S Ébastien Gontard

ON THE K ÄHLER-EINSTEIN METRIC AT STRICTLY PSEUDOCONVEX POINTS

Keywords: 2010 Mathematics Subject Classification. 32Q20, 32T15, 53C25 Strictly Pseudoconvex Domains, Kähler-Einstein Metrics, Holomorphic Sectional Curvatures. 1

We prove a local boundary regularity result for the complete Kähler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.

Introduction

In 1980, S.-Y. Cheng and S.-T. Yau proved that every bounded strictly pseudoconvex domain Ω ⊂ C n , n ≥ 2, with boundary of class C 7 , admits a complete Kähler-Einstein metric of negative Ricci curvature (for convenience, we will only work with Ricci curvature = -(n + 1)). Namely, they proved that there exists a (unique) solution g ∈ C ω (Ω) to the Monge-Ampère equation [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] Det g i j = e (n+1)g on Ω, satisfying the following boundary condition:

(2) g = +∞ on ∂Ω.

By comparing this solution to the approximate solutions constructed by C. Fefferman in [START_REF] Fefferman | Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains[END_REF], they proved that if Ω is bounded, strictly pseudoconvex with boundary of class C max(2n+9,3n+6) , then e -g ∈ C n+1+δ Ω for every δ ∈ 0, 1 2 , and the holomorphic sectional curvatures of this metric tend to -2, which are the curvatures of the unit ball equipped with its Bergman-Einstein metric, at any boundary point. Note that if the boundary is of class C ∞ , J. Lee and R. Melrose proved that e -g ∈ C n+1+δ Ω for every δ ∈ [0, 1[ (see [START_REF] Lee | Boundary behaviour of the complex Monge-Ampère equation[END_REF]), and this regularity is optimal in general. We prove a local version of the result of S.-Y. Cheng and S.-T. Yau. Namely, we prove the following theorem:

Theorem 1. Let Ω ⊂ C n , n ≥ 2, and q ∈ ∂Ω. Assume that there exists a neighborhood of q on which ∂Ω is strictly pseudoconvex and of class C k with k ≥ max(2n + 9, 3n + 6). Moreover, assume that Ω carries a complete Kähler-Einstein metric induced by a function g that satisfies conditions

(1) and [START_REF]The Einstein-Kähler metric on {|z| 2 + |w| 2p < 1}[END_REF]. Then there exists an open set U ⊂ C n containing q such that for every δ ∈ 0, 1 2 , we

have: e -g ∈ C n+1+δ Ω ∩ U .

Note that J. Bland already obtained this result in the case of "nice" strictly pseudoconvex boundary points, and also obtained that e -g ∈ C n 2 +δ Ω ∩ U in the general case (see [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF]). Regarding the curvature behavior, we prove the following: Theorem 2. Let Ω ⊂ C n , n ≥ 2, and q ∈ ∂Ω. Assume that there exists a neighborhood of q on which ∂Ω is strictly pseudoconvex and of class C k with k ≥ max (2n + 9, 3n + 6). Moreover, assume that Ω carries a complete Kähler-Einstein metric induced by a function g that satisfies conditions (1) and [START_REF]The Einstein-Kähler metric on {|z| 2 + |w| 2p < 1}[END_REF]. Then, [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] sup v,w∈S(0,1)

Bis g,z (v, w)

+ 1 + | v; w g,z | 2 v; v 2 g,z w; w 2 g,z -→ z→q 0.
Here and from now on, Bis g,z (v, w) (respectively v; w g,z ) stands for the holomorphic bisectional curvature (respectively the Hermitian scalar product) of the Kähler metric induced by the potential g, at point z, between the directions v and w (a more precise definition is given in Section 1).

Especially, using the results on the existence of Kähler-Einstein metrics of N. Mok and S.-T. Yau in the case of bounded pseudoconvex domains (see [START_REF] Mok | Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions[END_REF]), and of A. Isaev in the case of pseudoconvex tube domains with unbounded base (see [START_REF] Isaev | Kähler-Einstein metric on Reinhardt domains[END_REF]), we directly deduce:

Corollary 3. Let n ≥ 2.
Let Ω ⊂ C n be either a bounded pseudoconvex domain with boundary of class C 2 , or a tube domain whose base is convex and does not contain any straight line. Let g ∈ C ω (Ω) be the Kähler-Einstein potential on Ω that satisfies conditions (1) and [START_REF]The Einstein-Kähler metric on {|z| 2 + |w| 2p < 1}[END_REF]. Let q ∈ ∂Ω. Assume that there exists a neighborhood of q on which ∂Ω is strictly pseudoconvex and of class C k with k ≥ max(2n + 9, 3n + 6). Then there exists an open set U ⊂ C n containing q such that we have:

∀δ ∈ 0, 1 2 , e -g ∈ C n+1+δ Ω ∩ U .
Moreover, we have the following curvature behaviour: sup v,w∈S(0,1)

Bis g,z (v, w)

+ 1 + | v; w g,z | 2 v; v 2 g,z w; w 2 g,z -→ z→q 0.
The curvature behavior (3) can also be obtained in pseudoconvex domains, at boundary points for which the squeezing tends to one (for precise reference about the squeezing function, see for instance [START_REF] Yeung | Geometry of domains with the uniform squeezing property[END_REF]). Namely: Theorem 4. Let Ω ⊂ C n be a pseudoconvex domain, n ≥ 2, and q ∈ ∂Ω. Assume that the squeezing function of Ω tends to one at q. Moreover, assume that Ω carries a complete Kähler-Einstein metric induced by a function g solving equation [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] with condition (2) on Ω. Then, sup v,w∈S(0,1)

Bis g,z (v, w) + 1 + | v; w g,z | 2 v; v 2 g,z w; w 2 g,z -→ z→q 0.
We refer the reader to the proof of Theorem 4 for a more precise statement in terms of the squeezing function of the domain. In comparison with Theorems 1 and 2, Theorem 4 requires neither regularity assumptions on the boundary of the domain nor the strict pseudoconvexity at q, but gives no boundary regularity for the Kähler-Einstein potential. However, it is difficult to find geometric condition ensuring that the squeezing tends to one at a boundary point of a given domain. We can apply Theorem 4 at C 2 strictly pseudoconvex boundary points of a domain admitting a Stein neighborhood basis (see [START_REF] Joo | On boundary points at which the squeezing function tends to one[END_REF]), at C 2 strictly convex boundary points of bounded domains (see [START_REF] Kim | On the uniform squeezing property of bounded convex domains in C n[END_REF]), but also at every boundary point of the Fornaess-Wold domain, which is convex but not strictly pseudoconvex and has a boundary of class C 2 (see [START_REF] Erik | A non-strictly pseudoconvex domain for which the squeezing function tends to one towards the boundary[END_REF]). This paper is organized as follows. In Section 1, we introduce some notations and formulas that will be used in the other sections. In Section 2, we recall the construction of asymptotically Kähler-Einstein metrics developped by C. Fefferman in [START_REF] Fefferman | Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains[END_REF]. We provide details about the regularity of the functions involved in the construction and on their defining set. In Section 3, we first estimate the norm of the gradient of the difference between the potential of the Kähler-Einstein metric and the potential of an asymptotically Kähler-Einstein metric constructed in Section 2. Then we use these estimates to improve the C 0 estimate. We also derive the higher order estimates, and we use these to prove Theorems 1 and 2 at the end of the Section. In Section 4 we prove Theorem 4.

Acknowledgements. I would like to thank Professor S. Fu and Professor J. Bland for their kind hospitality during my visit in their institutions and the fruitful discussions.

Preliminaries and notations

Throughout the paper, we use Einstein summation notation.

1.1. Algebra. We denote by M n (C) the set of square matrices of size n, with complex coefficients. In this set, we denote by 0 the null matrix and by I the identity matrix.

