φ_g^D(g) = 0, we get 
    δ̃_gΦ_g^D(g) = (δ̃_gδ_g^*)δ̃_g^Dg.

Since for g close enough to g^D, (δ̃_gδ_g^*) is invertible on the image of δ̃_g^D by Lemma <ref>, we finally have δ̃_g^Dg = 0 and E(g)=0. In a C^2,α_β,*(g^D)-neighborhood of g^D the zero set of Φ_g^D is exactly the set of Einstein metrics in reduced divergence-free gauge with respect to g^D.


Let D_0,v_0>0, l∈ℕ, and β= β(v_0,D_0)>0 obtained in Corollary <ref>. Then, for all ϵ>0, there exists δ = δ(ϵ,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 above by D_0,
		
  * the Ricci curvature is bounded ||≤ 3.
	
	and such that for an Einstein orbifold (M_o,g_o),
	
    d_GH((M,g^ℰ),(M_o,g_o))⩽δ,

	then, there exists a naïve desingularization (M,g^D) of (M_o,g_o) by a tree of singularities, and a diffeomorphism ψ: M→ M such that 
    ψ^*g^ℰ-g^D_C^l_β,*(g^D)⩽ϵ,

	and
	
    δ̃_g^D(ψ^*g^ℰ)=0.

	In particular, we have 
    Φ_g^D(ψ^*g^ℰ)=0.





§ RESOLUTION OF THE EINSTEIN EQUATION MODULO OBSTRUCTIONS


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

We have seen in Corollary <ref> that up to a diffeomorphism, any Einstein metric g close to (M_o,g_o) in the Gromov-Hausdorff sense is a solution of

    Φ_g^D(g):= E(g) + δ_g^*δ̃_g^Dg =0.
 
To study this equation, we will naturally start by studying its linearization on volume preserving deformations at g^D, that is, on symmetric 2-tensors h satisfying ∫_M_g^D h dv_g^D = 0, for which we have the formula

    P_g^D(h):=  d_g^DΦ_g^D(h) =   1/2(∇^*_g^D∇_g^D h-2_g^D(h)
       -2 δ_g^D^*δ_g^Dh+2δ^*_g^Dδ̃_g^D h-(δ_g^Dδ_g^D h) g^D
       + (Δ_g^D_g^Dh) g^D - ∇_g^D^2 _g^D h
       +_g^D∘ h+h∘_g^D-_g^D h+ ⟨_g^D, h ⟩_g^Dg^D
       +1/2(g) h - 1/2(g)∫_M⟨(g^D)-(g^D)/2, h⟩_g^Dv_g^D),

in dimension 4. If g^D were an Einstein metric and h a divergence-free symmetric 2-tensor, then the operator P̅_g^D would reduce to 
    P_g^D:= 1/2∇_g^D^*∇_g^D - _g^D,

which is simpler to study. Since g^D is almost Einstein and h will be almost divergence-free, we will mostly study the operator P_g^D, and we will obtain results for P̅_g^D by approximation.



 §.§ Kernel and cokernel of the linearization on model spaces




  
Exceptional values for P_e:= 1/2∇_e^*∇_e on (ℝ^4Γ,g_e) and gauge constraints.

As described in the proof of Proposition <ref>, the elements of the kernel of P_e on ℝ^4Γ are sums of homogeneous symmetric 2-tensors whose coefficients in an orthonormal basis of ℝ^4 are homogeneous of order k or -2-k for k∈ℕ. 

However, some of these tensors cannot appear in our developments because they are not trace-free or in divergence-free gauge on our nontrivial quotient of ℝ^4Γ.
        
            On ℝ^4Γ for Γ≠{e}, there is no harmonic homogeneous symmetric 2-tensor whose coefficients are of order 1, -2 or -3 in divergence-free gauge.
        
     
        


	

  
Kernel of the operator P on the model spaces.
	Let us start by describing the kernel of P on our model spaces. 
	
	
		Let (N,g_b) be a Ricci-flat ALE orbifold, and denote P_b:= 1/2∇^*_b∇_b - _b, and 𝐎(g_b), the kernel of P_g_b on C^2,α_β,*(g_b).
		
		The elements of 𝐎(g_b) decay at least like r_b^-4 at infinity, and for all 𝐨_b∈𝐎(g_b), we have the following development coordinates at infinity,
		
    𝐨_b = O^4 + 𝒪(r_b^-5),

		with O^4∼ r_b^-4 a harmonic homogeneous symmetric 2-tensor. 
		
		Let also (M_o,g_o) be a compact Einstein orbifold, we denote 𝐎(g_o) the kernel of P_o on C^2,α_β,* for all 0<β<1. An element 𝐨_o∈𝐎(g_o) has a development 
		
    𝐨_o = O_0+ O_2+𝒪(r_o^3),

		for harmonic homogeneous symmetric 2-tensors O_i∼ r_o^i.
	
	
	Let us consider 𝐨∈𝐎(g_b), for which P_b𝐨=0, and 𝐨= 𝒪(r_b^-β) for some β>0. Such a symmetric 2-tensor is actually necessarily traceless and divergence-free. Indeed, we have
	δ_g_bP_g_b =  1/2∇_g_b^*∇_g_bδ_g_b,
	and
    _g_bP_g_b =  1/2∇_g_b^*∇_g_b_g_b.
    Therefore, if P_g_bh = 0 for h=𝒪(r_b^-δ) for some δ>0, then δ_g_bh=0, and _g_bh = 0 by the maximum principle. We deduce from Lemma <ref> that 𝐨 decays at least like r_b^-4 and its principal term is a harmonic symmetric 2-tensor.

	In the same way in the neighborhood of a singularity of an orbifold (M_o,g_o) or of a Ricci-flat ALE orbifold (N,g_b), since there is no harmonic symmetric 2-tensor with linear growth because of the action of the nontrivial group Γ, an element of the kernel admits a development
	
    𝐨 = O_0+O_2+𝒪(r_b^3),

	where O_0 and O_2 are harmonic homogeneous symmetric 2-tensors in r_o^0 and r_o^2 respectively.
	
	
	
	
	
	

  
Estimates on the inverses.
	
	Just like for the operator δδ^* in Proposition <ref>, the operators P_g_o and P_g_b_j are injective on the orthogonal of their respective kernels.
	
	
	    Let (N,g_b) be a Ricci-flat ALE orbifold and (M_o,g_o) an Einstein orbifold, and 0<β<1.
	    
	    Then, the operators 
    P_b: 𝐎(g_b)^⊥∩ C^2,α_β,*(g_b) →  r^-2_bC^α_β(g_b),

	    and
	    
    P_o: 𝐎(g_o)^⊥∩ C^2,α_β,*(g_o) →  r^-2_oC^α_β(g_o)

	    are injective and there exist C_o>0 and C_b>0, such that we have for any symmetric 2-tensor h_b⊥𝐎(g_b) on N and h_o⊥𝐎(g_o) on M_o,
	    
    C_b^-1P_bh_b_r^-2_bC^α_β(g_b)⩽h_b_C^2,α_β,*(g_b)⩽ C_b P_bh_b_r^-2_bC^α_β(g_b),
	    
	    and
	    
    C_o^-1P_oh_o_r^-2_oC^α_β(g_o)⩽h_o_C^2,α_β,*(g_o)⩽ C_o P_oh_o_r^-2_oC^α_β(g_o).

	    Moreover, their respective cokernels are 𝐎(g_b) and 𝐎(g_o).
	 
	 
	     By standard theory of elliptic operators between weighted Hölder spaces (see for instance <cit.>), the operators
	     
    P_o: C^2,α_-β(g_o) → r^-2_oC^α_-β(g_o),

	     and
	     
    P_b: r_b^-βC^2,α_0(g_b) → r^-2-β_bC^α_0(g_b)

	     are Fredholm for 0<β<1 because we avoid the exceptional values close to zero: -2 and 1. Let us study their kernels and cokernels. 
	     
	     Let us start by the case of an Einstein orbifold (M_o,g_o) and notice that _ C^2,α_-β(g_o)P_g_o⊂𝐎(g_o) because there is no exceptional value between 0 and -β. The kernel of P_o: C^2,α_-β(g_o) → r^-2_oC^α_-β(g_o) is therefore equal to 𝐎(g_o). Since P_o is self adjoint and since we are strictly between two exceptional values, its cokernel is the kernel of P_o on r^-2_oC^α_β(g_o) which is also reduced to 𝐎(g_o) by a similar argument.
	     
        Similarly, the kernel of  P_b: r_b^-βC^2,α_0(g_b) → r^-2-β_bC^α_0(g_b) is 𝐎(g_b), and its cokernel is the kernel of P_b on r^-2+β_bC^α_0(g_b) which is reduced to 𝐎(g_b).
	    
	    For the cokernels of 
    P_b: 𝐎(g_b)^⊥∩ C^2,α_β,*(g_b) →  r^-2_bC^α_β(g_b),

	    and
	    
    P_o: 𝐎(g_o)^⊥∩ C^2,α_β,*(g_o) →  r^-2_oC^α_β(g_o),
 like in the end of the proof of Proposition <ref>, we use the fact that 
    P_o(C^2,α_β,*(g_o)) =P_o( C^2,α_-β(g_o))∩ r^-2_oC^α_β(g_o),

	    and
	    
    P_b(C^2,α_β,*(g_b))=P_b( r_b^-βC^2,α_0(g_b_j))∩  r^-2_bC^α_β(g_b).

	 
	 
    


  
Approximation of kernels and cokernels on a naïve desingularization.

We wish to solve the equation (g) = Λ g for a metric g close to g^D modulo the kernel and the cokernel of the linearization of the Einstein operator. We will use approximate kernels and cokernels defined as the truncated infinitesimal deformations of each model space on the tree of singularities in order to obtain uniform controls as the singularities form, that is as the gluing parameters t tend to 0.

Let 𝐨_o∈𝐎(g_o) and 𝐨_j∈𝐎(g_b_j), and define 𝐨_o = 𝐨_o,* + ∑_k χ_B_o(p_k,ϵ_0)𝐨_o,k and 𝐨_j = 𝐨_j,* + ∑_k χ_B_j(p_k,ϵ_0)𝐨_j,k their respective decompositions as a symmetric 2-tensor of C^2,α_β and constant symmetric 2-tensors truncated in the neighborhoods of the singular points. Thanks to the cut-off functions of Definition <ref>, we define on M the following symmetric 2-tensors

    𝐨̃_o,t:=χ_M_o^t𝐨_o,* + ∑_k χ_𝒜_k(t,ϵ_0)𝐨_o,k,

and

    𝐨̃_j,t:=χ_N_j^t𝐨_j,* + ∑_k χ_𝒜_k𝐨_j,k.


    We have 𝐨̃_o,t = 𝐨_o on M_o^16t, and 𝐨̃_j,t =𝐨_j on N_j^16t.




    Let (M,g^D_t) be a naïve desingularization of a Einstein orbifold (M_o,g_o). On M, we will denote 
    𝐎̃(g^D):={𝐨̃_o,t + ∑_j𝐨̃_j,t,  𝐨_o∈𝐎(g_o),  𝐨_j∈𝐎(g_b_j)},
 the space of truncated obstructions.


    For 0<β<2, by elliptic regularity for the elements of 𝐎(g_o) and the 𝐎(g_b_j), and the C^4_0-control of the cut-off functions  we have
    
    𝐨̃_o,t + ∑_jT_j𝐨̃_j,t_C^0(g^D)≈𝐨̃_o,t + ∑_jT_j𝐨̃_j,t_C^2,α_β,*(g^D)≈sup(𝐨_o_C^0(g_o), 𝐨_j_C^0(g_b_j)),

    and
    
    𝐨̃_o,t + ∑_j𝐨̃_j,t_L^2(g^D)≈𝐨̃_o,t + ∑_j𝐨̃_j,t_r_D^-2C^α_β(g^D)≈sup(𝐨_o_L^2(g_o), 𝐨_j_L^2(g_b_j)).


    
    
We would like to produce Einstein metrics in reduced divergence-free gauge with respect to g^D. But the point is that it is not always possible because the space 𝐎̃(g^D) is an (approximate) obstruction space. We will show that we can perturb g^D+v for parameters t>0 and v∈𝐎̃(g^D) small enough to obtain a metric g_v=ĝ_t,v which will be in gauge with respect to g^D + v and solution of:

    Φ_g^D(ĝ_v) ∈𝐎̃(g^D),

hence the term Einstein modulo obstructions.




  
Control of the inverse of the linearization.

We can first show that the linearization is invertible and that we can control its inverse independently of the gluing scales thanks to Lemma <ref>. 

Let us start by showing that the operator π_𝐎̃(g^D)^⊥P_g^D is close to P_g^D for a sufficiently degenerate tree of singularities.


    There exists C>0 such that for any symmetric 2-tensor h∈ C^2,α_β,*(g^D), we have
    
    (π_𝐎̃(g^D)^⊥P_g^D-P_g^D)h_r_D^-2C^α_β(g^D)⩽ C t_max^1/2h_C^2,α_β,*(g^D).



    Here, the constant C only depends on the constants C_o and C_b_j of Lemma <ref>.


    The proof is similar to that of the estimate (<ref>). Thanks to the equivalence of the norms, see Remark <ref>, we only have to control the L^2(g^D)-norm of the projection on 𝐎̃(g^D).

    On M_o, since P_g^D is self adjoint, an integration by parts yields 
    
    |∫_M⟨ P_g^Dh, 𝐨̃_o,t⟩_g^Dv_g^D| =    |∫_M⟨ h,P_g^D(𝐨̃_o,t) ⟩_g^Dv_g^D|,

    and since g^D=g_o on M_o^16t and P_g_o(𝐨_o)=0 we use the decomposition
    
    P_g^D(𝐨̃_o,t) =    P_g_o((χ_M_o^16t-1)𝐨_o,*)
       + P_g_o(∑_k (χ_𝒜_k(t,ϵ_0)- χ_B_o(p_k,ϵ_0))𝐨_o,k)) 
       + (P_g_o-P_g^D)(∑_k χ_𝒜_k(t,ϵ_0)𝐨_o,k)

    in order to obtain the following  estimates (compare with the proof of (<ref>)) thanks to the control of the cut-off functions of Definition <ref> and to Remark <ref> 
    
    |P_g_o((χ_M_o^16t-1)𝐨_o,*)|_g_o⩽  C𝐨_o_L^2(g_o)1_{2t_k^1/4<r_D<4t_k^1/4} t_k^-1/2,

    
    |P_g_o(∑_k (χ_𝒜_k(t,ϵ_0)- χ_B_o(p_k,ϵ_0))𝐨_o,k))|⩽ C𝐨_o_L^2(g_o)1_{ϵ_0^-1t_k^1/2<r_D<2ϵ_0^-1t_k^1/2} t_k^-1,

    and
    
    |(P_g_o-P_g^D)(∑_k χ_𝒜_k(t,ϵ_0)𝐨_o,k)|⩽ C𝐨_o_L^2(g_o)1_{ϵ_0^-1t_k^1/2<r_D<2t_k^1/4} (1+t_k^2r_D^-6).

    Since for any C>0 independent on t, we have
    
        
  * _g_e({2t_k^1/4<r^D<4t_k^1/4})⩽ Ct_k,
        
  * _g_e({ϵ_0^-1t_k^1/2<r_D<2ϵ_0^-1t_k^1/2})⩽ Cϵ_0^-4t_k^2, and
        
  * ∫_{ϵ_0^-1t_k^1/2<r_D<2t_k^1/4} (1+t_k^2r_D^-6)dv_g^D⩽ Ct_max,
     
    integrating the controls (<ref>), (<ref>) and (<ref>) of the terms of (<ref>) yields
    
    |∫_M⟨ P_g^Dh, 𝐨̃_o,t⟩_g^Dv_g^D| ⩽ C t_k^1/2h_C^2,α_β,*(g^D)𝐨_o_L^2(g_o).

    Similarly, for the N_j, consider 𝐨_j∈𝐎(g_b_j). By invariance of the L^2-norm of 2-tensors in dimension 4 and since P_g/t=tP_g for any metric g and t>0, we have
    
    ∫_M⟨ P_g^Dh,𝐨̃_j,t⟩_g^Dv_g^D   = ∫_M⟨h/T_j,P_g^D/T_j𝐨̃_j,t⟩_g^D/T_jv_g^D/T_j.

    The control at the singular points is the same as in the case of M_o and at infinity we have
    
    P_g^D/T_j𝐨̃_j,t = (P_g^D/T_j-P_g_b_j) 𝐨̃_j,t +P_g_b_j𝐨_j+ P_g_b_j((χ_N_j^16t-1)𝐨_j,*),

    and therefore, since 𝐨_j = 𝒪(r_b_j^-4), we have
    
    |P_g^D/T_j𝐨̃_j,t|_g_b_j⩽ C𝐨_j_L^2(g_b_j)1_{1/2t_j^-1/4<r_b_j<t_j^-1/4}r_b_j^-6≈ C𝐨_j_L^2(g_b_j)1_{1/2t_j^-1/4<r_b_j<t_j^-1/4}t_j^3/2,

    and since (A_e(1/2t_j^-1/4,t_j^-1/4))⩽ Ct_j^-1 and at every point |χ_N^16t_jh/T_j|_g_b_j⩽h_C^2,α_β,*(g^D), we have
    
    |∫_M⟨ P_g^Dh, 𝐨̃_j,t⟩_g^Dv_g^D| ⩽ C t_max^1/2h_C^2,α_β,*(g^D)𝐨_j_L^2(g_b_j),

    and finally,
    
    (π_𝐎̃(g^D)^⊥P_g^D-P_g^D)h_r_D^-2C^α_β(g^D)⩽ C t_max^1/2h_C^2,α_β,*(g^D).



    Let 0<β<1, k∈ℕ, 0<α<1 and let (M,g^D) be a naïve desingularization of a compact or ALE Einstein orbifold by a tree of singularities. Then, there exists τ_D >0 and ϵ_D>0 and C_D>0 only depending on β, and the constants of Lemma <ref> and of Proposition <ref> such that for t_max<τ_D and for any metric g such that g-g^D_t_C^2,α_β,*(g^D)⩽ϵ_D, the operator 
    π_𝐎̃(g^D)^⊥P_g: 𝐎̃(g^D)^⊥∩ C^2,α_β,*(g^D) →𝐎̃(g^D)^⊥∩ r^-2_DC^α_β(g^D),
 where π_𝐎̃(g^D)^⊥ is the orthogonal projection for g^D on 𝐎̃(g^D)^⊥, is invertible and we have for any symmetric 2-tensor h⊥𝐎̃(g^D) on M,
    
    h_C^2,α_β,*(g^D)⩽ C_Dπ_𝐎̃(g^D)^⊥P_gh_r^-2_DC^α_β(g^D).



    
        The proof is similar to that of Lemma <ref>. The idea is again to extend the symmetric 2-tensors on the model spaces and to deduce a control on the whole tree of singularities. 
        
Let 0<ϵ<ϵ_D^1/2-β<ϵ_0 for ϵ_D and ϵ which we will choose small enough along the proof, and assume that t_max< ϵ^4 in order to have on each annulus 𝒜_k:= 𝒜_k(t,ϵ) between N_k and N_j, the existence of a diffeomorphism 
    Φ_k: A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j))⊂ℝ^4Γ_k→𝒜_k⊂ M,

        such that for any 0<β<1, there exists C>0, 
    Φ^*_kg^D-g_e_C^2,α_β(A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j)))⩽ C ϵ^2-β<Cϵ_D,

        by definition of g^D. Until the end of the proof, we will denote
        
    A_k:= A_e(ϵ^-1√(T_j)√(t_k),ϵ√(T_j)).

        
        Let h be a symmetric 2-tensor on M. Thanks to the estimate (<ref>) of Proposition <ref> and to its generalization to metrics close to g_e by Proposition <ref>, for ϵ_D and t_max small enough, we can choose constant and traceless symmetric 2-tensors H_k on ℝ^4Γ_k such that we have on A(√(T_j)t_k^1/4):=A_e((1/2)√(T_j)t_k^1/4,4√(T_j)t_k^1/4)
        
    Φ_k^*h-H_k_C^2,α_0(A(T_j^1/2t_k^1/4))⩽     C_e (ϵ^-βt_k^β/4P_g_eΦ_k^*h_r^-2_eC^α_β(A_k)
       + 4 ϵ^-1t_k^1/4Φ_k^*h-H_k_C^2,α_β(A_k))
    ⩽    2 C_e(ϵ^-βt_k^β/4(P_g^Dh)_|𝒜_k_r^-2_DC^α_β(g^D)
       + 4 ϵ^-1t_k^1/4(h-χ_𝒜_kH_k)_C^2,α_β(g^D)).

    Let us then consider the decomposition 
    h = h_* + ∑_k χ_𝒜_kH_k.

    
    We define a symmetric 2-tensor h_o extending h to M_o in the following way:
    
    h_o:= χ_M_o^t/16 h_* +∑_k∈ K_oχ_B_o(p_k,ϵ)H_k,

    where K_o is the set of k such that 𝒜_k∩ M_o^t≠∅. Denoting h_o,*:=χ_M_o^t/16 h_*, we have 
    
    h_o,*_C^2,α_β(g_o)⩾h_o,*_C^0_β(g_o)⩾(h_*)_|M_o^t_C^0_β(g^D),

    because M_o^t is outside of the region damaged by the gluing.
    
    Since for t_max small enough, the metric g^D is close to the metric g_o on M_o^t/16 by (<ref>) and
    
    P_g^Dh = P_g_oh_o + P_g_o(h-h_o) + (P_g^D-P_g_o)h,

    we moreover have the following control thanks to (<ref>),
    
    (P_g^Dh)_|M_o^t/16_r_D^-2C^α_β(g^D)⩾    P_g_oh_o_r_o^-2C^α_β(g_o)
       - CC_e∑_k∈ K_o(P_g^Dh)_|𝒜_k_r_D^-2C^α_β(g^D)
       -γ(t_max) h_C^2,α_β,*(g^D),

    where γ(t_max)→ 0 when t_max→ 0 (compare with (<ref>) for the vector fields).
    
    Since h⊥𝐎̃(g^D), we have π_𝐎(g_o)h_o_C^2,α_β,*/h_o_C^2,α_β,*→ 0, and by Lemma <ref>, this implies, for t_max small enough, the control
    
    h_o_C^2,α_β,*(g_o)⩽ 2C_o P_o h_o_r_o^-2C^α_β(g_o),

    and the estimate (<ref>) and Lemma <ref> imply that for t_max small enough, there exists C>0 only depending on the previous constants such that we have
    
    (h_*)_|M_o^t_C^0_β(g^D) + ∑_k∈ K_o |H_k|_g_e -Cγ(t_max)h_C^2,α_β,*(g^D)⩽ C π_𝐎̃(g^D)^⊥P_g^Dh_r_D^-2C^α_β(g^D).

    Indeed, on an orbifold (M_o,g_o), the constant symmetric 2-tensors of the decomposition in the definition of the norm ._C^2,α_β,* are determined, see Remark <ref>.
    

    
    Let us then consider the symmetric 2-tensor h_1:= h- ∑_k∈ K_o H_k which satisfies for C>0 depending on the previous constants, 
    
    π_𝐎̃(g^D)^⊥P_g^Dh_1 _r_D^-2C^α_β(g^D)⩽ C π_𝐎̃(g^D)^⊥P_g^Dh _r_D^-2C^α_β(g^D)+Cγ(t_max) h_C^2,α_β,*(g^D)

    thanks to the control (<ref>) of ∑_k∈ K_o |H_k|_g_e.
    
    Given j∈ K_o, the Ricci-flat ALE orbifold (N_j,g_b_j) is glued to M_o and we can extend the symmetric 2-tensor h_1 = h_* + ∑_l ∉ K_oχ_𝒜_lH_l to N_j by 
    h_j:= χ_N_j^t/16h_* + ∑_l∈ K_jχ_B_j(p_l,ϵ)H_l,
 where K_j is the set of l∉ K_o such that 𝒜_l has a nonempty intersection with the neighborhood of a N_j^t. 

    Denoting h_j,*:= χ_N_j^t/16h_*, we have
    
    h_j,*_C^2,α_β(g_b_j)⩾h_j,*_C^0_β(g_b_j)⩾(h_*)_|N_j^t_C^0_β(g^D),

    and by (<ref>) and since g^D/T_j is close to g_b_j on N_j^t/16 for t_max small enough, we moreover have the following control thanks to (<ref>),
    
    (P_g^Dh_1)_|N_j^t/16_r^-2_DC^α_β(g^D)⩾    P_g_b_jh_j_r^-2_b_jC^α_β(g_b_j)
       - CC_e ∑_k∈ K_j(P_g^Dh_1)_|𝒜_k_r_D^-2C^α_β(g^D)
       -γ(t_max) h_C^2,α_β,*(g^D),

    where γ(t_max)→ 0 when t_max→ 0. We then have a control on (h_*)_|N_j^t and on the H_k, for k∈ K_j thanks to Lemma <ref> by using again the fact that h⊥𝐎̃(g^D) which implies that for t_max small enough, we have
    
    h_j_r_jC^3,α_β,*(g_b_j)⩽ 2C_jP_g_b_jh_j_r^-2_jC^α_β(g_b_j).

    The estimates (<ref>), (<ref>) and Lemma <ref> then yield
    
    (h_*)_|N_j^t_C^0_β+ ∑_k∈ K_j|H_k|_g_e-Cγ(t_max) h_C^2,α_β,*(g^D)⩽ C π_𝐎̃(g^D)^⊥P_g^Dh _r_D^-2C^α_β(g^D).

    
    Iterating the above controls to the other Ricci-flat ALE orbifolds of the tree of singularities, we obtain controls similar to (<ref>) on all of the N_j and with the control (<ref>), we finally find
    
    (1-Cγ(t_max)) h_C^2,α_β,*(g^D)⩽ C π_𝐎̃(g^D)^⊥P_g^Dh _r_D^-2C^α_β(g^D),

    and therefore the stated result for g=g^D for t_max small enough.
    
    To obtain the same result for a metric g close to g^D, we simply apply Proposition <ref> to ensure that for g-g^D_C^2,α_β,*(g^D) arbitrarily small, P_g is arbitrarily close to P_g^D for the operator norm on C^2,α_β,*(g^D).
    
    The operator P being self adjoint on a compact manifold, its injectivity implies its surjectivity by integration by parts.
    
    
    



 §.§ Resolution modulo obstructions of the Einstein equation


Let us now show that we can always solve the Einstein equation modulo obstructions. Let us recall that being Einstein and in reduced divergence-free with respect to a naïve desingularization g^D is equivalent to being a zero of the operator

    Φ_g^D: g↦(g)-(g)/2g+λ g + δ_g^*δ̃_g^Dg.



    Let (M_o,g_o) be a compact or ALE Einstein orbifold of dimension 4 such that (g_o) = Λ g_o, for Λ∈ℝ, and let (N_j,g_b_j)_j be a tree of singularities desingularizing (M_o,g_o) with pattern D, and 0<β<1. 
    
    Then, there exists τ>0, ϵ>0 only depending on β and the constants of Lemma <ref> and of Proposition <ref> such that for any naïve desingularization g^D:=g^D_t, with t_max< τ, and for all v∈𝐎̃(g^D) satisfying v_C^0_β,*(g^D)<ϵ, there exists a unique solution ĝ_v=ĝ_t,v to the equation 
    
    Φ_g^D(ĝ_v)∈𝐎̃(g^D),

    satisfying the following conditions: 
    
        
  * ĝ_v-g^D_C^2,α_β,*⩽ 2ϵ,
        
  * ĝ_v-(g^D+v) is L^2(g^D)-orthogonal to 𝐎̃(g^D).
    


    Let (M,g^D_t)=(M,g^D) be a naïve desingularization of an orbifold (M_o,g_o) by a tree of singularities (N_j,g_b_j)_j.

    Define the operator Ψ: g^D + C^2,α_β,*(g^D)→𝐎̃(g^D)^⊥∩ r^-2_DC^α_β(g^D) by
    
    Ψ(g):= π_𝐎̃(g^D)^⊥Φ_g^D(g) = π_𝐎̃(g^D)^⊥((g) -(g)/2g + λ g + δ^*_gδ̃_g^D(g)),

where π_𝐎̃(g^D)^⊥ is the L^2(g^D)-orthogonal projection on 𝐎̃(g^D)^⊥. The conclusion of the theorem for v=0 then rewrites: there exists a unique solution g∈ g^D +(𝐎̃(g^D)^⊥∩ C^2,α_β,*) to the equation Ψ(g) = 0.

Let us apply the inverse function theorem, Lemma <ref>, to Ψ. The linearization of the operator Ψ at g^D for any symmetric 2-tensor h satisfying ∫_ M _g^Dh dv_g^D = 0 is
    
    d_g^DΨ(h)=    π_𝐎̃(g^D)^⊥P_g^D(h),

    where P_g^D is explicited in (<ref>). Let us show that this linearization is invertible at g^D and has an inverse which is uniformly bounded as t→ 0. We want to go back to the operator P_g = 1/2∇^*_g∇_g h-_g(h) for which the invertibility has been shown in Proposition <ref>.
    
    First, by the estimate (<ref>), we have
    
    δ̃_g^Dh-δ_g^D h_r_D^-1C^1,α_β,*(g^D)⩽ Ct_max^1/2h_C^2,α_β,*(g^D),

    and therefore the term -2 δ_g^*δ_gh+2δ^*_gδ̃_g^D h of (<ref>) is controlled in the following way 
    
    -2 δ_g^*δ_gh+2δ^*_gδ̃_g^D h _r_D^-2C^α_β(g^D)⩽ Ct_max^1/2h_C^2,α_β,*(g^D).

    
    Next, notice that the Ricci curvature of g^D is almost constant: 
    
    (g^D)-Λ g^D_r^-2_DC^α_β⩽ C t_max^2-β/4,

    because (g^D)-Λ g^D =0 for r_D>2t_max^1/4, and on the rest of the manifold, |(g^D)-Λ g^D| ⩽ C. Therefore, for t_max arbitrarily small, d_g^DΨ is close (as an operator from C^2,α_β,* to r_D^-2C^α_β) up a power of t_max to the operator π_𝐎̃(g^D)^⊥P̂̂̂_g^D, where for a symmetric 2-tensor h,
    
    P̂̂̂_g^D(h):=   1/2(∇^*_g^D∇_g^D h -2_g^D(h) 
       -(δ_g^Dδ_g^Dh)g^D
       - ∇^2_g^D_g^D h+ (Δ_g^D_g^Dh) g^D+ _g^D/4(_g^Dh)g^D),

    where we neglected the difference of the divergence terms by (<ref>), and simplified the terms involving the Ricci curvature by (<ref>).
    
    Let us use the following decomposition of a 2-tensor on (M,g^D_t): for any symmetric 2-tensor h∈ C^2,α_β,*(g^D), there exists a unique decomposition 
    
    h = h_T + δ^*_g^DX

    with 
    
        
  * a symmetric 2-tensor h_T∈ C^2,α_β,*(g^D) satisfying δ̃_g^D h_T=0, 
        
  * a vector field X∈ r_DC^3,α_β,*(g^D)∩𝐊̃_o^⊥.
    
    Indeed, according to Lemma <ref>, for any 2-tensor h∈ C^2,α_β,*(g^D), there exists a unique X∈ r_DC^3,α_β,*(g^D)∩𝐊̃_o^⊥ such that 
    
    δ̃_g^Dδ_g^D^*X = δ̃_g^Dh.

    The decomposition (<ref>) is then h =(h-δ_g^D^*X) + δ_g^D^*X and the sum is L^2(g^D)-orthogonal. 

    
        A simple integration by parts shows that a differentiable symmetric 2-tensor h on M satisfies δ̃_g^D if and only if it is L^2(g^D)-orthogonal to δ^*_g^D(C^∞(TM)∩𝐊̃_o^⊥).
    
    
    Similarly, for a symmetric 2-tensor v∈ r_D^-2C^α_β we have a unique decomposition
    
    v= v_T+δ^*_g^DY

    with a symmetric 2-tensor v_T orthogonal to δ^*_g^D(C^∞(TM)∩𝐊̃_o^⊥) and a vector field X⊥𝐊̃_o.
    
    Now, we have the following properties:
    
        
  * our metric has almost constant Ricci curvature by (<ref>),
        
  * the divergence of the obstructions is arbitrarily small, that is for 𝐨̃∈𝐎̃(g^D), we have 
    δ_g^D𝐨̃_r_D^-1C^1,α_β/𝐨̃_C^2,α_β,*→ 0

        and
        
    δ_g^D𝐨̃_r_D^-3C^0_β/𝐨̃_r_D^-2C^α_β→ 0

        as t_max→ 0 by elliptic regularity .
    
    These imply that for X∈ r_D C^3,α_β,*∩𝐊̃_o^⊥, one has
    
    π_𝐎̃(g^D)^⊥P̂̂̂_g^D (δ_g^D^*X) - δ^*_g^Dδ̃_g^Dδ^*_g^DX_r_D^-2C^α_β⩽γ(t_max)X_r_D C^3,α_β,*,

    for γ : ℝ^+→ℝ^+ with lim_0γ = 0, and that 
    
    Consider the decompositions (<ref>) and (<ref>) where we identify the symmetric 2-tensor of the form δ^*_g^DX with the vector field X since 
    δ^*_g^D:𝐊̃_o^⊥∩ r_DC^3,α_β,*→ C^2,α_β,*

    and
    
    δ^*_g^D:𝐊̃_o^⊥∩ r_D^-1C^1,α_β,*→ r_D^-2C^α_β,*

    are injective. Thanks to (<ref>), the operator π_𝐎̃(g^D)^⊥P̂̂̂_g^D: C^2,α_β,*→ r_D^-2C^α_β decomposes blockwise
    
    π_𝐎̃(g^D)^⊥P̂̂̂_g^D = [     δ̃_g^Dδ^*_g^D                 0;                 0 π_𝐎̃(g^D)^⊥P̂_g^D ] + A_g^D

    where A_g^D is an arbitrarily small (for t_max arbitrarily small) operator between the same spaces, and
    
    P̂_g^D(h):=   1/2(∇^*_g^D∇_g^D h -2_g^D(h) 
       - ∇^2_g^D_g^D h+ (Δ_g^D_g^Dh) g^D+ _g^D/4(_g^Dh)g^D).

    Given the shape of the matrix in (<ref>), it is enough to show that 
    δ̃_g^Dδ^*_g^D : 𝐊̃_o^⊥∩ r_DC^3,α_β,*→𝐊̃_o^⊥∩ r_D^-1C^1,α_β,*

    and 
    
    π_𝐎̃(g^D)^⊥P̂_g^D:𝐎̃(g^D)^⊥∩ C^2,α_β,*→𝐎̃(g^D)^⊥∩ r_D^-2C^α_β

    are invertible with inverse bounded independently of t_max. This is already the case for δ̃_g^Dδ^*_g^D thanks to Lemma <ref>.
    

    
    Let us now focus on π_𝐎̃(g^D)^⊥P̂_g^D. For any symmetric 2-tensor v, we have an L^2(g^D)-orthogonal decomposition into a conformal and traceless part 
    
    v = tr_g^Dv/4g^D + (v - tr_g^Dv/4g^D).

    Now, we have
    
    _g^D(π_𝐎̃(g^D)^⊥P̂_g^D(h)) = _g^DP̂_g^D(h) - _g^D(π_𝐎̃(g^D)P̂_g^D(h))

    and since for an element 𝐨̃ of 𝐎̃(g^D), we have
    
    (_g^D𝐨̃)g^D_r_D^-2C^α_β/𝐨̃_r_D^-2C^α_β→ 0

    and
    
    (_g^D𝐨̃)g^D_C^2,α_β,*/𝐨̃_C^2,α_β,*→ 0

as t_max→ 0, the operator _g^D(π_𝐎̃(g^D)^⊥P̂_g^D) is arbitrarily close to the operator _g^DP̂_g^D (as operators from C^2,α_β,* to r_D^-2C^α_β) which is itself arbitrarily close to Δ_g^D+Λ for t_max arbitrarily small because our metric is almost Einstein according to (<ref>). 
    
    For the traceless part of π_𝐎̃(g^D)^⊥P̂_g^D(h), recall that for any symmetric 2-tensor h, we have the following decomposition
    
    π_𝐎̃(g^D)^⊥P̂_g^D(h) - tr_g^D(π_𝐎̃(g^D)^⊥P̂_g^D(h))g^D/4=    π_𝐎̃(g^D)^⊥(P̂_g^D(h) - tr_g^D(P̂_g^D(h))g^D/4) 
       -  tr_g^D(P̂_g^D(h))π_𝐎̃(g^D)g^D/4
       +  (tr_g^D(π_𝐎̃(g^D)P̂_g^D(h)))g^D/4.

    Using the control (<ref>) once more together with (<ref>) and (<ref>), we neglect the last two terms of (<ref>) and see that the traceless part of π_𝐎̃(g^D)^⊥P̂_g^D(h) is arbitrarily close to the firsts term π_𝐎̃(g^D)^⊥(P̂_g^D(h) - tr_g^D(P̂_g^D(h))g^D/4), that is, to
    
    π_𝐎̃(g^D)^⊥(P_g^D(h)-(∇^2_g^Dtr_g^Dh-Δ_g^D(tr_g^Dh)g^D/4)).