Let A = A i j , B = B i j ∈ M n (C), v = (v i ) ∈ C n , w = (w j ) ∈ C n . If A is invertible, we note A i j = A -1 . It is characterized by the relations A i kA k j = A i kA k j = 1 if i = j, 0 otherwise. Especially, T r A -1 B = A i j B j ī,
where T r denotes the trace function. We denote by Det (A) the determinant of A. We denote by t A = A j ī the transpose matrix of A, and by A = A i j its conjugate. We denote by H n := {A ∈ M n (C) / t A = A} the space of Hermitian matrices of order n.

If A ∈ H n , we note v; w A := A i j v i w j . Recall that v; v A ∈ R. If A, B ∈ H n ,
we define the following relations:

B ≥ A ⇐⇒ ∀v ∈ C n \ {0}, v; v B ≥ v; v A , B > A ⇐⇒ ∀v ∈ C n \ {0}, v; v B > v; v A .
We note

H + n := {M ∈ H n /M ≥ 0} and H ++ n := {M ∈ H n /M > 0}. If A ∈ H + n , we note |v| A := v; v 1 2
A . We will need the following facts that we do not prove: 1.2.1. Functions. We denote by C k+ (U ) the set of real valued functions having derivatives up to order k and such that all these derivatives are Hölder of exponent , and by C ω (U ) the set of analytic functions in U . We simply note

Proposition 1.1. (1) Let A ∈ H + n .Then there exists R ∈ H + n such that R 2 = A. The matrix R is called a square root of A. (2) Let A ∈ H + n . Then 0 ≤ A ≤ T r (A) I. (3) Let A ∈ H ++ n . Then there exist 0 < λ ≤ Λ such that λI ≤ A ≤ ΛI. 1.2.
C k (U ) := C k+0 (U ). A function f ∈ C k+ (U ) is in C k+ U if all its derivatives up to order k extend continuously to the closure U of U . If f ∈ C k+ (U ) and (i 1 , j 1 , . . . , i n , j n ) ∈ N 2n satisfies s := n l=1 (i l + j l ) ≤ k, we denote by f i 1 j 1 ...injn := ∂ s f ∂z i 1
1 ∂z 1 j 1 . . . ∂z in n ∂z n jn . In particular, this notation is consistent with the notation of complex matrices introduced above. Also, observe that if f ∈ C 1 (U ), then for every 1 ≤ j ≤ n, we have fj = f j . A function f ∈ C 2 (U ) is plurisubharmonic at z, respectively strictly plurisubharmonic at z, if

f i j (z) ≥ 0, respectively f i j (z) > 0. A function f ∈ C 2 (U ) is (strictly) plurisubharmonic in U if
it is (strictly) plurisubharmonic at every point of U . We will need the following fact that we do not prove: Proposition 1.2. Let U ⊂ C n be an open bounded set and let f ∈ C 2 U be a strictly plurisubharmonic function. Then there exist constants 0 < λ ≤ Λ such that λI ≤ f i j ≤ ΛI on U .

1.2.2. Kähler metrics. A Kähler metric in U is an element of C (U, H ++ n ), that is, a matrix g i j with continuous coefficients in U and such that for every z ∈ U , g i j (z) ∈ H ++ n . We say that a Kähler metric g i j is induced by a function u ∈ C 2 (U ), called a (Kähler) potential for g i j , if

u i j = g i j in U . If v = (v i ) ∈ C n , w = (w j ) ∈ C n , f, g ∈ C 2 (U )
and g is a Kähler potential in U , we define the following quantities:

• v; w g := v; w (g i j ) = g i j v i w j : the scalar product of v and w, for the metric g i j .

• |v| g := v; v 1 2 g : the norm of v for g i j .

• Ric(g) := -Log Det g i j : the Ricci form of g i j .

• |∇f | g := |(f i )| (g i j ) = g i j f i fj 1 2 : the norm of the complex gradient of f for g i j .

• ∆ g f := T r g i j f i j = g i j f j ī: the Laplacian of f for g i j .

Moreover, if g ∈ C 4 (U ) and v, w = 0, we also define:

• ∀1 ≤ i, j, k, l ≤ n, R i jk l(g) := -g i jk l + 1≤p,q≤n
g ik pg pq g q jl: the curvature coefficients of g i j .

• Bis g (v, w)

:= R i jk l(g)v i v j w k w l |v| 2 g |w| 2 g
: the holomorphic bisectional curvature of g i j , between directions v and w.

• H g (v) := Bis g (v, v): the holomorphic sectional curvature of g i j , in the direction v.

If needed, we will specify the point z at which these quantities are computed by using the following notations: v; w g,z , Ric(g)(z), |∇f | g,z , ∆ g f (z), R i jk l(g)(z), Bis g,z (v, w), etc.

In the special case of the usual metric on C n , that is to say g = |•| 2 (or, equivalently, gi j = I), we simply note v; w , respectively |∇f |, instead of v; w g, respectively |∇f | g. We proceed likewise with the other notations.

Recall that the metric induces a distance function, that we denote by d g . We say that the metric is complete if the space (U, d g ) is complete. We say that a Kähler metric induced by a potential g ∈ C 4 (U ) is Kähler-Einstein if there exists λ ∈ R such that Ric(g) i j = λ g i j . We point out that in this paper, all the involved Kähler-Einstein metrics satisfy Ric(g) i j = -(n + 1) g i j .

Note that by definition every strictly plurisubharmonic function in U induces a Kähler potential in U . There is another way to construct Kähler potentials from strictly plurisubharmonic negative functions in U :

Proposition 1.3. Let ψ ∈ C 2 (U ) be a negative strictly plurisubharmonic function. Set g := -Log (-ψ). Then g is a Kähler potential in U , and the following formulas hold in U :

(-ψ) g i j = ψ i j + ψ i ψj -ψ , (4)
g i j = (-ψ) ψ i j -(-ψ) (ψ i j )(ψiψj)(ψ i j ) -ψ+|∇ψ| 2 ψ . ( 5
)
Proof of Proposition 1.3. The function g is well defined and of class C 2 in U by construction, and formula (4) directly comes from the chain rule. Let R be a square root of

ψ i j . Then R is invertible because Det (R) 2 = Det ψ i j = 0. Set B := R -1 ψ i ψj -ψ R -1 and A := (-ψ) R -1 g i j R -1 = I + B.
Since the rank of B is 1, we have

B 2 = T r (B) B. Since T r (B) = |∇ψ| 2 ψ -ψ = ψ i j ψ i ψj -ψ ≥ 0 > -1
, we can do the following computation:

A I - B 1 + T r (B) = I + -1 1 + T r(B) + 1 - T r(B) 1 + T r(B) B = I,
and likewise we have

I - B 1 + T r (B) A = I.
Hence A is invertible, and its inverse is

A -1 = I - B 1 + T r (B) = I -R -1 ψ i ψj -ψ + |∇ψ| 2 ψ R -1 .
Therefore we obtain the formula (5):

g i j = (-ψ)R -1 A -1 R -1 = (-ψ) ψ i j -(-ψ) ψ i j ψ i ψj ψ i j -ψ + |∇ψ| 2 ψ .
Let Ω ⊂ C n be a domain, let k ≥ 1 be an integer, and let q ∈ ∂Ω. We say that ∂Ω is of class C k in a neighborhood of q if there exists a defining function of class

C k of Ω in a neighborhood of q, that is, an open set V ⊂ C n containing q, and a function ψ ∈ C k (V ) satisfying Ω ∩ V = {ψ < 0} and ∀z ∈ ∂Ω ∩ V = {ψ = 0}, |∇ψ| z = 0. If k ≥ 2 and ∂Ω is of class C k in
a neighborhood of q, we say that ∂Ω is strictly pseudoconvex in a neighborhood of q if there exists a bounded open set V ⊂ C n containing q, and a function ψ ∈ C k (V ) satisfying Ω ∩ V = {ψ < 0}, ∀z ∈ ∂Ω ∩ V = {ψ = 0}, |∇ψ| z = 0 and ψ i j > 0 in V . The function ψ is a strictly plurisubharmonic defining function for ∂Ω ∩ V .

The two following results will be needed in Sections 2 and 3:

Proposition 1.4.
Let Ω ⊂ C n be a domain, let n ≥ 2 be an integer, and let q ∈ ∂Ω. Assume that there exists a neighborhood of q on which ∂Ω is of class

C 1 . Let V ⊂ C n be an open set containing q, let ψ ∈ C 1 (V ) be a defining function for ∂Ω ∩ V . Let U ⊂ U ⊂ V be a bounded open set containing q.
Then, there exists a constant > 0 such that inf

U ∩{|ψ|≤ } |∇ψ| > 0.
Proof of Proposition 1.4. We argue by contradiction. Then there exists a sequence

(z i ) i∈N ∈ U N such that lim i→+∞ ψ(z i ) = lim i→+∞ |∇ψ| z i = 0.
Since U is compact, we can assume, up to extracting a subsequence, that (z i ) i∈N converges in U . Denote by z its limit. By continuity of ψ at z, the condition lim Proposition 1.5. Let Ω ⊂ C n be a domain, and q ∈ ∂Ω. Assume that there exists a neighborhood of q on which ∂Ω is strictly pseudoconvex and of class C 2 . Let V ⊂ C n be a bounded domain containing q, ψ ∈ C 2 (V ) be a strictly plurisubharmonic defining function for ∂Ω ∩ V . Let g := -Log (-ψ).

i→+∞ ψ (z i ) = 0 implies ψ(z) = 0, which means that z ∈ ∂Ω ∩ U ⊂ ∂Ω ∩ V .
Then for every bounded open set U ⊂ U ⊂ V there exist 0 < λ ≤ Λ such that the following inequalities hold on Ω ∩ U :

(6) λ ψ 2 -ψ + |∇ψ| 2 I ≤ g i j ≤ Λ (-ψ) I.
Proof of Proposition 1.5. We use formula [START_REF] Fefferman | Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains[END_REF] and notations of Proposition 1.3 with U replaced with Ω ∩ U . We also use the notations introduced in the proof of Proposition 1.3.

According to Proposition 1.1, we have

B 1 + T r(B) ∈ H + n , hence 0 ≤ B 1 + T r(B) ≤ T r(B) 1 + T r(B) I. Since A -1 = I - B 1 + T r(B)
, we deduce

1 1 + T r(B) I = 1 - T r(B) 1 + T r(B) I ≤ A -1 ≤ I. Since -ψ > 0, we deduce the following: -ψ -ψ + |∇ψ| 2 ψ I ≤ 1 -ψ R g i j R ≤ I, ψ 2 -ψ + |∇ψ| 2 ψ ψ i j ≤ g i j ≤ (-ψ) ψ i j .
Moreover, since ψ i j is continuous on the compact set U , there exist 0 < λ ≤ Λ such that λI ≤ ψ i j ≤ ΛI on U . Hence:

λ ψ 2 -ψ + |∇ψ| 2 ψ I ≤ g i j ≤ Λ (-ψ) I.

Construction of local asymptotically Kähler-Einstein metrics

Let V be an open set. Let k ≥ 2 be an integer. If ψ ∈ C k (V ), its Fefferman functional is defined by

J(ψ) := (-1) n Det ψ (ψj) t (ψ i ) (ψ i j )
.

Then J(ψ) ∈ C k-2 (V ). We observe that

J(ψ) = ψ n+1 Det -Log(ψ) i j on {ψ > 0},
and that the function

F := Log (J(ψ)) = -Ric (-Log(ψ)) -(-(n + 1)Log(ψ))
is well defined on {ψ > 0} ∩ {Det -Log(ψ) i j > 0}. Especially, if -Log(ψ) i j > 0, F is well defined and measures the defect of (-Log(ψ)) to be the potential of a Kähler-Einstein metric: the metric -Log(ψ) i j is Kähler-Einstein if and only if J (ψ) = 1.

Let Ω ⊂ C n be a domain and q ∈ ∂Ω. Assume that there exists a neighborhood V of q such that ∂Ω ∩ V is strictly pseudoconvex and of class C k with k ≥ 2n + 4. Without loss of generality, we may assume that V is a bounded domain. We describe Fefferman's iterating process in V . Let ϕ ∈ C k (V ) be a strictly plurisubharmonic defining function for ∂Ω∩V . Let U 0 := {J(-ϕ) > 0}.

Since ϕ ∈ C 2 (V ) and J(-ϕ) > 0 on ∂Ω ∩ V , the set U 0 contains ∂Ω ∩ V and is open. Consider the following constructions on U 0 :

ϕ (1) := ϕ J(-ϕ) 1 n+1
and, for 2

≤ l ≤ n + 1, ϕ (l) := ϕ (l-1) 1 + 1 -J(-ϕ (l-1) ) l(n + 2 -l) .
Then, for every 1 ≤ l ≤ n + 1, ϕ (l) is well defined on U 0 and ϕ (l) ∈ C k-2l (U 0 ). Moreover, according to the computations done by C. Fefferman in [START_REF] Fefferman | Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains[END_REF], we have

J -ϕ (l) -1 (-ϕ) l ∈ C k-2l-2 (U 0 ).
This ensures that for every integer 1 ≤ l ≤ n + 1, the sets

U l := 1 -J(-ϕ (l) ) < 1 2 are
open and contain ∂Ω ∩ V . Consequently, there exist positive constants r and R such that the set

U := ∩ n+1 l=0 U l ∩((B(q, R) ∩ ∂Ω) + B(0, r)
) is open, contains q, satisfies U ⊂ V , and on which every ϕ (l) is a C k-2l defining function for ∂Ω ∩ U . Then according to Proposition 1.4, we can assume (by taking smaller r and R if necessary) that min Since ∂Ω ∩ V is strictly pseudoconvex, we can (by changing ϕ (l) to ϕ (l) 1 + tϕ (l) with t > 0 small and taking smaller r and R if necessary) assume that each ϕ (l) is strictly plurisubharmonic on U . Finally, the above construction gives, for every 1 ≤ l ≤ n + 1:

Log J -ϕ (l) (-ϕ) l = Log 1 + J -ϕ (l) -1 (-ϕ) l = J -ϕ (l) -1 (-ϕ) l 1 + +∞ m=1 (-1) m m + 1 J -ϕ (l) -1 m ∈ C k-2l-2 U .
Let us summarize all these facts: Proposition 2.1. Let Ω ⊂ C n be a domain and let q ∈ ∂Ω. Assume that there exists a neighborhood V of q such that ∂Ω ∩ V is strictly pseudoconvex and of class C k with k ≥ 2n + 4. Then there exists a bounded domain U containing q, and a collection of functions ϕ (l) 1≤l≤n+1 satisfying, for every 1 ≤ l ≤ n + 1:

(1) ϕ (l) ∈ C k-2l U , (2) Ω ∩ U = {ϕ (l) < 0} ∩ U , (3) inf z∈U |∇ϕ (l) | z > 0, (4) ϕ (l) is strictly plurisubharmonic on U , (5) 1 -J -ϕ (l) ≤ 1 2 on U , (6) J(-ϕ (l) )-1 (-ϕ) l ∈ C k-2l-2 U , ( 7
) ϕ (l) ϕ ∈ C k-2l U and is positive on U , ( 8
) Log(J(-ϕ (l) )) (-ϕ) l ∈ C k-2l-2 U .
Moreover, we have inf

z∈U |∇ϕ| z > 0.
Remark 2.2. • Especially, conditions (1) to (4) imply that for every integer

1 ≤ l ≤ n + 1, the function ϕ (l) is a strictly plurisubharmonic defining function of ∂Ω ∩ U of class C k-2l . • If k ≥ 3n + 5, then all the functions ϕ (l) , J -ϕ (l) -1 (-ϕ) l , ϕ (l) ϕ and Log J -ϕ (l) (-ϕ) l belong to C n+1 U . If k ≥ 3n+6, then all the aforementionned functions belong to C n+2 U ⊂ ∩ 0≤δ≤1 C n+1+δ U .
• The metrics -Log -ϕ (l) i j are called "asymptotically Kähler-Einstein" on ∂Ω ∩ U , since they

satisfy the condition J -ϕ (l) (z) -→ z→∂Ω∩U 1 (recall that -Log -ϕ (l) i j is Kähler-Einstein on Ω ∩ U if and only if J -ϕ (l) = 1 on Ω ∩ U ).