 
Let us therefore decompose h and the operator π_𝐎̃(g^D)P̂_g^D(h) in conformal and traceless parts. Block-wise, we obtain the following operator:
    
    π_𝐎̃(g^D)^⊥P̂_g^D = [                                     Δ_g^D + Λ                                             0; π_𝐎̃(g^D)^⊥(Δ_g^Dtr_g^Dhg^D/4-∇^2_g^Dtr_g^Dh)                              π_𝐎̃(g^D)^⊥P_g^D ] + A'_g^D

    where A'_g^D : C^2,α_β,*→ r_D^-2 C^α_β is an arbitrarily small operator for t_max arbitrarily small. There remains to show that the operator
    
    [                                     Δ_g^D + Λ                                             0; π_𝐎̃(g^D)^⊥(Δ_g^Dtr_g^Dhg^D/4-∇^2_g^Dtr_g^Dh)                              π_𝐎̃(g^D)^⊥P_g^D ]

from C^2,α_β,* to r_D^-2 C^α_β is invertible with a uniformly bounded inverse (independently of t) for t_max small enough. Given the shape of the matrix, it is enough to show it for the two diagonal blocks since the operator at the bottom left is uniformly bounded. The operator Δ_g^D + Λ is invertible with a uniformly bounded inverse thanks to Lichnerowicz eigenvalue estimate, see <cit.> for instance, and the estimate (<ref>). This is also the case for π_𝐎̃(g^D)^⊥P_g^D thanks to Proposition <ref>.
    
    We conclude that the linearization of Ψ at g^D is invertible with a bounded inverse as it is arbitrarily close to an invertible operator. 
    
    
    
        The operator P_g^D itself is not self adjoint because the metric g^D is not Einstein. Indeed, all of the terms are self adjoint except - ∇_g^2 _g h, -(δ_gδ_g h) g, and ⟨_g, h ⟩_g g, and we remark that the adjoint of h↦∇_g^2 _g h is h↦ (δ_gδ_g h) g. However, the term h↦⟨_g, h ⟩_g g whose adjoint is h↦ ( _g h )_g prevents P_g^D to be self adjoint when g^D is not Einstein.
    
    
    To apply the inverse function theorem, Lemma <ref> to the operator Ψ, there remains to control the non-linear terms. Since the variations of the Ricci curvature for a variation h of a metric g, are schematically, 
    
    (g+h) = (g)+(g+h)^-1*(g)+(g+h)^-2*∇^2h+(g+h)^-3*∇ h *∇ h,

    where * refers to diverse multilinear operations and by Remark <ref>, the non-linear terms Q_g^D(h):=Ψ(g^D+h)-Ψ(g^D)-d_g^DΨ(h) satisfy the control
    
    Q_g^D(h)-    Q_g^D(h')_r^-2_DC^α_β(g^D)
    ⩽     C((h_C^α_0(g^D)+h'_C^α_0(g^D))(h-h'^2_C^α_0(g^D))(g^D)_r^-2_DC^α_β(g^D)
       +h-h'_C^α_0(g^D)(∇^2 h_r^-2_DC^α_β(g^D)+∇^2 h'_r^-2_DC^α_β(g^D))
    
            +   (h_C^α_0(g^D)+h'_C^α_0(g^D))∇^2 (h-h')_r^-2_DC^α_β(g^D)
       +2∇ (h-h')_r^-1_DC^α_β(g^D)(∇ h_r^-1_DC^α_β(g^D)+∇ (h')_r^-1_DC^α_β(g^D)))
    ⩽     3C (h_C^2,α_β,*(g^D)+h'_C^2,α_β,*(g^D))h-h'_C^2,α_β,*(g^D).

    
    We moreover have the control 
    
    Ψ(g^D)_r^-2_DC^α_β(g^D)⩽ Ct_max^2-β/4.

    Hence, according to the inverse function theorem, Lemma <ref>,  for t_max small enough, there exists a unique solution ĝ with ĝ-g^D⊥𝐎̃(g^D), to the equation Ψ(ĝ)=0 satisfying moreover g^D-ĝ_C^2,α_β,*(g^D)⩽Ψ(g^D)_r^-2_DC^α_β(g^D)⩽ Ct_max^2-β/4.
    
    Now, we have only solved the equation in the neighborhood of g^D and on the orthogonal of 𝐎̃(g^D). For v∈𝐎̃(g^D) we study the operator g↦Ψ(g+v). The control of the non-linear terms is exactly the same for this operator, and for v arbitrarily small, its linearization at g^D is arbitrarily close to d_g^DΨ which is invertible. As a consequence, there exists ϵ>0 such that for all v_C^0_β(g^D)<ϵ, there exists a unique solution ĝ_v of Ψ(ĝ_v)=0 with ĝ_v-(g^D+v)⊥𝐎̃(g^D). 
    
    By adding v, we however damage the estimate on Ψ(g^D+v) which becomes
    
    Ψ(g^D+v)_r^-2_DC^α_β(g^D)⩽ C(v^2_C^2,α_β(g^D) + t_max^2-β/4).

    We will see later in Section <ref> that without an integrability assumption, we cannot hope for a better estimate.
    



        With the notations of Theorem <ref>, the map v↦ĝ_v is analytic.
   
   
    This is a consequence of the implicit function theorem for analytic functions, see <cit.> for instance. Let us define the map
       
    (v,h) ↦Ψ(ĝ_0+v+h)

       from 𝐎̃(g^D)×(𝐎̃(g^D)^⊥∩ C^2,α_β,*) to 𝐎̃(g^D)^⊥∩ r^-2_DC^α_β(g^D) where we denote
    
    Ψ(g) := π_𝐎̃(g^D)^⊥Φ_g^D(g) = π_𝐎̃(g^D)^⊥(E(g) + δ^*_gδ̃_g^D(g)),

    like in the proof of Theorem <ref> and where ĝ_0 is the solution of Theorem <ref> for v=0.
       
    The map (v,h) ↦Ψ(ĝ_0+v+h) is analytic since E and g↦δ_g^* are, and since δ̃_g^D and π_𝐎̃(g^D)^⊥ are linear. We know that ĝ_0-g^D⩽ C t_max^2-β/4, and that d_g^DΨ is invertible thanks to the proof of Theorem <ref>, hence d_ĝ_0Ψ is also invertible. 
    
    The implicit function theorem for analytic functions, see <cit.>, then implies, by the uniqueness of the solution ĝ_v of Theorem <ref> that for v small, the map v↦ĝ_v-(g^D+v) is analytic and that v↦ĝ_v is analytic too.
   
   
   
    The previous analysis of Theorem <ref> extends to partial desingularizations. 
    
    More precisely, let S_o be a subset of the singularities of M_o and for each j, S_j a subset of the singularities of N_j and denote S=(S_o,(S_j)_j) and M_S:= M_o#_jN_j (# means gluing the ALE spaces at orbifold singularities) where the gluings are given by some gluing pattern D. 
    
    For t_max,S and v small enough, the metric g^D_S=g^D_S,t iteratively defined just like in Definition <ref> can be perturbed to a unique solution ĝ_S,v=ĝ_S,t,v to the equation 
    
    Φ_g^D(ĝ_S,v)∈𝐎̃(g^D_S),

    satisfying the following conditions: 
    
        
  * ĝ_S,v-g^D_S_C^2,α_β,*(g^D_S)⩽ 2ϵ, where C^2,α_β,*(g^D_S) is the partial desingularization norm of Remark <ref>,
        
  * ĝ_S,v-g^D_S,v is L^2(g^D_S)-orthogonal.
    
   
   
 Thanks to Corollary <ref>, we have the following result.

   
   Let D_0,v_0>0, l∈ℕ, and β= β(v_0,D_0)>0 obtained in Corollary <ref>. Then, for all ϵ>0, there exists δ = δ(ϵ,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 above by D_0,
		
  * the Ricci curvature is bounded ||≤ 3.
	
	and such that for an Einstein orbifold (M_o,g_o), we have
	
    d_GH((M,g^ℰ),(M_o,g_o))⩽δ,

	then, there exists a naïve desingularization (M,g^D_t,v) of (M_o,g_o) by a tree of singularities, and a diffeomorphism ψ: M→ M such that 
    ψ^*g^ℰ = g_t,v,
 where g_t,v is the perturbation of g^D_t+v of Theorem <ref>. 
   


 §.§ Premoduli space of Einstein metrics around a singular one


Let us now explain how the Einstein modulo obstructions metrics of Theorem <ref> are analogous to the metrics in the set W of Theorem <ref>.

Let M a differentiable 4-manifold and consider an orbifold (M_o,g_o)∈∂_o𝐄(M)⊂𝐄(M)_GH\𝐄(M). According to Corollary <ref>, the Einstein metrics which are sufficiently close to (M_o,g_o) in the Gromov-Hausdorff are results of the gluing-perturbation procedure of Theorem <ref>.
    
    By analogy with the definition of <cit.> of the premoduli space of Einstein metrics around a smooth one, we define de directional premoduli space in the neighborhood of a singular metric in the following way.
     We define 𝔼^D_g_o(M), the premoduli space of Einstein metrics on M in the neighborhood of (M_o,g_o) and in the direction D, as the set of metrics g on M for which there exists t such that we have g-g^D_t_C^2,α_β,*(g^D_t)<ϵ for ϵ>0 the constant of Theorem <ref> and v∈𝐎̃(g^D_t) for which we have:
    
        
  * E(g) = 0, (g)= (g_o),
        
  * g-(g^D_t+v) ⊥_g^D_t𝐎̃(g^D_t),
        
  * δ̃_g^D_tg=0.
    
    
    
    
        The above premoduli space is directional in the sense that it does not cover all of the Gromov-Hausdorff desingularizations of (M_o,g_o), but only the ones whose tree of singularities is D. Given an arbitrary sequence (g_i)_i of smooth Einstein metrics on M d_GH-converging to (M_o,g_o), then, up to considering a subsequence, all of the metrics g_i are in a single directional premoduli space 𝔼^D_g_o(M).
    
    The result of <cit.> together with Theorem <ref> yield the following statement.
    
        The directional premoduli space 𝔼^D_g_o(M) is the zero-set of E on the set denoted Ŵ of metrics ĝ_t,v in the C^2,α_β,*(g^D_t)-neighborhoods of the metrics g^D_t of Theorem <ref>.
    
    Since E is analytic, Koiso's proof of Theorem <ref> reduces to proving that W is an analytic submanifold by implicit function theorem. Therefore the question of the regularity of the d_GH-completion of 𝐄(M^4), 𝐄(M^4)∪∂_o𝐄(M^4), reduces to the question of the regularity of the set Ŵ of the Einstein modulo obstructions metrics of Theorem <ref>.




§ OBSTRUCTIONS TO THE GROMOV-HAUSDORFF DESINGULARIZATION


Let us now come to the main application of this article, which is the obstruction to the desingularization of Einstein orbifolds. 



 §.§ Analysis of integrable Ricci-flat ALE spaces


In order to obtain an obstruction result we will need to assume that the Ricci-flat ALE metrics in the trees of singularities have integrable Ricci-flat deformations.



  §.§.§ Integrable Ricci-flat ALE


Since Corollary <ref> does not control the convergence speed towards the limit orbifold or the Ricci-flat ALE spaces, like in Theorem <ref>, we have to fix a gauge v∈𝐎̃(g^D) on the approximate kernel of the operator P̅.

Not to damage our controls, we cannot simply use g^D + v as an approximate metric. We need to find a better approximation to extend Proposition <ref> to the case when v≠ 0. It turns out that this will only be possible if we assume that the Ricci-flat ALE metrics are integrable.



    We will say that a Ricci-flat metric ALE g_b is integrable if for all v∈𝐎(g_b) small enough, there exists a Ricci-flat metric ALE g̅_b,v satisfying g̅_b,v-(g_b+v)⊥𝐎(g_b) and g_b,v-g_b_L^2(g_b)⩽ 2 v_L^2, and such that δ_g_bg̅_b,v=0.





All of the known examples of Ricci-flat ALE spaces are integrable since they are quotients of hyperkähler spaces. Moreover, any infinitesimal L^2-deformation of ALE Ricci-flat orbifolds is automatically divergence-free and trace-free by the proof of Lemma <ref>.





  §.§.§ Weighted Hölder spaces and asymptotics of Ricci-flat ALE spaces


Let us introduce yet another function space to control the asymptotics of our ALE metrics. This will be crucial to deduce obstructions in the following sections.


    Let (N,g_b) be an ALE orbifold, and let h be a symmetric 2-tensor on N, and assume that h = H^4 + 𝒪(r_b^-4-β) for β>0 and H^4 a homogeneous harmonic symmetric 2-tensor with |H^4|∼ r_e^-4. We define its C^2,α_β,**-norm by
    
    h_C^2,α_β,**:= sup r_e^4|H^4|_g_e + (1+r_b)^4(h-χ(ϵ r_b) H^4)_C^2,α_β,*.




    Let (N,g_b) be an ALE orbifold, and let X be a vector field on N, and assume that X = Y^3 + 𝒪(r_b^-3-β) for β>0 and Y^3 a homogeneous element of the kernel of δ_eδ_e^* with |Y^3|∼ r_e^-3. We define its r_bC^3,α_β,**-norm by
    
    X_r_bC^3,α_β,**:= sup r_e^3|Y^3|_g_e + (1+r_b)^4(X-χ(ϵ r_b) Y^3)_r_bC^3,α_β,*.



These norms are motivated by the following Lemmata.

    Let (N,g_b) be an ALE orbifold. Then, there exists C>0 such that we have, for any h⊥𝐎(g_b),
    
    C^-1(1+r_b)^4P̅_g_bh_r_b^-2C^α_β⩽h_C^2,α_β,**⩽ C (1+r_b)^4P̅_g_bh_r_b^-2C^α_β.



    By the theory of elliptic operators in weighted Hölder spaces (see for instance <cit.>), the operator P̅_g_b: (1+r_b)^-4C^2,α_-β→ (1+r_b)^-4r_b^-2C^α_-β is Fredholm with kernel 𝐎(g_b) and cokernel 𝐎(g_b) because there is no other exceptional value than 0 between 2 and -4. This implies that P̅_g_b: (1+r_b)^-4C^2,α_-β∩𝐎(g_b)^⊥→ (1+r_b)^-4r_b^-2C^α_-β∩𝐎(g_b)^⊥ is invertible with a bounded inverse. 
    
    Moreover, we have P̅_g_b^-1((1+r_b)^-4r_b^-2C^α_β) = C^2,α_β,** since -4 is the first negative exceptional value for P̅, and the stated inequality comes from the fact that the inverse is bounded.

Similarly for vector fields, we have the following result.


    Let (N,g_b) be an ALE orbifold. Then, there exists C>0 such that for any vector field X on N, we have
    
    X_r_bC^3,α_β,**⩽ C (1+r_b)^4δ_g_bδ_g_b^*X_r_b^-1C^1,α_β.



In particular the analysis of Theorem <ref> and Proposition <ref> extends to the case where (M_o,g_o) is a Ricci-flat ALE orbifold and where the norm C^2,α_β,*(g_o) is replaced by C^2,α_β,**(g_o) thanks to Lemma <ref> and r_oC^3,α_β,*(g_o) is replaced by r_oC^3,α_β,**(g_o) thanks to Lemma <ref>. Indeed, all of the controls are local around the singular points or coming from an estimate on the inverse on the rest of the orbifold exactly like (<ref>) and (<ref>). For the operator P̅, this yields the following control on the asymptotic terms on the ALE end.

 Let (N,g_b) be a Ricci-flat ALE orbifold, and denote (N^B,g^B_t) a naïve desingularization of (N,g_b) by a tree of Ricci-flat ALE orbifolds glued according to a pattern B with relative scales t.
    
    For v and t small enough depending on the constants of Lemma <ref>, let g̅_t,v be the unique metric (according to Theorem <ref>) satisfying for ϵ>0 small enough:
    
        
  * g^B_t- g_t,v_C^2,α_β,*(g^B_t)<2ϵ,
        
  * (g^B_t+v)- g_t,v is L^2(g^B_t)-orthogonal to 𝐎̃(g^B_t), and
        
  * π_𝐎̃(g^B_t)^⊥Φ_g^B_t(g_t,v)= 0.
    
    Then, for any 0<β<1, we have 
    g_t,v = g_e +H̅^4_t,v + 𝒪(r_B^-4-β),

    for |H̅^4_t,v|∼ r_B^-4 and H̅^4_t,v→H̅^4, the asymptotic terms of (N,g_b) as (t,v)→(0,0).

Similarly, using Lemma <ref>, we can put our ALE metrics in gauge with respect to each other. The nonlinear terms are taken care of like in Proposition <ref> by noting that the weight of our norm, (1+r_b)^4r_b^-1, is this time also larger than 1 at infinity.

Let 0<β<1, let (N,g_b) be a Ricci-flat ALE orbifold, and denote (N^B,g^B_t) a naïve desingularization of (N,g_b) by a tree of Ricci-flat ALE orbifolds glued according to a pattern B with relative scales t.

Then, there exist ϵ_B,τ_B,C_B>0 which only depend on the metrics g_b and the elements of B such that for t_max⩽τ_B and for any metric g satisfying g-g^B_t_C^2,α_β,**(g^B_t)⩽ϵ_B, there exists a unique vector field X on M for which, 
    
    δ̃_g^B_t(exp_X^*g)=0,

    where exp_X is the diffeomorphism exp_X: x∈ M ↦exp_x^g^B_t(X(x)).
    
    We moreover have, X_r_BC^3,α_β,**(g^B_t)⩽ C_B (1+r_B)^4δ̃_g^B_t(g-g^B_t)_r^-1_BC^1,α_β(g^B_t), and therefore, there exists η: ℝ^+→ℝ^+ with lim_0η=0 such that we have 
    
    exp_X^*g-g_C^2,α_β,**(g^B_t)⩽η(g-g^B_t_C^2,α_β,**(g^B_t)).





 §.§ Approximate Einstein modulo obstructions metric

Let (M_o,g_o) be an Einstein orbifold and let p be one of its singular points whose singularity model is ℝ^4Γ.

Let us now construct a good approximation of the metric ĝ_t,v of Theorem <ref>, which we will denote g^A_t,v. This is a crucial step to understand  and approximate the obstructions coming from the Ricci-flat ALE spaces appearing at a singular point p∈ M_o.

In all of this section, we will assume the following properties:

    
  * (M,g^D_t) is a naïve desingularization of (M_o,g_o),
    
  * at the singular point p∈ M_o, there is only one Ricci-flat ALE manifold (N,g_b) glued, and therefore no tree of singularity,
    
  * (N,g_b) is an integrable Ricci-flat ALE manifold.