Local boundary regularity

In this Section, we fix an integer n ≥ 2, a domain Ω ⊂ C n and a point q ∈ ∂Ω. We assume that Ω satisfies the hypothesis of Theorem 1. Namely, there exists a complete Kähler-Einstein metric induced by a potential w ∈ C ω (Ω) that satisfies conditions (1) and (2), and there exists a neighborhood V of q such that ∂Ω∩V is strictly pseudoconvex and of class C k with k ≥ max (2n + 9, 3n + 6). Thus, we can apply Proposition 2.1, and use the same notations introduced therein. One of the main ideas to prove Theorem 1 is to compare the complete Kähler-Einstein metric w i j to the aymptotically Kähler-Einstein metrics induced by the strictly plurisubharmonic defining functions ϕ (l) 1≤l≤n+1 as follows. Let 1 ≤ l ≤ n + 1, and set

η := ϕ (l) ϕ , w := -Log -ϕ (l) = -Log (-ηϕ) , F := Log J -ϕ (l) = Log (J (-ηϕ)) .
Then, according to points ( 5), ( 6), ( 7), (8

) of Proposition 2.1, η ∈ C k-2l U , w ∈ C k-2l (Ω ∩ U ), F ∈ C k-2l-2 U , F (-ϕ) l ∈ C k-2l-2
U and w and F are related on Ω ∩ U by the condition [START_REF] Isaev | Kähler-Einstein metric on Reinhardt domains[END_REF] Det w i j = e (n+1)w e F .

Let u := w -w. Then, on Ω ∩ U , u solves the Monge-Ampère equation ( 8)

Det w i j + u i j = e (n+1)u-F Det w i j .
Since w is real analytic in Ω and w ∈ C k-2l (Ω ∩ U ), then u ∈ C k-2l (Ω ∩ U ) . So, for each integer 1 ≤ l ≤ n + 1, we have an asymptotically Kähler-Einstein metric w i j on ∂Ω ∩ U , for which the defect of being Kähler-Einstein is encoded in the function F , and we study the difference between this metric and the Kähler-Einstein metric w i j on Ω ∩ U . More precisely, we study the boundary regularity of the difference of their potentials, namely the function u.

3.1. C 1 estimate and consequences. Whether global (see [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF]) or local (see [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF]), the study of the boundary behavior of u relies on its gradient estimate, which relies on the comparison between the metrics w i j and w i j (see condition [START_REF] Lee | Boundary behaviour of the complex Monge-Ampère equation[END_REF]). The gradient estimate enables to deduce the boundary behavior of u, and then the boundary behavior of the higher order derivatives of u by use of Schauder theory. All these estimates depend on the regularity of the gradient of F (-ϕ) l , for which we have the following result: Proposition 3.1. Under the hypothesis of Theorem 1, and with the notations introduced at the beginning of Section 3, we have

|∇F | 2 w (-ϕ) 2l-1 ∈ C k-2l-3 Ω ∩ U .
In particular, there exists a positive constant c ∇ , such that the following holds on Ω ∩ U :

(9) |∇F | 2 w ≤ c ∇ (-ϕ) 2l-1 . Proof of Proposition 3.1. Let 1 ≤ i, j ≤ n. Then, according to point (8) of Proposition 2.1, F i (-ϕ) l-1 = l F ϕ i (-ϕ) l + ϕ F (-ϕ) l i ∈ C k-2l-
w i j -ϕ = ψ ϕ w i j -ψ = ψ ϕ   ψ i j + ψ i j ψ i ψj ψ i j ij -ψ + |∇ψ| 2 ψ    ∈ C k-2l-2 Ω ∩ U ,
where ψ := ϕ (l) .

Hence |∇F | 2 w (-ϕ) 2l-1 = w i j -ϕ F i (-ϕ) l-1 Fj (-ϕ) l-1 ∈ C k-2l-3 Ω ∩ U .
We improve the gradient estimate obtained in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] by using the computations of [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] in a different way. Then we proceed exactly as in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] to obtain the estimates of the other derivatives of u. Proposition 3.2. Under the hypothesis of Theorem 1, and with the notations indroduced at the beginning of Section 3 and in Proposition 3.1, for every γ ∈]0; min(2n + 1, 2l -1)[, there exist positive constants c and such that |∇u| 2 w ≤ c (-ϕ) γ on Ω ∩ U ∩ {|ϕ| < }. Remark 3.3. • Proposition 3.2 improves the results obtained in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] in the sense that ∂Ω ∩ U is not required to be "nice". • Proposition 3.2 is a local version of Proposition 6.4 in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF].

• The proof of Proposition 3.2 will use the fact that |∇u| 2 w ∈ C 2 (Ω ∩ U ) and is bounded from above, which is true as long as k ≥ 2n + 5 (see page 297 of [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] for further details).

• It will also use the fact that Lemma II in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] actually works for C 2 functions that are bounded below (see Lemma 3.4 for a version that fits to our situation).

Proof of Proposition 3.2. The strategy of the proof of Proposition 6.4 in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] is first to show that there exists δ 0 > 0 such that for every 0 < α < n, 0 ≤ β < n + 1 and 0 < δ ≤ δ 0 satisfying α + β + δ ≤ 2l -1, there exist positive constants and c such that the following inequality holds on Ω ∩ {|ϕ| ≤ }:

∆ w |∇u| 2 w (-ϕ) β -c(-ϕ) α > n + 1 + nβ -β 2 2 |∇u| 2 w (-ϕ) β -c(-ϕ) α ,
and then to apply the generalized maximum principle and choose suitable constants α and β to get the conclusion.

In our case, we wish to follow the same strategy when we restrict our considerations to Ω ∩ U . We focus our attention on explaining the necessary modifications in the proof of Proposition 6.4 in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF], keeping in mind that we look for local estimates in a neighborhood of ∂Ω ∩ U . For that purpose, we first explain the dependence of the constants c 1 , . . . , c 9 with respect to the local data in order to obtain conditions (15) and ( 16). Then we use formulas ( 15) and (16) to complete the proof. For each constant, we refer precisely to the condition in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] where it is defined.

In the sequel, 0 < α < n and 0 ≤ β < n + 1.

• We apply the first Proposition of page 297 in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] to derive the existence of positive constants and δ 0 such that we have the following on Ω ∩ U ∩ {|ϕ| ≤ }:

(10)

w i j = 1 + O (-ϕ) δ 0 w i j ,
which means that there exists a positive constant c 1 such that:

1 -c 1 (-ϕ) δ 0 w i j ≤ w i j ≤ 1 + c 1 (-ϕ) δ 0 w i j .
Hence by inverting it we obtain:

1 + c 1 (-ϕ) δ 0 -1 w i j ≤ w i j ≤ 1 -c 1 (-ϕ) δ 0 -1 w i j . Since 1 1 -x = 1 + x 1 -x ≤ 1 + 2x if x ∈ 0, 1 2 , we have 1 1 -c 1 (-ϕ) δ 0 ≤ 1 + 2c 1 (-ϕ) δ 0 on the set Ω ∩ U ∩ {|ϕ| ≤ } whenever ≤ 1 2c 1 1 δ 0 .
Moreover, since 1 1 + x ≥ 1 -x ≥ 1 -2x for every x ∈ 0, 1 2 , we also have

1 1 + c 1 (-ϕ) δ 0 ≥ 1 -2c 1 (-ϕ) δ 0 on Ω ∩ U ∩ {|ϕ| ≤ }.
Thus, there exist positive constants and c 1 such that we have, on Ω ∩ U ∩ {|ϕ| ≤ }:

1 -c 1 (-ϕ) δ 0 w i j ≤ w i j ≤ 1 + c 1 (-ϕ) δ 0 w i j .
We also take ≤ 1 so that for every δ ≥ 0 we have |ϕ| δ ≤ 1. Consequently, we deduce the existence of constants ∈]0, 1], δ 0 , c 1 > 0 such that for every 0 ≤ δ ≤ δ 0 , we have the following on Ω ∩ U ∩ {|ϕ| ≤ }:

(11) 1 -c 1 |ϕ| δ w i j ≤ w i j ≤ 1 + c 1 |ϕ| δ w i j .
This is the same as condition (6.18) in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF], except that it holds in a neighborhood of ∂Ω ∩ U in our situation (in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF], due to the global assumption of strict pseudoconvexity of ∂Ω, the inequalities in [START_REF] Mok | Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions[END_REF] are valid in a neighborhood of ∂Ω).

From now on, we let δ ∈]0, δ 0 ].

• The constant c 2 (see condition (6.19)) depends only on c 1 .

• The constant c 3 (see conditions (6.22) and (6.23)) depends only on c 1 . Especially we have the following on Ω ∩ U ∩ {|ϕ| ≤ }:

1 -c 3 (-ϕ) δ ≤ |∇ϕ| 2 w ϕ 2 ≤ 1 + c 3 (-ϕ) δ .
In our situation we also assume that ≤ 1 2c 3 1 δ , so that we have the following on Ω ∩ U ∩ {|ϕ| ≤ }:

(12) 1 2 ≤ 1 -c 3 (-ϕ) δ ≤ |∇ϕ| 2 w ϕ 2 ≤ 1 + c 3 (-ϕ) δ .
• Set c 4 := 2nc 3 (see condition (6.24)).

• According to inequality (6.25), we have, on Ω ∩ U ∩ {|ϕ| ≤ }:

-∆ w (-ϕ) α ≥ α(-ϕ) α (n -α) |∇ϕ| 2 w ϕ 2 -c 4 (-ϕ) δ .
If we assume that < n-α 5c 4 1 δ , then we derive the inequality (n-α) 2 |∇ϕ| 2 w -c 4 (-ϕ) δ+2 > 0 on Ω ∩ U ∩ {|ϕ| ≤ }, which leads to the following:

(13) -∆ w (-ϕ) α > α(n -α) 2 |∇ϕ| 2 w ϕ 2 (-ϕ) α .
This is the same as inequality (6.26) in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF], but with c 5 = 0.

• Set c 6 := βc 4 + c 2 (see condition (6.28)).

• The constant c 7 depends only on c 6 (see condition (6.29)).

• The constant c 8 depends only on c 3 and c 7 (see condition (6.30)).

• If < n+1+nβ-β 2 2c 8 1
δ , then we have, on Ω ∩ U ∩ {|ϕ| ≤ }:

n + 1 + nβ -β 2 2 -c 8 (-ϕ) δ > 0,
so that in our case inequality (6.31) becomes the following:

(14) ∆ w |∇u| 2 w (-ϕ) β > n + 1 + nβ -β 2 2 |∇u| 2 w (-ϕ) β -|∇F | 2 w (-ϕ) -(δ+β) .
Combining ( 13) and ( 14), we obtain, on Ω ∩ U ∩ {|ϕ| ≤ } and for every c > 0:

∆ w |∇u| 2 w (-ϕ) β -c(-ϕ) α > n + 1 + nβ -β 2 2 |∇u| 2 w (-ϕ) β -|∇F | 2 w (-ϕ) -(δ+β) + c α(n -α) 2 |∇ϕ| 2 w ϕ 2 (-ϕ) α .
This is exactly the same as inequality (6.31) in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF], but with c 9 = 0.

• Using condition (9) (Proposition 3.1), we observe that |∇F | 2 w ≤ c ∇ (-ϕ) α+δ+β whenever |ϕ| ≤ 1 and α + δ + β ≤ 2l -1. Therefore, according to [START_REF] Yeung | Geometry of domains with the uniform squeezing property[END_REF], the following holds on Ω ∩ U ∩ {|ϕ| ≤ }:

-|∇F | 2 w (-ϕ) -(δ+β) + c α(n -α) 2 |∇ϕ| 2 w ϕ 2 (-ϕ) α ≥ -c ∇ (-ϕ) α + c α(n -α) 2 |∇ϕ| 2 w ϕ 2 (-ϕ) α ≥ -c ∇ + c α(n -α) 4 (-ϕ) α .
In particular if we take c > 4c ∇ α(n-α) the right-hand side is non-negative. This is exactly what is derived from relation (6.32) in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] (see the explanation below relation (6.33) in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF]), except that in our case it holds on Ω ∩ U ∩ {|ϕ| ≤ }.

For short, we have proved that there exists δ 0 > 0 such that for every 0 < α < n, 0 ≤ β < n + 1 and 0 < δ ≤ δ 0 satisfying α + β + δ ≤ 2l -1, there exist ∈]0, 1] and c > 0 such that the following inequalities hold on Ω ∩ U ∩ {|ϕ| ≤ }:

(15) ∆ w |∇u| 2 w (-ϕ) β -c(-ϕ) α > 0, (16) ∆ w |∇u| 2 w (-ϕ) β -c(-ϕ) α > n + 1 + nβ -β 2 2 |∇u| 2 w (-ϕ) β -c(-ϕ) α .
Inequality (15) implies that the function f := |∇u| 2 w (-ϕ) β -c(-ϕ) α cannot achieve its maximum on Ω ∩ U ∩ {|ϕ| ≤ }, provided it is bounded from above on the set D := Ω ∩ U ∩ {|ϕ| < }. Hence we can find a sequence

(z i ) i∈N ∈ D N such that lim i→+∞ f z i = sup D f and d w (z i , ∂D ) -→ z→+∞ +∞.
Note that this implies that there exists a positive number R and an integer i 0 ∈ N such that for every i ≥ i 0 we have d w (z i , ∂D ) ≥ R. The last step to conclude is to apply the local maximum principle due to J. Bland (see Lemma II in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF]) and use inequation (16). For completeness, we recall the local maximum principle in a version that fits our situation:

(z i ) i∈N ∈ D N such that lim i→+∞ f z i = sup D f, lim sup i→+∞ ∆ w f (z i ) ≤ 0.
We apply Lemma 3.4 to f = |∇u| 2 w (-ϕ) β -c(-ϕ) α with D = D and choose the suitable constants α, β, δ to conclude. We may argue as follows.