Consider S the complement of {p} among the singular points of M_o, let (M_S,g^D_S = g^D_S,t_S) be a naïve partial desingularization of (M_o,g_o) which only leaves the point p singular and let (M_S,ĝ_S = ĝ_S,t_S,v_S) be the perturbation of (M_S,g^D_S+v_S) orthogonally to 𝐎̃(g^D_S) of Theorem <ref> satisfying 

    Φ_g^D_S(ĝ_S) = 𝐨_S∈𝐎̃(g^D_S).

At p, the metric ĝ_S has the following development in local coordinates where it is in divergence-free gauge with respect to g_e,

    ĝ_S = g_e + H_S + 𝒪(r_o^3)

and we know that Φ_g^D_S(ĝ_S) = 𝐨_S = 𝐎_S + 𝒪(r_e^2) with |𝐎_S|_g_e∼ r_e^0, _g_e𝐎_S = 0 and δ_g_e𝐎_S = 0. We therefore have

    0=    Φ_g^D_S(ĝ_S)-𝐎_S = λ g_e + P̅_e(H_S)-𝐎_S+ 𝒪(r_e^2),

where λ = n-2/2n(ĝ_S).
Consequently P̅_eĤ_S + λ g_e = 𝐎_S, where 𝐎_S is the limit of 𝐨_S at p.

For some small v_p∈𝐎(g_b), we will glue a Ricci-flat deformation g̅_b,v_p of g_b at a scale t_p to ĝ_S. To obtain a better estimate, we will extend the quadratic terms of ĝ_S in order to minimize the error in the gluing.


	    Let (N,g_b) a Ricci-flat ALE orbifold asymptotic to ℝ^4Γ, Ĥ_S a quadratic symmetric 2-tensor on ℝ^4Γ, λ∈ℝ and 𝐎_S a constant symmetric 2-tensor on ℝ^4Γ such that we have: 
		
    P̅_eĤ_S+λ g_e = 𝐎_S.

		Then, there exists a 2-tensor ĥ_S, and for (𝐨_j)_j an L^2-orthonormal basis of 𝐎(g_b), real numbers λ̂_j such that (ĥ_S,λ̂_j) is a solution of 
		
    {[      P̅_bĥ_S+λ g_b = χ𝐎_S + ∑_j λ̂_j 𝐨_j,;   |ĥ_S - χĤ_S|_g_b =           𝒪(r_b^-2), ].

	    where χ is a cut-off function supported in a neighborhood of the infinity of (N,g_b) where the ALE coordinates are defined, and where 
	    
    λ̂_j:= -∫_𝕊^3Γ(3⟨Ĥ_S, O^4_j ⟩_g_e + O_j^4(∇_e_eĤ_S,∂_r_e) ) dv_𝕊^3Γ + ∫_N χ⟨𝐎_S, 𝐨_j ⟩_g_b dv_b.

		The set of solutions to the above equation (<ref>) is (ĥ_S + 𝐎(g_b),λ̂_j).
	
	
    We have P̅_b(χĤ_S)+λ g_b - χ𝐎_S∈ r_b^-2C^α_β(g_b) for 0<β<1 because in a neighborhood of infinity, g_b-g_e = 𝒪(r_b^-4) together with the derivatives. Lemma <ref> also holds by replacing the operator P_b by the operator P̅_b as a consequence of Theorem <ref> in the case where (M_o,g_o) = (N,g_b) is a Ricci-flat ALE manifold. Consequently, there exists ĥ'∈ C^2,α_β(g_b) such that P̅_b(χĤ_S + ĥ') + λ g_b = χ𝐎_S +∑_jλ̂_j 𝐨_j with
    
    λ̂_j = -∫_𝕊^3Γ(3⟨Ĥ_S, O^4_j ⟩_g_e +1/2 O_j^4(∇ trĤ_S,∂_r_e) )dv_𝕊^3Γ + ∫_N χ⟨𝐎_S, 𝐨_j ⟩_g_b dv_b,

    where O^4_j∼ r_b^-4 is the first term of the development of 𝐨_j at infinity. Indeed, by integration by parts, and using the fact that _b 𝐨_j = ⟨ g_b,𝐨_j ⟩_g_b = 0 and δ_b𝐨_j=0, we get,

    ⟨P̅_b(χĤ_S), 𝐨_j ⟩_L^2 =    1/2lim_ρ→∞∫_r_b⩽ρ⟨∇^*_b∇_b (χĤ_S)- ∇_b^2 (_b χĤ_S) , 𝐨_j ⟩ dv_g_b
    
    		    =    1/2lim_ρ→∞∫_r_b=ρ(⟨χĤ_S , ∇_n 𝐨_j ⟩-⟨∇_n (χĤ_S) , 𝐨_j ⟩
       + 𝐨_j(∇_b(_bχĤ_S) , ∂_r_e ))dS_ρ
    
    		    =     -∫_𝕊^3Γ(3⟨Ĥ_S, O^4_j ⟩_g_e + 1/2O_j^4(∇_e _eĤ_S,∂_r_e) )dv_𝕊^3Γ.

    Now, the integral ∫_N χ⟨𝐎_S, 𝐨_j ⟩_g_b dv_b converges even if ⟨𝐎_S, 𝐨_j ⟩_g_b = 𝒪(r_b^-4). Indeed, in ALE coordinates, r_b^4𝐨_j = ϕ_ijdx^idx^j + 𝒪(r_b^-1), where the ϕ_ij: 𝕊^3→ℝ are nonconstant eigenfunctions and therefore have zero mean values, hence, ∫_{r_b=ρ}⟨𝐎_S, 𝐨_j ⟩_g_bdS_ρ = 𝒪(ρ^-2) and the integral converges.  The values of λ̂_j from (<ref>) therefore ensure that we have 
    
    P̅_b(χĤ_S)+λ g_b - χ𝐎_S -∑_jλ̂_j 𝐨_j ⊥_g_b𝐎(g_b).


Choosing t_S,t_p>0, v_S∈𝐎̃(g^D_S,t_S) and v_p∈𝐎(g_b), by gluing  metric g̅_b,v_p at scale t_p to ĝ_S,v_S,t_S, we reach all of the gauges of Theorem <ref>, that is we attain some element of (g^D_t+v)+𝐎̃(g^D_t)^⊥ for any small v and t.



We define 𝐎̃_t_p(g̅_b,v_p) as the cut-off of the elements of 𝐎(g̅_b,v_p) as in Definition <ref> at scale t_p. We denote ĥ_S(t,v) the 2-tensor satisfying ĥ_S(t,v)⊥_g̅_b,v_p𝐎̃_t_p(g̅_b,v_p) obtained in Proposition <ref> with the Ricci-flat ALE metric g̅_b,v_p and Ĥ_S the quadratic terms of ĝ_S=ĝ_S,t_S,v_S.


    Let v∈𝐎̃(g^D_t) for (M,g^D_t) a naïve desingularization of (M_o,g_o).

    The Riemannian manifold (M,g^A =g^A_t,v) is obtained by naïve gluing (Definition <ref>) of (N,g̅_b,v_p + t_pĥ_S(t,v)) to (M_S,ĝ_S,t_S,v_S) at scale t_p>0.
    
    The numbers t_p,t_S>0, v_p∈𝐎(g_b) and v_S∈𝐎̃(g^D_S) are chosen in order to have g^A_t,v-(g^D_t+v)⊥𝐎̃(g^D_t).




 §.§ Better approximation and obstructions


The obstruction will come from the better controls of g^A_t,v and the following proposition.


    Let 0<α<1, and (M,g^D_t) be a naïve desingularization, and assume that for t_max<τ, the metric (M,g = g_t,v) obtained by Theorem <ref> is an Einstein metric (without obstructions). Then, there exists ϵ>0 and C>0 only depending on g_o and the g_b_j, such that, denoting Ψ (g^A_t,v ):=π_𝐎̃(g^D)^⊥Φ_g^D_t(g^A_t,v) and 𝐨^A_t,v := π_𝐎̃_t_p(g̅_t_p,v_p)Φ_g^D_t(g^A_t,v ), we have
    
    ĝ_t,v -g^A_t,v_C_β,*^2,α(g^D_t)⩽ CΨ (g^A_t,v ) _r^-2_DC_β^α(g^D_t),

    and
    
    𝐨^A_t,v_L^2(g^D)⩽ (   Ψ (g^A_t,v ) _r^-2_DC_β^α(g^D)+ t_p^1/2+β/4) Ψ (g^A_t,v ) _r^-2_DC_β^α(g^D).



    Let us denote g^A := g^A_t,v, ĝ:=ĝ_t,v and h^A:= ĝ-g^A as well as g̅_b = g̅_b,v_p for simplicity. The inequality (<ref>) is a direct consequence of the proof of Theorem <ref> because the proof uses an inverse function theorem. Indeed, at the linear level, by Lemma <ref>, we have
    
    ĝ -g^A _C_β,*^2,α(g^D)⩽ Cπ_𝐎̃(g^D)^⊥P̅_g^D(ĝ-g^A) _r^-2_DC_β^α(g^D).

    Since Ψ(ĝ) = 0, the controls of the nonlinear terms of Ψ imply that
    
    Ψ(g^A)_r^-2_DC_β^α(g^D)⩾    π_𝐎̃(g^D)^⊥P̅_g^D(ĝ-g^A) _r^-2_DC_β^α(g^D)
       - C(ĝ-g^D _C_β,*^2,α(g^D) +g^A-g^D _C_β,*^2,α(g^D))ĝ -g^A _C_β,*^2,α(g^D).

    We moreover know that g^A-g^D _C_β,*^2,α(g^D)⩽2ϵ  and ĝ-g^D _C_β,*^2,α(g^D) = 𝒪 (t_max^2-β/4) thanks to the proof of Theorem <ref> and by assumption. Therefore, choosing ϵ and t_max small enough and putting the estimates (<ref>) and (<ref>) together yields the expected control (<ref>).
    
    Let us focus on the zone N^t where the elements of 𝐎̃_t_p(g̅_b,v_p) are supported. For 𝐨̃∈𝐎̃_t_p(g̅_b,v_p), we have
    
    |⟨P_g^Ah^A , 𝐨̃⟩_L^2(g^D)|   ⩽|⟨(P_g^A - P_t_pg̅_b)h^A , 𝐨̃⟩_L^2(t_pg̅_b)|+|⟨P_t_pg̅_bh^A ,𝐨̃⟩_L^2(t_pg̅_b)|
       = |⟨(P_g^A - P_t_pg̅_b)h^A , 𝐨̃⟩_L^2(t_pg̅_b)|+|⟨ t_p^-1h^A ,P_g̅_b𝐨̃⟩_L^2(g̅_b)|,

  since P̅_g̅_b is self dual and since the L^2-product of 2-tensor is invariant by rescaling in dimension 4 as well as the rescaling behavior of P̅. Now, we have the following estimates on N^t = {r_b<2t_p^-1/4}⊂{r_D<2t_p^1/4}, for k∈{0,1,2}, there exists C>0 such that:
  
      
  * r_D^k|∇^k_t_pg̅_b(g^A-t_pg̅_b)|_t_pg̅_b⩽ C r_D^2 since g^A=t_pg̅_b+t_p^2ĥ_S on N^16t by construction and because of the controls of the cut-off functions,
      
  * r_b^k|∇^k_g̅_b𝐨̃|⩽ C r_b^-4 as well as P̅_g̅_b𝐨̃= 0 on N^16t⊂{r_D<t_p^1/4}, and
      
  * r_D^k|∇^k_t_pg̅_b h_A|_t_pg̅_b⩽ C(t_p^-1/2r_D)^-βΨ(g^A)_r^-2_DC_β^α(g^D) by (<ref>).
  
  We can therefore control the terms of (<ref>) in the following way:
  
    |⟨P_g^Ah^A , 𝐨̃⟩_L^2(g^D)|   ⩽ CΨ(g^A)_r^-2_DC_β^α(g^D)∫_t_p^1/2^t_p^1/4(r/√(t_p))^-β(r/√(t_p))^-4 r^3dr
       + Ψ(g^A)_r^-2_DC_β^α(g^D)∫_t_p^-1/4^2t_p^-1/4t_p^-1 r^-β r^-6 r^3dr
       ⩽ CΨ(g^A)_r^-2_DC_β^α(g^D)( t_p^2+β/4+ t_p^1/2+β/4)
       ⩽ C Ψ(g^A)_r^-2_DC_β^α(g^D) t_p^1/2+β/4.

    Let 𝐨^A= 𝐨^A_t,v:=π_𝐎̃_t_p(g̅_b,v_p)Φ_g^D(g^A), since Φ_g^D(ĝ)=0, and d_g^AΦ_g^D = P_g^A. We have
    
    -𝐨^A_L^2(g^D)^2 = ⟨Φ_g^D(g )-𝐨^A,𝐨^A⟩= ⟨Ψ (g^A ) + P_g^A h^A +Q_g^A (h^A ), 𝐨^A⟩,

    where the non-linear terms Q_g^A (h^A )= Φ_g^D(ĝ )-Φ_g^D(g^A ) - P_g^A (h^A ) satisfy
    
    Q_g^A (h^A )_r_D^-2C^0(M)⩽ C h^A _C_β,*^0^2 ⩽ C Ψ (g^A ) _r^-2_DC_β^0(g^D)^2,

    and by definition of the weighted norms, since r_D^-2C^α_β(g^D)⊂ L^2(g^D), we therefore have for 𝐨∈𝐎̃(g^D), 
    
    |⟨𝐨, Q_g^A (h^A ) ⟩_L^2(g^D)|⩽ C𝐨_L^2Ψ (g^A )_r^-2_DC_β^0(g^D)^2.

    Hence, since Ψ (g^A )⊥_g^D𝐨^A by definition of Ψ, we have by (<ref>), and thanks to (<ref>) and (<ref>) we therefore have 
    
    𝐨^A_L^2(g^D)⩽  C(   Ψ (g^A ) _r^-2_DC_β^α(g^D)
        + t_p^1/2+β/4) Ψ (g^A ) _r^-2_DC_β^α(g^D).
 



    The inequality (<ref>) means that if we are able to construct a metric g^A such that Ψ(g^A) is small, then g^A is a good approximation of ĝ, the only zero of Ψ. This allows us to approximate the metrics ĝ, and therefore degenerating Einstein metrics with an arbitrarily good precision.
    
    The inequality (<ref>) is an obstruction result. Indeed, if we construct a good approximation g^A, for which Ψ(g^A ) is small, but without having 𝐨^A small, then ĝ cannot be Einstein, and the metric g^A (and therefore g^D) cannot be perturbed to an Einstein metric orthogonally to 𝐎̃(g^D).


Let us now control the above quantities of (<ref>) for g^A_t,v.


    For (t_S,t_p)>0 small enough and k∈ℕ, denoting (𝐨̃_j)_j an orthonormal basis of 𝐎̃_t_p(g̅_t_p,v_p) there exist real numbers (λ̂_j = λ̂_j(t,v))_j and C_k>0 such that we have
    
    π_𝐎̃(g^D)^⊥Φ_g^D(g^A_t,v) _r^-2_DC^k_β(g^D)⩽ C_k t_p^3-β/4,

    and, for any i_0, we have
    
    ⟨Φ_g^D(g^A_t,v)-t_p ∑_jλ̂_j 𝐨̃_i, 𝐨̃_i_0⟩_L^2(g^D)⩽ C_0t_p^5/4.



    If we did not use the partial Einstein modulo obstructions desingularization ĝ_S, we would only have a control with powers of t_max instead of t_p. In particular, we would not be able to later prove that an obstruction holds at all of the singular points, but just at one of them.


    Let us again use the following notations along this proof: g^A := g^A_t,v, g̅_b = g̅_b,v_p, ĝ_S=ĝ_S,t_S,v_S for simplicity.

    On M_S^16t, we have by assumption
    
    Φ_g^D_S(ĝ_S) = 𝐨_S∈𝐎̃(g^D_S),

    and on N^16t, we have
    
    Φ_t_pg̅_b(t_p(g̅_b,v_p + t_pĥ_S)) = χ𝐎_S +t_p∑_jλ̂_j𝐨_j + 𝒪(r_D^2).

    Now, since at the point p, the development of ĝ_S in local coordinates where the metric is divergence-free gauge with respect to g_e is
    
    ĝ_S = g_e + Ĥ_S + 𝒪(r_o^3),
 and since t_p(g̅_b,v_p+t_pĥ_S) has the following development for t_p^-1/4⩽ r_b⩽ 2t_p^-1/4
    
    t_p(g̅_b,v_p+t_pĥ_S) = g_e + Ĥ_S + 𝒪(t_p^2r_b^-2 +t_pr_b^-4),

    with corresponding controls for the derivatives up to order 2. On the annulus of radii r_D=t_p^1/4 and r_D=2t_p^1/4, recalling that r_D = r_o = √(t_p)r_b, for k∈ℕ, we have 
    
    r_D^k|∇_g^D^k(ĝ_S - (t_pg̅_b,v_p+t_pĥ_S)) |_g^D   = 𝒪(r_D^3 + t_p^2(t_p^-1/2r_D)^-2+ t_p(t_p^-1/2r_D)^-4)
       = 𝒪(t_p^3/4 + t_p^3/2 + t_p)
       =𝒪(t_p^3/4)

    thanks to (<ref>) and (<ref>). By definition of the gluing, this yields
    
    r_D^2+k|∇^k_g^D_t(π_𝐎̃(g^D_t)Φ_g^D_t(g^A_t,v))| = 𝒪(t_p^3/4),

    According to (<ref>), (<ref>) and (<ref>), we have the estimate (<ref>). 
    
    Finally, we have the control (<ref>) thanks to (<ref>), (<ref>) and (<ref>):
    
    t_p|λ̂_j(t,v)|⩽ C (t_p^3-β/4+t_p^2+β/4)t_p^3-β/4⩽ C t_p^5/4





    Let λ̂_j(t,v) be the real numbers of Proposition <ref> obtained by extending Ĥ_S, the quadratic terms of ĝ_S = ĝ_S,t_S,v_S on (N,g̅_b,v_p), and let λ̂_j be the ones obtained by extending the divergence-free quadratic terms of g_o on (N,g_b). Then as t,v→ 0, we have
    
    |λ̂_j(t,v)-λ̂_j| 0.



    We have the expression
    
    λ̂_j(t,v) =     -∫_𝕊^3Γ(3⟨Ĥ_S(t_S,v_S), O^4_j(v_p) ⟩_g_e + O_j^4(v_p)(∇_e_eĤ_S(t_S,v_S),∂_r_e) ) dv_𝕊^3Γ
       + ∫_N χ⟨𝐎_S(t_S,v_S), 𝐨_j(v_p) ⟩ dv_b

     thanks to Proposition <ref>. Since we want to show that it converges, as (t,v)→ 0, to 
    
    λ̂_j:= -∫_𝕊^3Γ(3⟨Ĥ_S, O^4_j ⟩_g_e +1/2 O_j^4(∇_e _eĤ_S,∂_r_e) )dv_𝕊^3Γ.

    We therefore just have to show that as (t,v)→ 0, we have
    
        
  * Ĥ_S(t_S,v_S)→Ĥ_S in r_e^2C^1_0(g_e),
        
  * |𝐎_S(t_S,v_S)|_g_e→ 0, and
        
  * O_j^4(v_p)→ O_j^4 in r_e^-4C^0_0(g_e).
    
    
    Thanks to Theorem <ref>, we know that ĝ_S,t_S,v_S converges smoothly to g_o on compacts of M_o\ S as (t_S,v_S)→ (0,0), in particular, we have smooth convergence on small neighborhoods of p. Therefore, the quadratic terms of the expansion of ĝ_S,t_S,v_S converge to those of g_o as (t_S,v_S)→ (0,0). Consequently, the obstruction 𝐎_S(t_S,v_S) = P̅_eĤ_S(t_S,v_S)+λ(t_S,v_S) g_e also converges to P̅_eĤ_S+λ g_e=0.
    
    Similarly, thanks to Corollary <ref>, the asymptotic term of g̅_b,v_p converge to the asymptotic terms of g_b as v_p→ 0. 
    
    
    
        We needed to consider partial desingularizations to obtain these controls. 
    

    Thanks to the computations of <cit.>, it turns out that generically (when the self-dual part of the curvature at p, 𝐑_+ is of rank 2 and Λ≠ 0), the difference λ̂_j(t,v)-λ̂_j= ⟨ v,𝐨_j⟩(𝐨_j(𝐑_+)-Λ𝐨_j) + 𝒪(v^2_C^2,α_β,*) does not vanish.



    If there were non integrable infinitesimal deformations of g_b, we a priori could not expect to prove an obstruction result by the above techniques. Indeed, the metric has an expansion g̅_b,v_p = g_b+v_p+w + 𝒪(|v_p|^3), where w satisfies:
    
    Q̅^(2)_g_b(v_p,v_p)+P̅_g_b(w) = π_𝐎(g_b)Q̅^(2)_g_b(v_p,v_p) ∈𝐎(g_b),
 and we potentially have π_𝐎(g_b)Q̅^(2)_g_b(v_p,v_p) = 𝒪(|v_p|^2) if v_p is not integrable. By considering the metric 1/t_pg^A_t,v, we have the following development of Φ_g_b on N^16t,
    
    Φ_g_b(1/t_pg^A_t,v) = Φ_g_b(g_b,v_p) +t_pP̅_b(ĥ_S) +  t_pQ^(2)_g_b(v_p,ĥ_S) + 𝒪(t_p^2).

    Up to the order t_p^2 there are three different sources of obstructions: 
    
        
  * the projection of Φ_g_b(g_b,v_p)=𝒪(|v_p|^2) on 𝐎(g_b),
        
  * the projection of t_pP_b(ĥ_S) = 𝒪(t_p) on 𝐎(g_b), and
        
  * the projection of t_pQ^(2)_g_b(v_p,ĥ_S)=𝒪(t_p|v_p|) on 𝐎(g_b).
    
    Hence, we can only "see" the obstructions which are 𝒪(t_p) if (g̅_b,v_p)=0, or if |v_p|^2≪ t_p.


By adapting the end of the proof of <cit.>, we get the following useful result.

    Let H_2 be a quadratic symmetric 2-tensor satisfying d_e(H_2)=Λ g_e, and let V∼ r_e^3 be a homogeneous vector field which satisfies δ_eδ^*_eV = -δ_e H_2, and define Ĥ_S:= H_2+δ^*_eV which satisfies δ_e Ĥ_S = 0. Then, we have 
    λ̂_j = λ_j,
 where
    
    λ̂_j:= -∫_𝕊^3Γ(3⟨Ĥ_S, O^4_j ⟩_g_e +1/2 O_j^4(∇_e _eĤ_S,∂_r_e) )dv_𝕊^3Γ,

    and
    
    λ_j:= -∫_𝕊^3Γ(3⟨ H_2, O^4_j ⟩_g_e + O_j^4(B_eH_2,∂_r_e)) dv_𝕊^3Γ






 §.§ Obstruction to the Gromov-Hausdorff desingularization


We can finally conclude that there are obstructions to the desingularization of some Einstein orbifolds.

  Let (M_o,g_o) be an Einstein orbifold, and (M_i,g_i)_i a sequence of Einstein manifolds converging to (M_o,g_o) in the Gromov-Hausdorff sense and assume that, at a singular point p∈ M_o, the possible non-flat blow-up limits are integrable Ricci-flat ALE manifolds (which implies that there are no tree of singularities forming).

    Then, if we denote H_2 the quadratic terms of the development of g_o in geodesic coordinates at p, and (O_j^4)_j the r_b^-4-terms of a basis of 𝐎(g_b), we have:
    
    ∫_𝕊^3(3⟨ H_2,O_j^4 ⟩+O_j^4(B_eH_2,∂_r_e))dv_𝕊^3 = 0.

    


Let (M_o,g_o) be an Einstein orbifold, and assume that there exists a sequence of Einstein metrics (M_i,g_i)_i converging to (M_o,g_o) in the Gromov-Hausdorff sense whose blow-ups at the singular point p satisfy the assumptions of the theorem. According to Corollary <ref>, for i large enough, there exists a naïve desingularization of (M_o,g_o), (M,g
^D_t_i), v_i∈𝐎̃(g^D_t_i), t_i,v_i→ 0 and a diffeomorphism ϕ_i:M→ M such that ϕ_i^*g_i =g_t_i,v_i is the Einstein modulo obstructions perturbation of g^D_t_i+v_i of Theorem <ref>. Let us fix p a singular point of M_o and denote S the rest of the singularities of M_o. Assume that no tree of singularities forms at p and denote (N,g_b) the Ricci-flat ALE manifold limit of blow-ups at p.
    
    According to Proposition <ref>, there exists an approximation g^A_t_i,v_i satisfying
    
    π_𝐎̃(̃g̃^̃D̃_̃t̃_̃ĩ)̃Φ_g^D_t_i(g^A_t_i,v_i)_r^-2_DC^α_β(g^D_t_i)⩽ C t_p,i^3-β/4,

    and consequently
    
    t_p,i∑_jλ̂_j(t_i,v_i) 𝐨̃^j_t_i,v_i_L^2⩽ t_p,i^5/4

    where the (𝐨̃^j_t_i,v_i)_j  form an L^2-orthonormal basis of elements of 𝐎̃_t_p(g̅_b,v_p,i), and 
    
        
  * g^A_t_i,v_i-g^D_t_i_C^2,α_β,*(g^D_t_i)⩽ 2ϵ,
        
  * g^A_t_i,v_i-(g^D_t_i+v_i) is L^2(g^D_t_i)-orthogonal to 𝐎̃(g^D_t_i).
    
    The estimate (<ref>) implies that we have
    
    t_p,i|λ̂_j(t_i,v_i)|⩽ C t_p,i^5/4≪ t_p,i.

    
    Now, we know that t_i,v_i→ 0, and according to Lemma <ref>, this implies that the λ̂_j(t_i,v_i) converge to λ̂_j. Since the λ̂_j are only constants depending on the geometry of (M_o,g_o) and that of (N,g_b), they must necessarily vanish to satisfy the inequality (<ref>) for t_i arbitrarily small. By coming back to the expression of λ̂_j of (<ref>) we find the obstruction. We can finally extend it in geodesic coordinates (for example) to obtain (<ref>) thanks to Corollary <ref>.




§ OBSTRUCTIONS FOR KNOWN SINGULARITY MODELS


The description of the previous section allowed us to find obstructions to the desingularization of Einstein orbifolds by smooth Einstein manifolds. We will now test them on the known examples and first show that the obstruction to the desingularization by gluing-perturbation of an Eguchi-Hanson metric of <cit.> also holds for any Gromov-Hausdorff desingularization of a finite number of singularities by trees of Kähler ALE Ricci-flat orbifolds in  Theorem <ref>. This is conjecturally the only possible way for Einstein metrics to degenerate in a noncollapsed setting. 

In dimension 4, the 2-forms decompose into self-dual and anti self-dual 2-forms which are elements of the eigenspaces of Hodge star operator * (which satisfies *^2 =) respectively associated to the eigenvalues 1 and -1.
Thanks to this direct sum, the symmetric endomorphism on 2-forms, 𝐑 given by the Riemannian curvature decomposes into blocks,

    𝐑=:
      [ 𝐑^+  ^0;  ^0 𝐑^- ],

 where the ^0 is the traceless part of the Ricci curvature, and where 𝐑^± are the self-dual and anti self-dual parts of the curvature.



 §.§ Kähler Ricci-flat ALE metrics and obstructions



The first obstructions to the desingularization of an Einstein orbifold (M_o,g_o) by a Ricci-flat ALE manifold (N,g_b) come from the infinitesimal deformations of g_b decaying as r_b^-4 at infinity. We will show that for any Kähler Ricci-flat ALE orbifold, there is a common obstruction to the desingularization which was already found in the case of the gluing of an Eguchi-Hanson metric for a particular gluing-perturbation procedure in <cit.>:

    𝐑^+ =0,

at the singular point. We will moreover see that generically (see Remark <ref>), there are additional obstructions corresponding to 
    𝐑^+ = 0.



 If we glue the Kähler Ricci-flat ALE metrics with the opposite orientation, that is with with a gluing parameter in O(4)\ SO(4) the common obstruction becomes 𝐑^- =0 and therefore in general, since the Einstein equation implies that the curvature is block diagonal (^0=0), the obstruction is 
    𝐑 =0.



The only known examples of ALE Ricci-flat orbifolds are Kähler. They have been classified and we have a satisfactory parametrization of the moduli space of these quotients of hyperkähler (hence Ricci-flat) ALE metrics.

Let us precise what the deformations of these known Ricci-flat ALE orbifolds are, in order to extend the obstruction to the desingularization by any Kähler Ricci-flat ALE space.


    
        Let Γ be a finite subgroup of SU(2), (N,g_b) a hyperkähler ALE manifold asymptotic to ℝ^4Γ and (O_i^4)_i a basis of the r_b^-4-terms of the elements of 𝐎(g_b). 
        
        Then, the conditions ∫_𝕊^3(3⟨ H_2,O_i^4⟩_g_e+O_i^4(B_eH_2,∂_r_e))dv_𝕊^3 = 0 for a quadratic symmetric 2-tensor H_2 satisfying d_e H_2 = Λ g_e imply that 
    𝐑_+(H_2) =0,

        where 𝐑_+(H_2) is the common selfdual part of the curvature of metrics with a development g_e + H_2 + 𝒪(r_e^3).
    
    
     
Denote by (x,y,z,t) the coordinates in an orthonormal basis of ℝ^4, and define a basis of invariant 1-forms on the sphere 𝕊^3, (α_1,α_2,α_3) by,

    α_1:=1/r_e^2(xdy-ydx+zdt-tdz)
    α_2:=1/r_e^2(xdz-zdx+ydt-tdy)
    α_3:=1/r_e^2(xdt-tdx+ydz-zdy).

    Manifestly, from Theorem <ref> these obstructions do not particularly depend on the Eguchi-Hanson metric, but on the r_b^-4-terms of the development of its deformations, O_i^4, which are, by <cit.>:
    
        
  * O^4_1:= 2 dr_e^2 + r_e^2α_1^2-r_e^2α_2^2-r_e^2α_3^2/r_e^4,
        
  * O^4_2:= r_e^2α_1.α_2+r_e dr_e.α_3/r_e^4,
        
  * O^4_3:= r_e^2α_1.α_3-r_e dr_e.α_2/r_e^4.
    
    
        Let Γ be a finite subgroup of SU(2). Then, according to <cit.>, there exists k_Γ∈ℕ^* and D_Γ, a finite union of vector subspaces of ℝ^3k_Γ of codimension at least 3 containing 0 such that the set of smooth hyperkähler metrics asymptotic to ℝ^4Γ is parametrized as (X_ζ,g_ζ)_ζ∈ℝ^3k_Γ\ D_Γ. Moreover, by <cit.>, for each ζ = (ζ_1,ζ_2,ζ_3) ∈ℝ^3k_Γ\ D_Γ, there exists a diffeomorphism Φ_ζ from a neighborhood of the infinity of ℝ^4Γ to the infinity of X_ζ such that Φ_ζ^*g_ζ = g_e + h_ζ +𝒪(r_e^-6), where,
    
    h_ζ =    - ∑_j,k,l|ζ_j|^2  dr_e^2 + 
        r_e^2α_j^2-r_e^2α_k^2-r_e^2α_l^2/r_e^4
       -⟨ζ_1, ζ_2 ⟩r_e^2α_1.α_2-r_e dr_e.α_3/r_e^4
       -⟨ζ_1, ζ_3 ⟩r_e^2α_1.α_3+r_e dr_e.α_2/r_e^4
       -⟨ζ_2, ζ_3 ⟩r_e^2α_2.α_3-r_e dr_e.α_1/r_e^4,

    where the first sum is taken on the (j,k,l) satisfying l≡ k+1 ≡ j+2  3.
        

        
        Without loss of generality, we can assume that the first coordinates of ζ_1,ζ_2 and ζ_3 ∈ℝ^k_Γ are 1, 0 and 0. Indeed, there exists l∈{1,...,k_Γ} such that the l-th coordinate of (ζ_1,ζ_2,ζ_3), (ζ^l_1,ζ^l_2,ζ^l_3) does not vanish. Just like for the homothetic deformations of the Eguchi-Hanson metric, thanks to an action of SO(4) on the asymptotics of the metric and a homothetic transformation (which yields an action of SO(3) and a common rescaling on all the ζ^l), we are able to reach another metric g_ζ̃ with (ζ̃^l_1,ζ̃^l_2,ζ̃^l_3)= (1,0,0).
    
        By differentiating the above expression of h_ζ, we see that the infinitesimal variations associated to the variations of (ζ̃^l_1,ζ̃^l_2,ζ̃^l_3) are then asymptotic to O^4_1, O^4_2 and O^4_3. The obstructions they induce by Theorem <ref> are therefore the same as for the Eguchi-Hanson metric, and by <cit.> they imply the condition 𝐑_+ =0 which is independent of the above actions of SO(4) and scaling.
    
    
    
        The case of the Eguchi-Hanson metric, or when the ζ^l are parallel, is actually the least obstructed case, and by the formula <cit.> the obstruction condition is 𝐑_+ =0 for ζ generic when k_Γ>1. 
    
    We find the same obstruction for Kähler Ricci-flat ALE orbifolds which are all asymptotic to ℝ^4Γ∼ℂ^2Γ for a group Γ =1/dn^2(1,dnm-1)⊂ U(2), that is the cyclic group generated by

    [         e^i2π/dn^2                  0;                  0 e^i2π (dnm-1)/dn^2 ],

where d⩾ 1, n⩾ 2 and, n and m are mutually prime.
    
        Let Γ be a group 1/dn^2(1,dnm-1)⊂ U(2) for d⩾ 1, n⩾ 2 and n and m mutually prime, and let (N,g_b) be a Kähler Ricci-flat ALE metric asymptotic to ℝ^4Γ, and (O_i^4)_i a basis of the r_b^-4-terms of the elements of 𝐎(g_b). 
        
        Then, for a quadratic symmetric 2-tensor H_2 such that d_e H_2 = Λ g_e, for i∈{1,2,3} we have the conditions
        
    ∫_𝕊^3(3⟨ H_2,O_i^4⟩_g_e+O_i^4(B_eH_2,∂_r_e))dv_𝕊^3= 0

        which imply that 
    𝐑_+(H_2)= 0 .

    
    
    Let 1/dn^2(1,dnm-1)⊂ U(2) be a finite subgroup of U(2), and (N,g_b) a non flat Kähler Ricci-flat ALE orbifold asymptotic to ℝ^4Γ. According to <cit.>, (Ñ,g̃_b) the universal cover of (N,g_b) is a hyperkähler orbifold asymptotic to Γ̃ = 1/dn(1,-1)⊂1/dn^2(1,dnm-1). Let H_2 be a quadratic symmetric 2-tensor on ℝ^4Γ, we can also lift it to ℝ^4Γ̃ as H̃_2. 
    
    Let us come back to the origin of the obstruction in Proposition <ref>, and more precisely the existence of a symmetric 2-tensor h_2 asymptotic to H_2 such that d_g_b(h_2)=Λ g_b. If such a symmetric 2-tensor exists on (N,g_b), we can lift it as a symmetric 2-tensor h̃_2 on (Ñ,g̃_b) asymptotic to H̃_2 and satisfying d_g̃_b(h̃_2)=Λg̃_b which implies, again according to Proposition <ref> and to Proposition <ref>, the condition 𝐑_+(H̃_2)= 0, and finally 𝐑_+(H_2)= 0.
    



    The above proof (or any proof in this article really) does not use the Kähler nature of the studied metric. It would also apply to any Ricci-flat ALE metric whose order r_e^-4 terms do not vanish <cit.>. It is however not clear if these terms can vanish (in well chosen coordinates) on a non flat Ricci-flat ALE metric.




 §.§ Trees of Kähler Ricci-flat ALE orbifolds and obstructions


Let us now treat the case of trees of ALE Kähler Ricci-flat orbifolds.



  §.§.§ Uniform controls of gravitational instantons

Let us use the notations of the proof of Proposition <ref>. Let Γ_1∈ SU(2) and ζ_1∈ D_Γ_1 fixed such that the ALE orbifold (X_ζ_1,g_ζ_1) has a singularity ℝ^4Γ for Γ⊂ SU(2). For v∈𝐎(g_ζ_1) which we will choose small, let us then denote g̅_ζ_1,v the Ricci-flat (and even hyperkähler) of g_ζ_1 given by Definition <ref>. The goal of this section is to show that there exists a common scale τ>0 such that for any ζ∈ℝ^3k_Γ with |ζ| = 1, for any t<τ, the naïve gluing of g_ζ to g̅_ζ_1,v at scale t, denoted g^B_v,t,ζ can be perturbed into a Ricci-flat modulo obstructions metric which we will denote ĝ_v,t,ζ.