(1) If 2n + 1 ≤ 2l -1, we first apply Lemma 3.4 with β = 0, α = n -δ 4 and δ ∈ ]0, min (δ 0 , 4n)[ to deduce the existence of constants ∈]0, 1] and c > 0 for which we have

|∇u| 2 w -c(-ϕ) n-δ 4 ≤ 0 on Ω ∩ D . Since (-ϕ) < ≤ 1 on Ω ∩ D , this directly implies: |∇u| 2 w -c(-ϕ) n-δ 2 ≤ 0 on Ω ∩ D .
(2) Hence we may apply Lemma 3.4 with α = β = n -δ 2 and δ ∈ ]0, min (δ 0 , 2n)[ to deduce the existence of constants ∈]0, 1] and c > 0 for which

|∇u| 2 w (-ϕ) n-δ 2 -c(-ϕ) n-δ 2 ≤ 0 on Ω ∩ D . Again, since (-ϕ) < ≤ 1 on Ω ∩ D , this directly implies: |∇u| 2 w -c(-ϕ) n+1-δ 2 ≤ 0 on Ω ∩ D .
(3) Hence we may apply once more Lemma 3.4 with β = α + 1 = n + 1 -δ 2 and δ ∈ ]0, min (δ 0 , 2n)[ to deduce the existence of c, > 0 for which

|∇u| 2 w (-ϕ) n+1-δ 2 -c(-ϕ) n-δ 2 ≤ 0 on Ω ∩ D . Finally, we directly deduce that |∇u| 2 w ≤ c(-ϕ) 2n+1-δ on Ω ∩ D . (4) If 2l -1 < 2n + 1, we can proceed likewise: first taking β = 0, α = min n, l - 1 2 - δ 8 with δ ∈ 0, min δ 0 , 8 min n, l -1 2 , then considering α = β = min n, l - 1 2 - δ 4 with δ ∈ 0, min δ 0 , 4 min n, l -1 2
, and finally taking α

= β = l - 1 2 - δ 2 with δ ∈ ]0, min (δ 0 , 2l -1)[.
In both cases, we obtain the desired conclusion by letting δ tend to 0. Hence the result.

In the rest of Subsection 3.1, we use Proposition 3.2 first to derive the estimates of u of order 0 (Proposition 3.5), second to derive estimates of higher order (Proposition 3.7), and finally to obtain a regularity result for ϕe -u (Proposition 3.8). Proposition 3.5. Under the hypothesis and notations of Proposition 3.2, we have:

(1) For every γ ∈]0, min (2n + 1, 2l -1) [, there exist positive constants and c such that |∇u| ≤ c (-ϕ)

γ 2 -1 on the set Ω ∩ U ∩ {|ϕ| < }. In particular, if γ > 2, one has u ∈ C 1 Ω ∩ U . (2) For every z ∈ ∂Ω ∩ U , ∇e -w z = 0.
(3) For every γ ∈]0, min(2n + 1, 2l -1)[ there exist positive constants c and such that |u| ≤ c (-ϕ)

γ 2 on Ω ∩ U ∩ {|ϕ| < }.
Remark 3.6. • Observe that relation [START_REF] Lee | Boundary behaviour of the complex Monge-Ampère equation[END_REF] already gives a control on u. Indeed, by applying Log • Det on both sides, using equation ( 8), and simplifying both sides, we may successively obtain, on Ω ∩ U ∩ {|ϕ| ≤ }:

e (n+1)u-F Det w i j = 1 + O |ϕ| δ 0 n Det w i j , u = n n + 1 Log 1 + O |ϕ| δ 0 + F n + 1 .
Thus, part (3) of Proposition 3.5 only improves the exponent δ 0 .

• Part (3) of Proposition 3.5 is exactly as in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF], the only difference being that we have it for every γ ∈ ]0, min (2n + 1, 2l -1)[. We prove it a slightly different way by first proving part ≤ M 2 on U . Thus, we have the following on Ω ∩ U :

|∇u| 2 ≤ 1 λ -ψ + |∇ψ| 2 ψ ψ 2 |∇u| 2 g ≤ c λ (-ψ + |∇ψ| 2 ψ ) ϕ ψ 2 (-ϕ) γ-2 , = c λ (-ψ + |∇ψ| 2 ψ ) 1 η 2 (-ϕ) γ-2 , ≤ c λ M 1 M 2 (-ϕ) γ-2 .
Therefore we obtain the conclusion by setting c = c λ M 1 M 2 . Especially, if γ > 2, then all the derivatives of u of order 1 extend continuously to Ω ∩ U (and equal 0 on

∂Ω ∩ U ), hence u ∈ C 1 Ω ∩ U .
(2) To prove part (2) of Proposition 3.5, we let l = n+1. Then by construction e -w = -ϕ (n+1) e -u . Moreover, according to point (1) of Proposition 2.1 and to point (1) of Proposition 3.5, we have

ϕ (n+1) , u ∈ C 1 Ω ∩ U . Thus e -w ∈ C 1 Ω ∩ U so that we can differenciate in Ω ∩ U and let z tend to any point in ∂Ω ∩ U to deduce lim z→∂Ω∩U ∇e -w z = lim z→∂Ω∩U ∇ϕ (n+1) z = 0, because of points (2), (3) of Proposition 2.1. (3) Fix γ ∈ ]0, min (2n + 1, 2l -1)[. Let z ∈ U ∩ {|ϕ| < }. Let z 0 ∈ ∂Ω ∩ U ∩ {|ϕ| < } such that d(z, ∂Ω) = |z -z 0 | =: s. Set - → v := z -z 0 .
Define the following function:

f : [0, 1] -→ R t -→ u (z 0 + t - → v ) .
According to point (1) of Proposition 3.5 we have f ∈ C 1 ([0, s]). Moreover, by the Cauchy-Schwarz inequality we have

|f (t)| ≤ |∇u| z 0 +t -→ v | - → v | = s |∇u| z 0 +t -→ v .
From point (1) of Remark 3.6 we also have u(z 0 ) = 0. Using the fundamental theorem of calculus we deduce:

|u(z)| = |f (1) -f (0)| = 1 0 f (t) dt , ≤ s 1 0 |∇u| z 0 +t -→ v dt, ≤ cs 1 0 (-ϕ (z 0 + t - → v )) γ 2 -1 dt, ≤ cs inf [0,1] h (t) 1 0 h (t) (h(t)) γ 2 -1 dt, = 2cs γ inf [0,1] h (t) 1 0 h γ 2 (t) dt, = 2cs γ inf [0,1] h (t) (-ϕ(z)) γ 2 , ≤ 2cs γ inf [0,1] h (t)
,

where h := -ϕ (z 0 + • - → v ) ∈ C 1 ([0, 1]
). According to point (3) of Proposition 2.1 we have inf

[0,1] h > 0.
Hence the result.

Proposition 3.7 is exactly as in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF], the only difference being that we have the estimates for every γ ∈ ]0, min (2n + 1, 2l -1)[. Proposition 3.7. Under the hypothesis and notations of Proposition 3.2, we have: for every γ ∈ ]0; min(2n+1, 2l-1)[, there exist positive constants and c such that for every integer 0 ≤ p ≤ k-2l, the following holds on Ω ∩ U ∩ {|ϕ| < }:

|D p u| w ≤ c |ϕ| γ 2 ,
where |D p u| w is the length of the p-th covariant derivative of u with respect to w i j .

Proof of Proposition 3.7. We fix γ ∈ ]0, min (2n + 1, 2l -1)[ and follow line by line the proof at the beginning of page 300 in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF], the only thing that changes being the range in which γ can be choosen. Namely, we apply Log • Det to equation ( 8) to obtain the following partial differential equation of second order:

(17) (n + 1)u -F = h i j u j ī,
where h i j :=

1 0 (w + tu) i j dt ∈ C k-2l-2 Ω ∩ U ∩ {|ϕ| < }, H ++ n .
We use equation [START_REF] Mok | Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions[END_REF] with δ = 0 to deduce the existence of constants , c > 0 such that we have, on Ω ∩ U ∩ {|ϕ| < }:

1 c w i j ≤ h i j ≤ c w i j .
Moreover u ∈ C k-2l (Ω ∩ U ∩ {|ϕ| < }), and according to Proposition 2.1 we have F, F (-ϕ) l ∈ C k-2l-2 Ω ∩ U ∩ {|ϕ| < } . We conclude by applying Schauder theory.