The naïve gluing g^B_v,t,ζ is Ricci-flat everywhere except on the annulus where the gluing takes place. We therefore have the control

    Φ_g^B_v,t,ζ(g^B_v,t,ζ)_r_B^-2C^α_β(g^B_v,t,ζ)⩽ Ct^2-β/4

where C>0 only depends on ζ_1 and v because g_ζ for |ζ|= 1 is controlled at infinity in the ALE coordinates of <cit.>.

Going back to the proof of Theorem <ref> by inverse function theorem, in order to perturb g^B_v,t,ζ to a Ricci-flat modulo obstructions metric to to show the existence of ĝ_v,t,ζ, it would be enough to show that the linearization of π_𝐎̃(g^B_v,t,ζ)^⊥Φ_g^B_v,t,ζ, 
    π_𝐎̃(g^B_v,t,ζ)^⊥P_g^B_v,t,ζ : 𝐎̃(g^B_v,t,ζ)^⊥∩ C^2,α_β,*(g^B_v,t,ζ) →𝐎̃(g^B_v,t,ζ)^⊥∩ r_B^-2C^α_β(g^B_v,t,ζ)

is uniformly bounded and has a uniformly bounded inverse for |ζ|=1 and t small enough and to control the nonlinear terms of Φ_g^B_v,t,ζ.
Fixing ζ_1 and v, It would therefore be enough thanks to Proposition <ref> to show that 
    P_g_ζ : 𝐎(g_ζ)^⊥∩ C^2,α_β,*(g_ζ) →𝐎(g_ζ)^⊥∩ r_ζ^-2C^α_β(g_ζ)

is invertible with bounded inverse. This is however not the case when ζ→ D_Γ, that is when g_ζ degenerates to an orbifold. In this situation, we replace the norms with respect to g_ζ by norms with respect to a naïve desingularization g^B_ζ (with additional decay in the neck regions) close to g_ζ in order to keep a uniform gluing scale.


    For any Γ⊂ SU(2) and ϵ>0, there exists C = C(Γ,ϵ)>1 such that for any ζ∈ℝ^3k_Γ with |ζ|=1,
    there exists a naïve desingularization (partial if ζ∈ D_Γ and total if ζ∉ D_Γ, see Definition <ref>) g^B_ζ of an orbifold g_ζ_o with ζ_o∈ℝ^3k_Γ and |ζ_o|= 1 by hyperkähler orbifolds g_ζ_j for |ζ_j|= 1 and ζ_j∈ℝ^k_Γ_j with |Γ_j|<|Γ|, for which, denoting
    
    R(g^B_ζ) := sup_h∈𝐎̃(g^B)^⊥∩ C^2,α_β,*(g^B)h_C^2,α_β,*(g^B)/P_g^Bh_r_B^-2C^α_β(g^B) + sup_h∈ C^2,α_β,*(g^B)P_g^Bh_r_B^-2C^α_β(g^B)/h_C^2,α_β,*(g^B) ,

    we have
    
        
  * g^B_ζ - g_ζ_C^2,α_β,*(g^B_ζ)<ϵ/R(g^B_ζ), 
        
  * R(g^B_ζ) ⩽ C.
    


    The norms in the above statement are the naïve desingularizations norms (Definition <ref>) with respect to g^B_ζ. They are needed in order to obtain uniform estimates.


    What is crucial in this statement is the fact that the constant C is independent of the metric g_ζ and in particular allows ζ to approach D_Γ with uniform controls. The constant ϵ essentially decides from which proximity we replace the metric g_ζ by a naïve desingularization. Indeed, the trivial naïve desingularization g^B_ζ = g_ζ always satisfies the first property of the conclusion.


    Let ϵ >0 and let us show the result by induction on the order k of the group at infinity. It holds for ℤ_2 of order 2 because all the metrics g_ζ for ζ∈ℝ^3 are isometric for |ζ| =1, and we can take g^B_ζ = g_ζ.
    

    
    Assume now that the conclusion is satisfied for any group of order less than or equal to k-1⩾ 2 and consider  Γ of order k.
    
    Let us again work by induction, this time on the value of the square of the L^2-norm of the curvature of the ALE orbifolds asymptotic to ℝ^4Γ. This quantity is proportional to the dimension of the (orbifold) L^2 cohomology in degree 2 for these gravitational instantons.
    

    
    Consider, the orbifolds g_ζ, ζ∈ℝ^3k_Γ, |ζ|= 1 which have the smallest energy. Assume by contradiction that there exists a sequence of orbifolds g_ζ_i, |ζ_i|= 1 with the minimal energy such that for any naïve desingularization (Definition <ref>) g^B of a hyperkähler orbifold asymptotic to ℝ^4Γ, we have
    
        
  * either, g_ζ_i-g^B_C^2,α_β,*(g^B)⩾ϵ/R(g^B),
        
  * either, R(g^B)>i.
     
    In the coordinates of <cit.>, all of the metrics g_ζ_i admit coordinates of order 4 with a uniform control in r_ζ_i^-4C^3(g_ζ_i) in a uniform neighborhood of infinity, where r_ζ_i is the function of Definition <ref>. Therefore, there can only be L^2-concentration of the curvature in a compact of diameter uniformly bounded and volume uniformly bounded below by  Bishop-Gromov inequality. By minimality of the L^2-norm of the curvature, no tree of singularities can form with |ζ_i|= 1 because the limit orbifold would then have a smaller L^2-norm of the curvature. Therefore, there exists  g_ζ_∞ such that the g_ζ_i converge smoothly (considering local covering at the singular points) to g_ζ_∞. Since the asymptotic terms converge by <cit.>, we have g_ζ_i-g_ζ_∞_C^2,α_β,*(g_ζ_∞)→ 0 when i→∞ where we have ._C^2,α_β,*(g_ζ_∞)≈(1+r_ζ_∞)^β._C^2,α_0(g_ζ_∞)
    here since we use local coverings at the singularities instead of weights at the singularities of g_ζ_∞. The trivial naïve desingularization g^B =g_ζ_∞ therefore contradicts the assumptions. Indeed, the convergence happens in C^2,α_β,*(g_ζ_∞) and since P_g_ζ_∞ is elliptic, there exists C>0 such that for any h⊥𝐎(g_ζ_∞), we have
    
    h_C^2,α_β,*(g_ζ_∞)⩽ CP_g_ζ_∞h_r_ζ_∞^-2C^α_β(g_ζ_∞).

    
    Consider then a higher value for the L^2-norm of the curvature E>0 and assume that our property holds for the hyperkähler orbifolds asymptotic to ℝ^4Γ whose L^2-norm of the curvature is strictly less than E. Let us assume by contradiction that there exists a sequence ζ_i∈ℝ^3k_Γ |ζ_i|=1 satisfying _g_ζ_i_L^2(g_ζ_i)=E for which for any naïve desingularization g^B of a hyperkähler orbifold asymptotic to ℝ^4Γ, we have
    
        
  * either, g_ζ_i-g^B_C^2,α_β,*(g^B)⩾ϵ/R(g^B),
        
  * or, R(g^B)>i.
     
    Like in the above argument for the minimal energy, if no singularity was forming, up to taking a subsequence, we would have a C^2,α_β,*(g_ζ_∞)-convergence to a orbifold g_ζ_∞ with  R(g_ζ_∞) bounded. A tree of singularities is therefore forming. More precisely, up to taking a subsequence, there exists a subsequence naïve desingularizations g^B'_ζ_i of a hyperkähler orbifold asymptotic to ℝ^4Γ, composed of orbifolds g_ζ_o with |ζ_o|= 0 asymptotic to ℝ^4Γ and g_ζ_j with |ζ_j|=1 (up to changing the scale of the gluing, we can always assume that since g_tζ is isometric to tg_ζ), ζ_j∈ℝ^3k_Γ_j, at scales t_i,j, for which we have g_ζ_i-g^B'_ζ_i_C^2,α_β,*→ 0 as i→∞. Moreover, all of the metrics g_ζ_o and g_ζ_j are hyperkähler according to <cit.>. The L^2-norm of the curvature of g_ζ_o is strictly inferior to E since some of the total L^2-norm is lost in the singularities by <cit.>, and we have |Γ_j|<|Γ|=k by Bishop-Gromov inequality. Up to replacing g_ζ_o and the g_ζ_j by the naïve desingularizations g^B_ζ_o and g^B_ζ_j of the previous steps of our inductions, we obtain a naïve desingularization g^B_ζ_i for which we uniformly (depending on the constants of the previous steps of the induction only) control the operator P. Thanks to Lemma <ref>, we obtain a uniform control on the inverse of the operator P_g^B_ζ_i.
    This contradicts the initial assumptions and proves the statement.
    
    
    The following lemma lets us approximate the kernel of the operator P_g_ζ thanks to the approximate kernel 𝐎̃(g^B_ζ).
 
Let P and P' be two operators between Banach spaces X and Y for which there exists C>0, 1/100 C>ϵ>0, a finite dimensional space K'⊂ X and S' a complement of K' in X such that we have:
    
        
  * for any x∈ X, 
    (P-P')x_Y⩽ϵx_X,

        
  * for any x ∈ S', 
    x_X⩽ CP'x_Y,

        
  * for any x∈ K', 
        
    P'x_Y⩽ϵx_X,

        
  * and dim( P) = dim (K').
    
    Then, for any k∈ P, there exists k'∈ K' such that
    
    k-k'_X⩽2Cϵ/1-Cϵk'_X.



    Let k∈ P, and consider its decomposition k = k'+s' in the direct sum X = K' ⊕ S'. Thanks to the first hypothesis, we have
    
    P'k_Y⩽ϵk_X,

    thanks to the second,
    
    s'_X⩽ CP's'_Y

    and thanks to the third one, we have
    
    P'k'_Y⩽ϵk'_X.

    Putting (<ref>), (<ref>) and (<ref>) together, we find
    
    ϵk_X⩾P's'_Y-P'k'_Y⩾s'_X/C-ϵk'_X,

    hence, since k = k' + s',
    
    s'_X⩽ Cϵ(k_X+k'_X)⩽ 2Cϵk'_X+Cϵs'_X,

    and finally
    
    k-k'_X=s'_X⩽2Cϵ/1-Cϵk'_X.




    With the three first assumptions of Lemma <ref>, we still have 
    dim  P ⩽ dim K',

    and any element of P is close to an element of  K' (its projection on K' parallel to S') in the sens of (<ref>).


We conclude from Lemma <ref> and the estimates 1 and 2 of Proposition <ref> that for any Γ⊂ SU(2) and any ϵ>0 small enough depending on Γ only, for any ζ∈ℝ^3k_Γ and for the metric g^B_ζ obtained by Proposition <ref>, we have the following control for C= C(Γ,ϵ)>0 : for any
    h⊥𝐎(g_ζ),
    
    1/Ch_C^2,α_β,*(g^B_ζ)<P_g_ζh_r_B^-2C^α_β(g^B_ζ)<Ch_C^2,α_β,*(g^B_ζ).


We therefore conclude that we can define Ricci-flat modulo obstructions perturbations of our naïve desingularizations.


    Let (X_ζ_1,g_ζ_1) be a Ricci-flat orbifold for ζ_1∈ D_Γ_1 having a singularity ℝ^4Γ, Γ⊂ SU(2). Then, there exists ϵ>0 and τ>0 such that for any 0<t<τ, ζ∈ℝ^3k_Γ with |ζ| = 1 and v∈𝐎(g_ζ_1) with v_C^2,α_β,*(g_ζ_1)<τ, if we denote 
    
        
  * g^BB_v,t,ζ the naïve gluing of the metric (X_ζ,g^B_ζ), of Proposition <ref> for the constant ϵ, at the singularity ℝ^4Γ of g̅_ζ_1,v (defined above), and
        
  * g^B_v,t,ζ the naïve gluing of (X_ζ,g_ζ) at the scale t,
    
    there exists a unique metric ĝ_v,t,ζ satisfying
    
        
  * g^B_v,t,ζ-ĝ_v,t,ζ⊥𝐎(g^B_v,t,ζ),
        
  * g^B_v,t,ζ-ĝ_v,t,ζ_C^2,α_β,*(g^BB_v,t,ζ)⩽ 2τ (notice the naïve desingularization norm g^𝐁𝐁_v,t,ζ), and
        
  * Φ_g^B_v,t,ζ(ĝ_v,t,ζ)∈𝐎(g^B_v,t,ζ).
    
Moreover, the metric  ĝ_v,t,ζ depends smoothly on v, t and ζ. 


    As discussed at the beginning of this section, for the naïve gluing g^B_v,t,ζ, we have a control

    Φ_g^B_v,t,ζ(g^B_v,t,ζ)_r_B^-2C^α_β(g^BB_v,t,ζ)⩽ Ct^2-β/4

where C>0 only depends on ζ_1. Now, using the control (<ref>) to replace the controls of Lemma <ref>, we adapt the proof of Proposition <ref> to show that for τ small enough depending on Γ and ζ_1 alone, the operator

    π_𝐎̃(g^B_v,t,ζ)^⊥P_g^B_v,t,ζ : 𝐎̃(g^B_v,t,ζ)^⊥∩ C^2,α_β,*(g^BB_v,t,ζ) →𝐎̃(g^B_v,t,ζ)^⊥∩ r_B^-2C^α_β(g^BB_v,t,ζ),

is uniformly bounded (independently of ζ and of t and v small enough) and invertible with a uniformly bounded inverse. Notice the norms g^BB_v,t,ζ again. The proof of Theorem <ref> by inverse function theorem extends here, and there exists a unique solution ĝ_v,t,ζ to the equations of the statement.

Like in Corollary <ref>, the smooth dependence of the metric is a consequence of the implicit function theorem applied to 
    Ψ : (v,t,ζ,h)↦π_𝐎(g^B_v,t,ζ)^⊥Φ_g^B_v,t,ζ(g^B_v,t,ζ+h)

for h⊥𝐎(g^B_v,t,ζ).




  §.§.§ Gluing-perturbation of gravitational instantons


Let us first make sure that all of the gluing gauges given by the isometries of ℝ^4Γ (see Remark <ref>) are equivalent to gluing a gravitational instanton but with a different parameter ζ.


    Let (M_o,g_o) be an orbifold (compact or ALE) with a singularity ℝ^4Γ at p with Γ⊂ SU(2), and let
ζ∈ℝ^3k_Γ. Let moreover ψ be an isometry of  ℝ^4Γ preserving the orientation.

Then, there exists ζ'∈ℝ^3k_Γ such that the naïve gluing of (X_ζ,g_ζ) at p with the ALE coordinates of <cit.> composed with ψ when identifying with the coordinates of (M_o,g_o) (see Remark <ref>)
is isometric to the gluing of (X_ζ',g_ζ') to (M_o,g_o) at p
without composing with an isometry.


    Let Γ⊂ SU(2) be a finite subgroup. Having ψ∈(ℝ^4Γ) implies that ψ is in the normalizer of Γ in SO(4) and therefore that ψ∈ϕ(𝕊^3× N_Γ), where N_Γ is the normalizer of Γ in SU(2) (see <cit.> for an explicit expression) and where ϕ: 𝕊^3×𝕊^3→ SO(4) is the double covering of SO(4) where the first 𝕊^3 is the left multiplication by a unit quaternion and the second a right multiplication. In this identification, we have SU(2) = ϕ({1}×𝕊^3) and we will denote  SU(2) = ϕ( 𝕊^3×{1}).
    

    
    Let us now study the action of the normalizer of Γ, which is ϕ(𝕊^3× N_Γ), on a metric of <cit.> in the coordinates of <cit.>. Let us come back from their construction in <cit.> starting with 
    P:=ℂ^2⊗ End(R),

    where ℂ^2 is the standard representation of SU(2) and where R is its regular representation SU(2). Denote P^Γ the set of Γ-invariant elements of P, F the set of elements commuting with Γ seen as a subset of the unitary transformations of R, and finally denote T the subgroup of complex numbers of unit norm acting trivially. The gravitational instantons asymptotic to ℂ^2Γ are the hyperkähler quotients of P^Γ by F/T. 
    
    By definition, the normalizer ϕ(𝕊^3× N_Γ) acts on P^Γ and commutes to the action of F. It consequently acts on the set of solutions of the moment map μ : P^Γ→ (𝔣/𝔱)^*⊗ℝ^3 where 𝔣 and 𝔱 are the Lie algebras of F and T respectively. More precisely, denoting I_k, k∈{1,2,3} the 3 complex structures of P^Γ given by the identification of ℂ^2 with the quaternions, for all ξ∈𝔣/𝔱, the three coordinates of μ satisfy the equations
    
    grad(μ_k.ξ) = I_k(V_ξ),

    where V_ξ is the vector field on P^Γ generated by the action of ξ. The SU(2) in which Γ and N_Γ commutes to the three complex structures since it is identified to the right multiplication by a unit-norm quaternion. We conclude that N_Γ acts by isometry on the hyperkähler metric μ^-1(ζ)/(F/T) for all ζ∈ (𝔣/𝔱)^*⊗ℝ^3. The part SU(2) of the normalizer, acts by rotation on the three complex structures. And more precisely, an element n_-∈SU(2) sends μ^-1(ζ)/(F/T) to μ^-1(Ad(n_-)ζ)/(F/T) where Ad(n_-) is the standard action of SU(2)/±≈ SO(3) on the factor ℝ^3 of (𝔣/𝔱)^*⊗ℝ^3.
    

    
    To conclude there remains to ensure that this action of the normalizer is the standard action on the asymptotic cone ℝ^4Γ in the coordinates of <cit.>. To prove this, we use the identification ℝ^4Γ≈μ^-1(0)/(F/T) of <cit.>. The correspondence between the infinities of μ^-1(ζ)/(F/T) and μ^-1(0)/(F/T) for the coordinates of <cit.> given by <cit.> lets us conclude that the action of the normalizer is indeed the standard action on the asymptotic cone ℝ^4Γ.
    
    Therefore, the gluing (X_ζ,g_ζ)≈μ^-1(ζ)/(F/T) for ζ∈ℝ^3k_Γ≈ (𝔣/𝔱)^*⊗ℝ^3 composed with the isometry ψ=ϕ(ψ_-,ψ_+) of ℝ^4Γ is isometric to the gluing of (X_ζ',g_ζ') with ζ' = Ad(ψ_-) ζ.


Let us then remark that any ALE hyperkähler orbifold can be desingularized by a sequence of ALE hyperkähler manifolds.


    Let (X_ζ_1,g_ζ_1) be an ALE hyperkähler orbifold with a singularity ℝ^4Γ. Then, for any t>0, there exists ζ_0^t∈ℝ^3k_Γ\ D_Γ and v_t∈𝐎(g_ζ_1) satisfying lim_t→ 0v_t = 0 and lim_t→ 0 X_ζ_0^t = X_ζ_0∈ℝ^3k_Γ\ D_Γ, such that the Einstein modulo obstructions desingularization (X_ζ_1#X_ζ_0,ĝ_v_t,t,ζ_0^t) of Corollary <ref> is hyperkähler.


    The result of <cit.> shows that for t small enough, there exists ζ_0 for which there is a hyperkähler metric g_t satisfying 
    
    g_t-g^B_0,t,ζ_0_(1+r_B)^-βC^2,α_0(g^B_0,t,ζ_0)⩽γ(t),

    where lim_t→ 0γ(t)=0. We can then apply our construction of coordinates of <cit.> to (X_ζ_1#X_ζ_0,g_t) in order to obtain a diffeomorphism ϕ : X_ζ_1#X_ζ_0→ X_ζ_1#X_ζ_0 thanks to which we have the better control
    
    ϕ^*g_t-g^B_0,t,ζ_0_C^2,α_β,*(g^B_0,t,ζ_0)⩽ Cγ(t).

    Thanks to Proposition <ref>, there exists a small diffeomorphism ψ : X_ζ_1#X_ζ_0→ X_ζ_1#X_ζ_0 such that δ̃_g^B_0,t,ζ_0ψ^*ϕ^*g_t= 0. The uniqueness of Theorem <ref> and the fact that the infinitesimal deformations of (X_ζ_0,g_ζ_o) integrate to curves t↦(X_ζ_0(t),g_ζ_0(t)) by the classification of <cit.> ensures that there exists v_t∈𝐎(g^B_0,t,ζ_0) and ζ_0^t such that we have
    
    ψ^*(ϕ^*g_t)=ĝ_v_t,t,ζ_0^t,

    with ĝ_v_t,t,ζ_0^t one of the metrics of Theorem <ref>.



    Note that <cit.> provides such gluings-perturbations in all directions ζ∈ℝ^3k_Γ\ D_Γ. The admissible scales are however not independent on |ζ|=1, and arbitrary trees of singularities are not allowed.  



    There exists τ>0 such that for all 
    
    (v,t,ζ)∈(B_C^2,α_β,*(g_ζ_1)(0,τ)∩𝐎(g_ζ_1))× (0,τ)× (𝕊^3k_Γ-1\ D_Γ),

    the metric ĝ_v,t,ζ of Proposition <ref> is isometric to a hyperkähler metric of <cit.>.


    Let us start by noting that the set 
    (B_C^2,α_β,*(g_ζ_1)(0,τ)∩𝐎(g_ζ_1))× (0,τ)× (𝕊^3k_Γ-1\ D_Γ)
 is connected since D_Γ is a finite union of spaces of codimension at least 2 in 𝕊^3k_Γ-1. We therefore just need to prove that the set
    
    E = {(v,t,ζ)∈(B_C^2,α_β,*(g_ζ_1)(0,τ)∩𝐎(g_ζ_1))× (0,τ)× (𝕊^3k_Γ-1\ D_Γ),  (ĝ_v,t,ζ)=0},

    which is non empty by Lemma <ref> is open and closed. It is therefore isometric to g_ξ(v,t,ζ) for ξ(v,t,ζ)∈ℝ^3k_Γ_1\ D_Γ_1.
    
    The set E is closed by the continuity of 
    (v,t,ζ)∈(B_C^2,α_β,*(g_ζ_1)(0,τ)∩𝐎(g_ζ_1))× (0,τ)× (𝕊^3k_Γ-1\ D_Γ)↦(ĝ_v,t,ζ)
 proven in Corollary <ref>.
    
    For the openness, let us assume that for (v,t,ζ) fixed, we have (ĝ_v,t,ζ) = 0. The metric ĝ_v,t,ζ is therefore one of the metrics of <cit.>. There exists a space of the same dimension as 𝐎̃(g^B_v,t,ζ) of hyperkähler deformations in the neighborhood of g_ξ(v,t,ζ) and therefore the Ricci-flat modulo obstructions deformations of ĝ_v,t,ζ are hyperkähler. Indeed, since the metrics g̅_ζ_1,v, g_ζ_2 and g_ξ(v,t,ζ) are hyperkähler, they have a hyperkähler deformation space of dimension three times that of their L^2-cohomology in degree 2, and the dimension of this cohomology is additive for our gluings.
    
    According to Lemma <ref> applied to P = P_ĝ_v,t,ζ, P' =P_g^B_v,t,ζ, K' = 𝐎̃(g^B_v,t,ζ) and S' = 𝐎̃(g^B_v,t,ζ)^⊥, the Ricci-flat deformations of ĝ_v,t,ζ are arbitrarily close to elements of 𝐎̃(g^B_v,t,ζ). By Corollary <ref>, the metrics g^B_v',t',ζ' approximate the small Ricci-flat deformations ĝ_v',t',ζ' of ĝ_v,t,ζ staying in B_C^2,α_β,*(g^B_v,t,ζ,2ϵ), for ϵ>0 the constant of Theorem <ref>. We therefore reach metrics isometric to all the ĝ_v',t',ζ' for (v',t',ζ') in a neighborhood of (v,t,ζ) in (B_C^2,α_β,*(g_ζ_1)(0,τ)∩𝐎(g_ζ_1))× (0,τ)× (𝕊^3k_Γ-1\ D_Γ). The set E is therefore open.





  §.§.§ Obstruction for trees of Kähler Ricci-flat ALE orbifolds



 Let us use the notations of the proof of Proposition <ref>, and parametrize the set of Kähler Ricci-flat manifolds asymptotic to ℝ^4Γ as (X_ζ,g_ζ)_ζ∈ℝ^3k_Γ\ D_Γ in the following statement.


Let (X_ζ_0,g_ζ_0) be a Kähler Ricci-flat orbifold asymptotic to ℝ^4Γ, and let ζ∈ℝ^d_Γ\ D_Γ be close to ζ_0. Then, there exists a naïve desingularization g^B_t of (X_ζ_0,g_ζ_0) by Kähler Ricci-flat ALE orbifolds glued in the same orientation and v∈𝐎̃(g^B_t) such that (X_ζ,g_ζ) = (N,g̅_b,t,v) is the (iterated) perturbation of Lemma <ref> of (N,g^B_t+v).
    
    Moreover, there exists ϵ>0 such that for ζ close enough to ζ_0, there exists a diffeomorphism Φ_ζ between neighborhoods of the infinities of (X_ζ,g_ζ) and of ℝ^4Γ such that there exists 𝐨_1(ζ), 𝐨_2(ζ) and 𝐨_3(ζ) elements of 𝐎(g_ζ) satisfying for all i∈{1,2,3},
    
    Φ_ζ^*𝐨_i(ζ) = O_i^4 +𝒪(r_B^4+β),

    where O_i^4 = 𝒪(r_B^-4) is the homogeneous symmetric 2-tensor used in the proof of Proposition <ref>, and with 𝐨_i(ζ)_L^2(g_ζ)⩾ϵ.


According to Corollary <ref>, (X_ζ,g_ζ) = (N,g̅_b,t) is a Kähler Ricci-flat perturbation of a naïve desingularization (N,g^B_t) for some small t depending on ζ. Moreover, the r_e^-4 terms of (X_ζ,g_ζ) converge to those of (X_ζ_0,g_ζ_0) as ζ→ζ_0 by Corollary <ref>. By assumption, there exists l∈{1,...,k_Γ} such that ζ_0^l≠ 0, hence, for ζ close enough to ζ_0, ζ^l ≠ 0 by continuity. Just like in the proof of Proposition <ref>, this implies that there exists a diffeomorphism Φ_ζ between neighborhoods of the infinities of (X_ζ,g_ζ) and of ℝ^4Γ and infinitesimal deformations of g_ζ, 𝐨_1(ζ), 𝐨_2(ζ) and 𝐨_3(ζ) such that there exists C>0 independent of ζ for which we have for all i∈{1,2,3},

    |Φ_ζ^*𝐨_i(ζ) - O_i^4| ⩽ C r_B^-4-β

by Corollary <ref>. In particular, since O_i^4≠ 0, there exists ϵ>0 depending on C and β, but independent of ζ such that we have 𝐨_i(ζ)_L^2(g_ζ)⩾ϵ.





    Let g^B_t be a naïve gluing of Kähler Ricci-flat ALE orbifolds, and g̅_b,t its Kähler Ricci-flat pertubation of Lemma <ref>.

    Then, for any symmetric 2-tensor w∈ r_B^-2C^α_β(g^B_t), there exists a unique symmetric 2-tensor u∈𝐎(g̅_b,t)^⊥_g̅_b,t∩ C^2,α_β,*(g^B_t), such that 
    
    P̅_g̅_b,t u = π_𝐎(g̅_b,t)^⊥w.

    
    We moreover have the following control for C = C(g^B_t)>0, 
    
    u_C^2,α_β,*(g^B_t)⩽ Cπ_𝐎(g̅_b,t)^⊥w_r_B^-2C^α_β(g^B_t).


 
    The crucial part of this lemma is the fact that the solution is controlled in the tree of singularities norm C^2,α_β,*(g^B_t) which behaves well as t→ 0.
 

    According to Theorem <ref>, we have

    g̅_b,t-g^B_t_C^2,α_β,*(g^B_t)⩽ C t_max^2-β/4,

which, combined with the proof of Theorem <ref> implies that, for t_max small enough, the operator P̅_g̅_b,t is injective on 𝐎̃(g^B_t)^⊥∩ C^2,α_β,*(g^B_t). 

Moreover, for 0<β<1, its cokernel on r_D^-2C^α_β(g^B_t) is equal to its kernel on r_D^-2C^α_-β(g^B_t) which is equal to 𝐎(g̅_b,t). Indeed, for any g_b a Ricci-flat ALE metric, the kernel and the cokernel of P̅_g_b: C^2,α_β,*(g_b)→ r_b^-2C^α_β(g_b) are equal to 𝐎(g_b) because taking the divergence of P̅_g_b(h) = 0 for h∈ C^2,α_β,*(g_b), yields δ_g_bδ_g_b^*(δ_g_bh) = 0, and finally δ_g_bh=0 by Proposition <ref>. By taking the trace of the remaining of the equation, we find that ∇^*_g_b∇_g_b(_g_bh)=0, and since h decays at infinity, _g_bh = 0. Finally P_g_b(h)=0, and we conclude that the kernel of P̅_g_b: C^2,α_β,*→ r_b^-2C^α_β is 𝐎(g_b), and similarly, its cokernel is also 𝐎(g_b).




    Let t=(t_1 =1,...,t_k)>0, and let
    
        
  * (N_k,g_b_k)_k be a tree of ALE Kähler Ricci-flat orbifolds desingularizing ℝ^4Γ,
        
  * (N,g^B_t) the naïve gluing of the (N_k,g_b_k) at the relative scales t_k to (N_1,g_b_1), small enough for k≠ 1, and
        
  * (N,g̅_b,t) be the Kähler Ricci-flat ALE perturbation of (N,g^B_t) of Lemma <ref>.
    
       Let us assume that (N_1,g_b_1) is asymptotic to ℝ^4Γ, consider Ĥ_S a quadratic symmetric 2-tensor on ℝ^4Γ such that P̅_e Ĥ_S +λ g_e = 𝐎_S for a constant symmetric 2-tensor 𝐎_S.
    
    Then, there exists C>0 independent of the t_k and χ, a cut-off function supported in a neighborhood of infinity of (N,g̅_b,t) independent of the t_k, and there exists ĥ_2 a symmetric 2-tensor on N such that we have
    
    P_g̅_b,tĥ_2 + λg̅_b,t - χ𝐎_S =∑_iλ̂_i𝐨_i∈𝐎(g̅_b,t),

    and 
    ĥ_S-χĤ_S_C^2,α_β,*(g^B_t)⩽ C Ĥ_S_r^2_eC^0(g_e).



    Let us consider Ĥ_S a quadratic symmetric 2-tensor on ℝ^4Γ such that P̅_eĤ_S + λ g_e = 𝐎_S, and let χ be a cut-off function on N_1 supported in a neighborhood of infinity where (N_1,g_b_1) has ALE coordinates we will also denote χ on N the cut-off function extended by 0 on the deeper ALE orbifolds.
    
    We then have 
    P_g̅_b,t(χĤ_S) + λg̅_b,t -χ𝐎_S _r_D^-2C^α_β(g^B_t)⩽ C Ĥ_S_r_e^2C^0_0.

    Indeed, in a neighborhood of infinity where χ≡ 1, since g̅_b,t-g_e = 𝒪(r_B^-4), we have P_g̅_b,t(χĤ_S) + λg̅_b,t = 𝐎_S+ 𝒪(r_B^-4), and on the rest of the manifold, we have the expected control by definition of the norm r_D^-2C^α_β(g^B_t). According to Lemma <ref> applied to g = g̅_b,t, there exists a unique symmetric 2-tensor h'∈ C^2,α_β,*(g^B_t) ∩𝐎(g̅_b,t)^⊥, such that we have
    
    P_g̅_b,t(χĤ_S + h') + λg̅_b,t- χ𝐎_S ∈𝐎(g̅_b,t).

    Moreover, according to Proposition <ref>, the element of 𝐎(g̅_b,t) is explicit. More precisely, consider (𝐨_i)_i an orthonormal basis of 𝐎(g̅_b,t), and thanks to the diffeomorphism Φ_t of Lemma <ref>, let us assume that the three first elements are asymptotic to c_iΦ_t,*O_i^4 for c_i>1/ϵ. We have
    
    P_g̅_b,t(χĤ_S + h') + λg̅_b,t- χ𝐎_S = ∑_i λ̂_i𝐨_i∈𝐎(g̅_b,t),

    where, for i = 1,2,3,
    
    λ̂_i:= -∫_𝕊^3Γ(3⟨Ĥ_S, O_i^4 ⟩_g_e + O_i^4(∇_e_eĤ_S,∂_r_e) ) dv_𝕊^3Γ + ∫_N χ⟨𝐎_S, 𝐨_i ⟩_g_b dv_b.



Let (M_o,g_o) be an Einstein orbifold and p one of its singular points of singularity ℝ^4Γ, S the set of singularities of M_o\{p}, and let (N_k,g_b_k)_k be a tree of ALE Kähler Ricci-flat orbifolds desingularizing ℝ^4Γ. Let moreover g_S be a naïve desingularization modulo obstructions of (M_o,g_o,S) and Ĥ_S the quadratic terms of a development in divergence-free gauge at p∈ M_o, t_1>0, g̅_b,t a Kähler Ricci-flat gluing of the (N_k,g_b_k)_k at relative scales t=(t_k)_k>0 produced by Lemma <ref>, and a symmetric 2-tensor h_2 on N and the real numbers λ_i of Lemma <ref>. 

    Let us define the approximate metric g^A as the naïve gluing (Definition <ref>) of g_S and t_1(g̅_b,t+t_1ĥ_S).

We have the following control whose proof is the same as Theorem <ref>.


    Let β>0, and let us use the above notations.
    For t_max>0 small enough we have the following controls: for all k∈ℕ there exists C_k>0,
    
    π_𝐎̃(g^D)^⊥Φ_g^D(g^A) _r^-2_DC^k_β(g^D)⩽ C_k t_1^3-β/4,

    and for all 𝐨̃∈𝐎̃_t_1(g̅_b,t), and denoting (𝐨̃_i)_i and orthonormal basis of 𝐎̃_t_1(g̅_b,t)
    
    ⟨Φ_g^D(g^A)-t_1 ∑_iλ̂_i 𝐨̃_i, 𝐨̃⟩_L^2(g^D)⩽ C_0𝐨_L^2(g_b)t_1^5/4,

    while satisfying, 
    
        
  * g^A-g^D_C^2,α_β,*(g^D)⩽ 2ϵ,
        
  * g^A-g^D is L^2(g^D)-orthogonal to 𝐎̃(g^D),
    



    The crucial part here is that, by considering the right weighted spaces, C^2,α_β,*(g_t^B), and Kähler Ricci-flat perturbations of our tree of singularities, we obtain a control by powers of t_1 only.


We then conclude, exactly like in Theorem <ref> that the obstruction is satisfied in the limit at every singular point of (M_o,g_o) where the trees of singularities appearing are composed of Kähler Ricci-flat orbifolds ALE.


    Let (M_o,g_o) an Einstein orbifold, and assume that there exists (M_i,g_i)_i a sequence of Einstein manifolds such that 
    (M_i,g_i) (M_o,g_o).

    
    Then, (M_o,g_o) satisfies 𝐑(g_o)=0 at every singular point where the trees of singularities forming in the Gromov-Hausdorff sense according to Corollary <ref> are composed of ALE Kähler Ricci-flat orbifolds glued in the same orientation.



    The result is optimal in the sense that it is the only local obstruction to the desingularization of a ℝ^4ℤ_2 singularity. Indeed, together with the existence of Einstein desingularizations of <cit.>, proven in the case of rigid asymptotically hyperbolic Einstein metrics with a singularity ℝ^4ℤ_2 singularity, we see that there exists a desingularization in the Gromov-Hausdorff sense by Eguchi-Hanson metrics if and only if the condition 𝐑(g_o)=0 is satisfied.



    For now, we cannot prove any obstruction result if trees of non Kähler Ricci-flat ALE orbifolds were to appear. The reason is that it might not be possible to glue and perturb them into a single Ricci-flat ALE manifold. The obstructions to such a gluing could possibly compensate the ones coming from the gluing to the orbifold.



    Like in Example <ref>, let us consider the sphere 𝕊^4 as 𝕊^4⊂ℝ^5 = ℝ×ℝ^4. We define 𝕊^4Γ, the orbifold obtained as the quotient of 𝕊^4 by the action of Γ for the first 4 coordinates of ℝ^5. 𝕊^4Γ has its sectional curvatures constant equal to 1, and two singularities modeled on ℝ^4Γ. The condition 𝐑= 0 is therefore not satisfied for this orbifold.




§ OBSTRUCTIONS UNDER TOPOLOGICAL ASSUMPTIONS


Let us now give topological conditions which will ensure that the Ricci-flat ALE orbifolds appearing as blow ups in our degenerations are Kähler and glued in the same orientation, and therefore that the obstruction 𝐑=0 holds.

All of these topological conditions come from the topological characterization of <cit.>, see also <cit.> for a generalization. Basically, if a desingularization has the topology of a minimal resolution of a SU(2)-singularity (or a quotient for the U(2) singularities) in a neighborhood of a singularity, then, all of the bubbles are Kähler and glued in the same orientation and we can apply Theorem <ref>. We will state more global topological conditions based on Hitchin-Thorpe inequality in Theorem <ref>, and a spin condition in Theorem <ref>. We will then finally comment on the desingularization of Einstein orbifolds with various pinching conditions and bound on the Ricci curvature in Corollary <ref> and Remark <ref>.