In particular, we deduce the following, exactly as was done in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF]: Proposition 3.8. Under the notations and hypothesis of Proposition 3.2, for every number γ ∈ ]0, min (2n + 1, 2l -1) [ and for every 0 ≤ δ < γ 2 -γ 2 (where γ 2 denotes the integral part of γ 2 ), we have: u, e -u ∈ C γ 2 +δ Ω ∩ U . Moreover, if γ > 2, we have:

ϕe -u ∈ C γ 2 +1+δ Ω ∩ U .
Proof of Proposition 3.8. This is exactly as in [START_REF] Bland | Local boundary regularity of the canonical Einstein-Kähler metric on pseudoconvex domains[END_REF] (or [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF] for a global version). Observe that since

k -2l ≥ 3n + 6 -2(n + 1) ≥ n + 2 ≥ γ 2 , u ∈ C n+2 (Ω ∩ U ) and ϕ ∈ C n+2 U (see Proposition 2.1
), it is enough to prove the existence of a positive constant such that for every 0

≤ δ < γ 2 -γ 2 , one has u ∈ C γ 2 +δ Ω ∩ U ∩ {|ϕ| < } and ϕe -u ∈ C γ 2 +1+δ Ω ∩ U ∩ {|ϕ| < } .
Let γ ∈]0, min (2n + 1, 2l -1) [. According to Proposition 3.7, there exist positive constants and c such that for every integer 0 ≤ p ≤ k -2l, the following holds on Ω ∩ U ∩ {|ϕ| < }:

|D p u| w ≤ c |ϕ| γ 2 .
Moreover, according to Proposition 1.5, there exist positive constants λ ≤ Λ such that the following holds on Ω ∩ U :

λ -ψ -ϕ -ψ -ψ + |∇ψ| 2 I ≤ w i j -ϕ ≤ Λ -ψ -ϕ I.
Since -ψ -ϕ ∈ C U is a positive function (see Proposition 2.1) and U is a compact set, we deduce that there exist positive constants M and M such that the following holds on Ω ∩ U :

λM -ψ -ψ + |∇ψ| 2 I ≤ w i j -ϕ ≤ ΛM I.
Together with the expression of |D p u| w in terms of the derivatives of u and of w, this implies the existence of positive constants and c such that for every integer 0 ≤ p ≤ k -2l and every multiindex (i 1 , j 1 , • • • , i n , j n ) ∈ N 2n satisfying n k=1 (i k +j k ) ≤ p, the following holds on Ω∩U ∩{|ϕ| < }:

u i 1 j 1 •••injn , e -u i 1 j 1 •••injn ≤ c |ϕ| γ 2 -p .
• Let p = γ 2 . Then the derivatives of u of order p extend continuously to Ω ∩ U ∩ {|ϕ| < } (and are equal to 0 on ∂Ω ∩ U ), and these extensions are Hölder of exponent δ for every 0 ≤ δ < γ 2 -γ 2 . This gives the desired regularity of u and e -u .

• According to the chain rule and the regularity of ϕ and e -u , we have the existence of a constant c > 0 such that the following holds on Ω ∩ U ∩ {|ϕ| < }:

ϕe -u i 1 j 1 •••injn -ϕ i 1 j 1 •••injn e -u ≤ c |ϕ| γ 2 -(p-1)
.

Moreover, we have

ϕ i 1 j 1 •••injn e -u ∈ C 1 Ω ∩ U ∩ {|ϕ| < } ⊂ ∩ 0≤δ≤1 C δ Ω ∩ U ∩ {|ϕ| < } be- cause we assume that γ 2 > 1. Let p = γ 2 + 1.
Then the derivatives of ϕe -u of order p extend continuously to Ω ∩ U ∩ {|ϕ| < } and these extensions are Hölder of exponent δ for every 0 ≤ δ < γ 2 -γ 2 . This gives the desired regularity of ϕe -u .

3.2. Proof of Theorems 1 and 2. We deduce Theorem 1 by using Proposition 3.8:

Proof of Theorem 1. We take l = n + 1. Then, according to Proposition 3.1, the range of γ is ]0, 2n + 1[. Let α ∈ ]0, 1[ and take γ := 2n + α so that γ 2 = n. We apply Proposition 3.8 to obtain

ϕe -u ∈ C n+1+δ Ω ∩ U for every 0 ≤ δ < α 2 . Since k -2(n + 1) ≥ n + 2, then ϕ (n+1) ϕ ∈ C n+2 U
by point (7) of Proposition 2.1. We directly deduce that -w = ϕ (n+1) e -u = ϕ (n+1) ϕ ϕe -u ∈ C n+1+δ Ω ∩ U . This holds for every 0 ≤ δ < α 2 < 1 2 , hence the result. We can also prove Theorem 2:

Proof of Theorem 2. By definition,

Bis g,z (v, w) = 1≤i,j,k,l≤n R i jk l(g)v i v j w k w l |v| 2 g,z |w| 2 g,z ,
where the curvature coefficients satisfy the following formula which follows from the definition by direct calculations:

(18)

R i jk l(g) = -(g i j g k l + g i lg k j ) + 1 -ψ     R i jk l(ψ) - 1 |∇ψ| 2 ψ -ψ ψ ik -ψ ik pψ pq ψ q ψ ,ik := ψ jl -ψ pψ pq ψ q jl ψ , jl:=     .
Therefore, if v, w ∈ S(0, 1), we have the following on Ω ∩ U :

Bis g (v, w) = -1 + | v; w g | 2 |v| 2 g |w| 2 g =:T 1 (v,w) + 1 -ψ |v| 2 ψ |w| 2 ψ |v| 2 g |w| 2 g Bis ψ (v, w) =:T 2 (v,w) - 1 -ψ 1 |∇ψ| 2 ψ -ψ ψ ,ik ψ , jlvivj w k w l |v| 2 g |w| 2 g =:T 3 (v,w)
.

Using the proof of Proposition 3.8 with γ 2 = n + δ ≥ 2 + δ for some fixed 0 < δ < 1 2 we have the existence of positive constants c, > 0 such that the following holds on Ω ∩ U ∩ {|ϕ| < } for every 1 ≤ i, j, k, l ≤ n:

ψ i jk l ≤ c |ϕ| -1+δ ,
and we also have

ψ ∈ C 3 Ω ∩ U ∩ {|ϕ| < } .
The rest of the proof consists of estimating |T 2 (v, w)| and |T 3 (v, w)|. This will directly follow from formulas ( 19) and (20).

• Using the notations of Proposition 1.3 and of the proof of Proposition 1.3, we have 0 ≤ B, hence I ≤ A, hence ψ i j =: R 2 ≤ RAR = (-ψ)(g i j ). This means that for every v ∈ C n , the following holds on Ω ∩ U :

|v| 2 ψ ≤ (-ψ)|v| 2 g . (19)
• Since Ω ∩ U {|ϕ| < } is compact and ψ ∈ C 2 Ω ∩ U ∩ {|ϕ| < } , we also have the existence of a positive constant 0 < λ -such that the following inequality holds on Ω ∩ U ∩ {|ϕ| < }:

(20) λ -I ≤ (ψ i j ).
We complete the proof as follows. According to inequality (19), we have the following on Ω ∩ U for every vectors v, w ∈ S(0, 1):

1 -ψ |v| 2 ψ |w| 2 ψ |v| 2 g |w| 2 g ≤ (-ψ) = (-ϕ)e -u .
Moreover, there exists a constant c > 0 such that for all 1 ≤ i, j, k, l ≤ n we have

R i jk l (ψ) ≤ c |ϕ| δ-1 on Ω∩U ∩{|ϕ| < }. Hence there exists a positive constant c > 0 such that |T 2 (v, w)| ≤ c|ϕ| δ on Ω ∩ U ∩ {|ϕ| < }.
Likewise, using inequalities (19) and (20) we obtain, on Ω∩U and for every vectors v, w ∈ S(0, 1):

- 1 -ψ 1 |∇ψ| 2 ψ -ψ 1 |v| 2 g |w| 2 g ≤ (-ψ) |∇ψ| 2 ψ -ψ 1 λ 2 - = (-ϕ) |∇ψ| 2 ψ -ψ e -u λ 2 - .
Note that (up to taking a smaller positive ) |∇ψ| 2 ψ -ψ ∈ C Ω ∩ U ∩ {|ϕ| < } and is a positive function thanks to point (2) of Proposition 3.5. Moreover, there exists a constant c > 0 such that for all 1 ≤ i, j, k, l ≤ n we have ψ ,ik ψ , jl ≤ c on Ω ∩ U ∩ {|ϕ| < }. Hence there exists a positive constant c > 0 such that |T Moreover, since each g (ν) is holomorphic and injective, the linear map ∂g Moreover, since each g (ν) is holomorphic and injective, the linear map ∂g for every vectors v, w ∈ C 2 \ {0}. We combine this remark with condition (23) to deduce that the third term of the right hand side tends to 0 as ν tends to +∞. Hence the result.