 §.§ Hitchin-Thorpe inequality and desingularization of Einstein orbifolds

Let us first notice that desingularizing an Einstein orbifold by smooth Einstein manifolds necessarily damages the Hitchin-Thorpe inequality satisfied by the orbifold, see Theorem <ref>. The equality case is exactly when all the Ricci-flat ALE orbifolds are Kähler and glued in a common orientation.

For an Einstein manifold of dimension 4, Chern-Gauss-Bonnet formula implies,

    χ(M)= 1/8π^2∫_M ||^2dv = 1/8π^2∫_M (^2/24+|W^+|^2+|W^-|^2)dv,

and Hirzebruch's signature formula gives,

    τ(M)= 1/12π^2∫_M (|W^+|-|W^-|^2)dv.

Simply because ∫_M |W^±|^2dv⩾ 0 and ∫_M ^2dv⩾ 0, thanks to (<ref>) and (<ref>), we have the following Hitchin-Thorpe inequality for Einstein 4-manifolds,

    2χ(M)⩾ 3|τ(M)|.

with equality if and only if (M,g) is a quotient of the flat torus or of a hyperkähler metric on the K3 surface.

In the case of orbifolds and ALE metrics, to be consistent with Chern-Gauss-Bonnet and Hirzebruch formulas, (<ref>) and (<ref>) for compact Einstein manifolds of dimension 4, we have to modify the Euler characteristic and the signature thanks to a boundary term. The integral quantities (<ref>) and (<ref>) above are topological invariants for Einstein orbifolds and Ricci-flat ALE orbifolds. We will denote them χ̃ and τ̃.


For Ricci-flat ALE manifolds, Nakajima obtained an Hitchin-Thorpe inequality.

    Let (N,g_b) be a Ricci-flat ALE manifold of dimension 4. Then, we have the following inequality between the modified Euler characteristic and the modified signature of Ricci-flat ALE orbifolds,
    
    2χ̃(N)⩾ 3 |τ̃(N)|,

    with equality if and only if (N,g_b) is a Kähler Ricci-flat ALE orbifold.



    In particular, the only Ricci-flat ALE manifolds diffeomorphic to a minimal resolution of a singularity ℂ^2Γ for Γ⊂ SU(2) or one of its quotients are Kähler.


The topological invariants τ̃ and χ̃ are additive by gluing ALE spaces to orbifold singularities like in Definition <ref>. If M = M_o#_jN_j, we then have

    τ(M) = τ̃(M_o) + ∑_jτ̃(N_j),

and

    χ(M) = χ̃(M_o)+ ∑_jχ̃(N_j).

This directly implies:

    2χ(M) - 3|τ(M)| =     2(χ̃(M_o) + ∑_jχ̃(N_j)) - 3 |τ̃(M_o) + ∑_jτ̃(N_j)|
    ⩾     2χ̃(M_o)- 3 |τ̃(M_o)| + ∑_j (2χ̃(N_j) - 3|τ̃(N_j)|)
    ⩾     2χ̃(M_o)- 3 |τ̃(M_o)|.

Since every term is nonnegative by Hitchin-Thorpe inequality and Lemma <ref>, we see that there is equality if and only if for all j we have 2χ̃(N_j) - 3|τ̃(N_j)|= 0 and that the gluings are done in the same orientation for which τ̃(M_o) and all the τ̃(N_j) have the same sign.


    If (M_o,g_o) is a hyperkähler orbifold, then the only Gromov-Hausdorff desingularizations preserving the inequality are hyperkähler and correspond to gluing hyperkähler ALE in the same orientation. 



	For Γ⊂ SU(2), an Einstein desingularization of 𝕊^4Γ preserving Hitchin-Thorpe inequality is diffeomorphic to M = 𝕊^4Γ#X_Γ#X_Γ for X_Γ a minimal resolution of the singularity ℂ^2Γ. 


By studying the equality case in the previous inequalities, we get a quite restrictive situation.

    Let (M_o,g_o) be an Einstein orbifold oriented so that τ̃(M_o)⩾ 0, and assume that (M,g_i)_i is a sequence of Einstein metrics converging in the Gromov-Hausdorff sense to (M_o,g_o).
    
    We then have the following inequality,
    
    2χ(M)-3|τ(M)|⩾ 2χ̃(M_o)-3τ̃(M_o).

    Moreover, we have equality, if and only if
    M is a desingularization of M_o by gluing trees of  Kähler Ricci-flat ALE orbifolds in the same orientation (that is with gluing parameters in SO(4)), and in this equality case we have the condition 
    𝐑_+(g_o)=0
 at all of the singular points of M_o.


    The equality condition limits the possible group actions of the singularities.


This for example implies the following.

	Let Γ⊂ SU(2), (M_i,g_i)_i be a sequence of Einstein manifolds converging in the Gromov-Hausdorff sense to the spherical orbifold 𝕊^4Γ. Then, for i large enough, we have
	
    2χ(M_i)-3|τ(M_i)| > 2χ̃(M_o)-3|τ̃(M_o)|.





 §.§ Spin manifolds


Another way to ensure that the Ricci-flat ALE orbifolds appearing are Kähler and glued in the same orientation is to impose that the sequence of differentiable manifolds is spin. Our result is essentially an application of the following Lemma of Nakajima.

    Let (N,g_b) be a Ricci-flat ALE metric on a spin manifold which is asymptotic to ℝ^4Γ for Γ a finite subgroup of SU(2), then, (N,g_b) is a hyperkähler metric.


As a consequence, there is also an obstruction to the desingularizations of Einstein orbifolds by smooth Einstein metrics on spin manifolds. The proof of Theorem 1.1 of <cit.> whose main tool is Lemma <ref> implies that the limit orbifold and the Ricci-flat ALE metrics are spin and glued in the same orientation for a degeneration of Einstein metrics on spin manifolds. If the group at infinity of the ALE spaces, which are also the groups of the singularities of the orbifold are in SU(2), we use Lemma <ref> to get the following obstruction.


    Let (M_i,g_i)_i be a sequence of Einstein spin manifolds of dimension 4 converging to an Einstein orbifold (M_o,g_o). Then, (M_o,g_o) is spin and at its singular points whose groups are in SU(2), we have 
    𝐑 = 0.




    There is no restriction on the group singularities in <cit.>. This comes from their additional assumption on the kernel of the Dirac operator of the sequence which actually implies that all singularities have their group in SU(2).





 §.§ Pinched Ricci curvature and the Einstein condition


Our result shows that there is a fundamental difference between the Einstein condition and some pinching conditions on the Ricci curvature once we authorize the formation of singularities. From Theorems <ref> and <ref>, we deduce that there exists an obstruction to the desingularization of Einstein orbifolds by smooth Einstein metrics which is not there if we consider pinching conditions on the Ricci curvature. Let us illustrate this with the simple example of a spherical orbifold, even though a similar result obviously holds for general orbifolds with singularity groups in SU(2).


    Let Γ be a finite subgroup of SU(2), and M = 𝕊^4Γ#X_Γ#X_Γ (# means gluing at both orbifold singularities in an orientation), where X_Γ is the minimal resolution of the singularity ℂ^2Γ. Then, for all 1⩽ p <+∞,
    
        
  * there exists a sequence of metrics (M,g_i)_i such that we have 
        
            
  * (g_i)-3g_i_L^p(g_i)⩽1/i
            , and
            
  * ( M,g_i ) ( 𝕊^4Γ, g_𝕊^4Γ),
        
        but,
        
  * there  does not exist any sequence of Einstein metrics (M,g_i^ℰ) such that 
        
            
  * (g^ℰ_i)=3g^ℰ_i, and
            
  * ( M,g_i^ℰ) (𝕊^4Γ, g_𝕊^4Γ).
        
    


    The second part is a consequence of Theorem <ref> because the curvature of the sphere never satisfies the condition 𝐑 =0 since 𝐑= Id for such a metric.
    
    For the first part, we can just remark that our approximation metric g^A_t with fixed Kähler Ricci-flat ALE metrics satisfies (g^A_t)-Λ g^A_t_L^∞(g^A_t)= 𝒪(1) and that (g^A_t)-Λ g^A_t is supported in regions with a volume of order t, therefore, if we choose t small enough, we have the control in L^p-norm for p<+∞.




  
Question: Can we desingularize 𝕊^4ℤ_2 thanks to the Eguchi-Hanson metric by metrics with Ricci curvature converging to 3 in the L^∞-sense?


    By being more precise in the expression of the obstructions to the desingularization of 𝕊^4ℤ_2 by two Eguchi-Hanson metrics, for t_max small enough, 
    
    |(g^A_t)-3 g^A_t|_g^A_t⩽ 1+η(t_max)

     where η(t_max)→ 0 when t_max→ 0. 



    It is possible to desingularize a spherical orbifold 𝕊^4Γ for Γ⊂ SU(2) by metrics with ⩾ 3 (or ⩽ 3) while  is pinched in L^p. 
    
    Consider for ϵ>0 and b>1, choose a cut-off function, χ_b,ϵ, supported on [0,bϵ] and equal to 1 on [0,ϵ] whose k-th derivatives are 𝒪(1/log(b)ϵ^-k), and define the metric
    
    g_b,ϵ:= dr^2+ sin((1+χ_b,ϵ)r)g_𝕊^3Γ.

    Assume that ϵ→ 0, b→ +∞ and bϵ→ 0, the orbifold metric therefore becomes arbitrarily close in the Gromov-Hausdorff sense to 𝕊^4Γ. Moreover, the sectional curvatures of g_b,ϵ are bounded below by 1- C/log b→ 1 for some uniform C>0. Let us finally glue t(g_EH+2th_2), where h_2 is asymptotic to -1/3r_e^2g_𝕊^3Γ at the singular points for t_max≪ϵ^2, so that the gluing happens in 0⩽ r<ϵ where the metric g_b,ϵ equals dr^2+ sin^2(2r)g_𝕊^3Γ just like on the sphere of radius 1/2 whose sectional curvatures are constant equal to 4. For r<ϵ, the controls are the same as on g^A_t/4, and therefore the metric satisfies ⩾ 3 by (<ref>) since 4(3-(1-η(t_max)))>3. Since the metric satisfies ⩾ 3-C/log(b) for larger r, we can simply rescale it a little to ensure that we have >3 everywhere.






§ A GENERAL OBSTRUCTION FOR SPHERICAL AND HYPERBOLIC ORBIFOLDS


Let us finally exhibit an obstruction to the desingularization of spherical and hyperbolic orbifolds by general Ricci-flat orbifolds (not necessarily Kähler) in Theorem <ref>. We will deduce from it that there does not exist any smooth desingularization of spherical or hyperbolic orbifolds whose blow ups are integrable Ricci-flat ALE spaces in Corollary <ref>.



 §.§ A general infinitesimal deformation for Ricci-flat ALE spaces


On (ℝ^4Γ,g_e), the vector field 2r_e∂_r_e is a conformal Killing vector field. It is moreover the gradient of the function u:=r_e^2 which is a solution to -∇_e^*∇_e u = 8, and we have ℒ_∇_e ug_e = _g_eu = 4g_e. On a Ricci-flat ALE we can approximate this by an infinitesimal deformation.


    Let (N,g_b) be a Ricci-flat ALE orbifold asymptotic to ℝ^4Γ, and consider a diffeomorphism Φ between neighborhoods of the infinities of N and ℝ^4Γ.
    
    Then, there exists a unique vector field X on (N,g_b) such that Φ^*X = 2r_b∂_r_b + o(r_b), and ∇^*∇ X =0. We actually have X=∇ u, where u is the unique solution of -∇^*∇ u = 8, such that u = r_b^2 + o(1).
    
    Moreover, (ℒ_Xg_b)^∘ = ℒ_Xg_b-4g_b, the traceless part of ℒ_Xg_b is an infinitesimal Ricci-flat deformation of g_b which is trace-free and divergence-free.


    The proof of the existence and the uniqueness of the function u can be found in the proof of Theorem B of <cit.>. The symmetric 2-tensor (ℒ_∇ ug_b)^∘ is indeed an infinitesimal deformation of g_b because the equation = 0 is invariant by scaling and pull-back by diffeomorphisms, and the divergence and the trace of (ℒ_∇ ug_b)^∘ = 2_g_bu-4g_b vanish because -∇^*∇ u=8. 
    
    Moreover, (ℒ_∇ ug_b)^∘ vanishes exactly for flat cones. Indeed, if it vanishes, then ∇ u is a conformal Killing vector field and therefore generates a family of conformal diffeomorphisms. By considering the maximum of the pointwise norm of the curvature of (N,g_b) which is preserved by this family of diffeomorphism, we see that it has to vanish.


    This deformation is integrable because it simply comes from a rescaling and a change of coordinates. 





 §.§ Obstructions to the desingularization of spherical and hyperbolic orbifolds


Let us now use the above deformation 𝐨_1:= (2_g_bu-4g_b) in order to deduce some general obstructions to the desingularization of spherical and hyperbolic orbifolds.

    Let (N,g_b) be a Ricci-flat ALE orbifold and H_2 be the quadratic terms of a spherical or hyperbolic metric in geodesic coordinates, and O_1^4 terms of order r_b^-4 of the deformation 𝐨_1= (2_g_bu-4g_b).
    
    Then, we have
    
    ∫_𝕊^3(3⟨ H_2, O_1^4⟩_g_e + O_1^4(B_eH_2,∂_r_e))dv ≠ 0,

    and therefore the perturbation of g^D_t to an Einstein metric orthogonally to 𝐎̃(g^D_t) is always obstructed.



Let (N,g_b) be a Ricci-flat ALE orbifold asymptotic to a flat cone ℝ^4Γ, and let 𝐨_1= (2_g_bu-4g_b) = O^4_1 + 𝒪(r_b^-5) be the infinitesimal deformation of Proposition <ref>. Let us start by proving that O_1^4(∂_r_e,∂_r_e) does not vanish. There exists a compact K⊂ M such that M\ K is foliated by hypersurfaces Σ_ρ whose mean curvature is constant equal to 3/ρ. If we denote Ω_ρ the interior of Σ_ρ, then, by <cit.> the following limit exists and is finite:
    
    𝒱:= lim_ρ→∞[_g_b(Ω_ρ)-_g_e(B(0,ρ)Γ) ],

    and we actually have 𝒱⩽ 0, with equality if and only if (N,g_b)=(ℝ^4Γ,g_e).
    
    Moreover, let u be the unique solution of -∇^*∇ u = 8 with u = r_b^2  +o(1), then, we actually have
    
    u = r_b^2 +b/r_b^2 + 𝒪(r_b^-3),

    and by the proof of <cit.>, we have the explicit value
    
    b= -4𝒱/|∂ B(0,1)Γ|⩾ 0.

    We also deduce the following development of 𝐨_1= (2_g_bu-4g_b),
    
    𝐨_1(∂_r_b,∂_r_b) = 8b/r_b^4 +𝒪(r_b^-5)

    which is strictly positive if g_b is not flat.
    

    Now, for a hyperbolic metric, we have H_2 = r_e^4/3(α_1^2+α_2^2+α_3^2) in geodesic coordinates, and for a spherical metric, H_2 = -  r_e^4/3(α_1^2+α_2^2+α_3^2). Notice moreover that, since g_e = dr_e^2 +r_e^2(α_1^2+α_2^2+α_3^2), we have 0=_g_e O_1^4 = O_1^4(∂_r_e,∂_r_e) + ⟨ r_e^2(α_1^2+α_2^2+α_3^2), O_1^4⟩ and therefore

    ⟨ r_e^2(α_1^2+α_2^2+α_3^2), O_1^4⟩ =   _g_e O_1^4 - O_1^4(∂_r_e,∂_r_e)
    
        =    - O_1^4(∂_r_e,∂_r_e).

For the other part of the obstruction, we have B_e(r_e^4(α_1^2+α_2^2+α_3^2)) = 6 r_e ∂_r_e. Indeed, r_e^4(α_1^2+α_2^2+α_3^2) = r_e^2g_e-r_e^2dr_e^2, and we have

    B_e(r_e^2g_e)    = δ_e(r_e^2g_e) + 1/2d_e(r_e^2g_e)
        = -2r_e g_e(∂_r_e,.) + 4rdr_e
       = 2r_e dr_e,

and

    B_e(r_e^2dr_e^2)    = δ_e(r_e^2dr_e^2) + 1/2d_e(r_e^2dr_e^2)
        = δ_e(∑_ijx^ix^jdx^idx^j) + r_e dr_e
       = - ∑_i≠ j x^jdx^j - 2∑_j x^jdx^j+ r_e dr_e
       =-4r_e dr_e.

Finally, for r_e = 1,

    O_1^4(B_e(r_e^4(α_1^2+α_2^2+α_3^2)),∂_r_e) = 6 O_1^4(∂_r_e,∂_r_e).

The obstruction generated by 𝐨_1, that is ∫_𝕊^3(3⟨ H_2, O_1^4⟩_g_e + O_1^4(B_eH_2,∂_r_e))dv therefore never vanishes by (<ref>) and (<ref>).



    It is also possible to extend the deformations given by the Killing vector fields at infinity to generate more obstructions, but it is not clear if a Ricci-flat ALE space can have vanishing terms of order r_b^-4. Indeed, the quantity 𝒱 is global and does not tell anything on the asymptotics of the metrics, but as we just saw, it tells something about their derivatives along the deformation (ℒ_∇_b ug_b)^∘.


We deduce that we get a general obstruction to a Gromov-Hausdorff desingularization if we assume that the Ricci-flat ALE spaces are integrable.


    Let (M_o,g_o) be a compact spherical or hyperbolic orbifold. Then, there does not exist any sequence of Einstein manifolds (M_i,g_i) such that 
    (M_i,g_i) (M_o,g_o),

    while the non-flat limits of (M_i,g_i/t_i,p_i) for t_i→ 0, t_i>0 and p_i∈ M_i converge to smooth and integrable Ricci-flat ALE manifolds (which means that there are no trees of singularities forming).


    According to Theorem <ref>, if the quadratic terms of the development of g_o are H_2, the obstruction induced by the deformation 𝐨_1 is 

    ∫_𝕊^3(3⟨ H_2, O_1^4⟩_g_e + O_1^4(B_eH_2,∂_r_e))dv =0,

which is never satisfied according to Theorem <ref>.

The obstruction of Theorem <ref> is therefore never satisfied for spherical and hyperbolic metrics, and it is impossible to desingularize it by Ricci-flat ALE manifolds which are integrable.


The obstruction to the desingularization of spherical and hyperbolic manifolds is therefore identified, but we need the technical integrability assumption to deduce a Gromov-Hausdorff obstruction thanks to it. We believe that this is only a technicality and conjecture the following statement.


    Singular spherical and hyperbolic orbifolds cannot be Gromov-Hausdorff limits of smooth Einstein manifolds.
















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