3 (v, w)| ≤ c|ϕ| on Ω ∩ U ∩ {|ϕ| < }.

  Functions and Kähler geometry in open sets of C n . We work with the usual topology on C n , induced by the usual Euclidean norm, that we note |•|. For p ∈ C n and r > 0, we note S(p, r), respectively B(p, r) the Euclidean sphere, respectively the Euclidean open ball, of center p and radius r. Let U ⊂ C n be a non-empty open set, and let z ∈ U . Let k ∈ N, and let ∈ [0, 1].

  On the one hand, it implies that |∇ψ| z > 0 because ψ is a defining function for ∂Ω ∩ V . One the other hand, the continuity of the function |∇ψ| at z implies that |∇ψ| z = lim i→+∞ |∇ψ| z i = 0. Hence the contradiction.

  1≤l≤n+1 inf z∈U |∇ϕ (l) | z > 0 and also inf z∈U |∇ϕ| z > 0.

  3 U , and according to equation (5) as well as point (7) of Proposition 2.1,

Lemma 3 . 4 .

 34 Let Ω ⊂ C n be a domain. Assume that there exists a Kähler-Einstein metric induced by a potential w on Ω. Let D ⊂ Ω be a domain. Let f ∈ C 2 (D) bounded from above. If there exists a sequence (z i ) i∈N ∈ D N such that lim i→+∞ f z i = sup D f and there exists R > 0 such that for every integer i, d w (z i , ∂D) ≥ R, then there exists an other sequence

( 1 ) 1 η 2 ∈

 112 of Proposition 3.5. Proof of Proposition 3.5. (1) We apply Proposition 3.2, and use Proposition 1.5 with ψ = ηϕ, g = w and U replaced with U ∩ {|ϕ| < }. With notations of Propositions 3.2 and 1.5, we have c λ > 0. Moreover we know that -ψ + |∇ψ| 2 ψ , C U and are positive functions. Hence they are bounded from above, so that there exist positive constants M 1 , M 2 such that -ψ + |∇ψ| 2 ψ ≤ M 1 and 1 η 2

  strictly pseudoconvex domains with smooth boundary, so that according to Lemma 4.2 there exists a strictly pseudoconvex domain D ν with boundary of class C ∞ that satisfies Ω ν-1 ⊂ D ν ⊂ Ω ν and sup v,w∈C n \{0}

z

  (ν) (v), ∂g (ν) z (ν) (w)) -Bis Dν 0 (∂g (ν) z (ν) (v), ∂g(ν)z (ν) (w)) ≤ 1 2 ν . (22) Because of the property Ω ν ⊂ D ν+1 ⊂ Ω ν+1 for every ν ∈ N, the sequence (D ν ) ν∈N isan increasing sequence of strictly pseudoconvex domains with boundary of class C ∞ . Since lim ν→∞ s Ω z (ν) = 1 we have ∪ ν∈N D ν = B(0, 1), that is (D ν ) ν∈N is an exhaustion of the unit ball by strictly pseudoconvex domains with boundary of class C ∞ . Therefore according to Lemma 4.2 we deduce the following up to extracting a subsequence from (D ν ) ν∈N : sup v,w∈C n \{0}

  sup v,w∈C n \{0} Bis Dν 0 ∂g (ν) z (ν) (v), ∂g

  (ν)z (ν) (w) -Bis B(0,1) 0 ∂g (ν) z (ν) (v), ∂g

  (ν)z (ν) (w) -→ ν→∞ 0. (24)Using triangle inequality we obtain for every integer ν ∈ N and every vectors v, w ∈ C n \ {0}:Bis Ω z (ν) (v, w) 22), respectively condition (24), condition (21), the first term of the right hand side, respectively the second, the fourth, tends to 0 as ν tends to +∞. Moreover the Kähler-Einstein metric we work with satisfies Bis

Using the triangle inequality, we deduce the existence of positive constants , c > 0 such that the following inequality holds on Ω ∩ U ∩ {|ϕ| < }: sup v,w∈S(0,1)

We obtain the result since lim z→q ϕ(z) = 0 and δ > 0.

Remark 3.9. Regardless that n ≥ 2, the asymptotic curvature behavior (3) does not persist if we remove the hypothesis of strict pseudoconvexity. For instance, if m ∈ N * , in the "egg domain"

we can easily deduce from the computations done in [START_REF]The Einstein-Kähler metric on {|z| 2 + |w| 2p < 1}[END_REF] that, at q = (1, 0):

This differs from (3) if m ≥ 2. We also notice that the same approach as in [START_REF]The Einstein-Kähler metric on {|z| 2 + |w| 2p < 1}[END_REF] may be adapted to obtain the same estimates in tube domains

Proof of Theorem 4

We recall the definition of the squeezing function of a domain.

In [START_REF] Deng | Some properties of squeezing functions on bounded domains[END_REF] the authors prove that the supremum in Definition 4.1 is achieved. In the rest of this Section, every domain that appears possesses a unique complete Kähler-Einstein potential which is solution to Equation (1) with condition (2) and we only work with this one. Moreover, given a domain D with complete Kähler-Einstein potential g solving Equation (1) with condition (2), we use the notations

, Bis g,z to avoid confusions. We need the following Lemma, which is a direct consequence of the proof of Theorem 7.5. in [START_REF] Cheng | On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equation[END_REF]: Lemma 4.2. Let D ⊂ C n be a bounded pseudoconvex domain. Let (D ν ) ν∈N be an exhaustion of D by strictly pseudoconvex domains with boundary of class C ∞ . Then, up to extracting a subsequence from (D ν ) ν∈N , the following holds for every compact set K ⊂ D:

We prove Theorem 4: ν) and set Ω ν := g (ν) (Ω). Since g (ν) is a biholomorphic mapping from the pseudoconvex domain Ω to Ω ν , Ω ν is a bounded pseudoconvex domain. By construction of g (ν) , for every integer ν ∈ N we have Ω ν ⊂ B(0, 1). Moreover we have lim ν→∞ s Ω z (ν) = 1 hence up to taking a subsequence we may assume that Ω ν ⊂ Ω ν+1 . Let ν ∈ N * . Since Ω ν is a bounded pseudoconvex domain, there exists an exhaustion of Ω ν by


Jérôme Weiss

Maurine Montagnat

Long-range spatial correlations and scaling in dislocation and slip patterns

Keywords: Dislocation, dislocation density, slip line, scale invariance, long-range correlations, strain gradient plasticity

Although the intermittent and heterogeneous nature of plastic flow has been known for several decades, only recently did observations performed on the surface of deformed samples by AFM or scanning white-light interferometry, or in the bulk by X-Ray topography, reveal the scale invariant character of dislocation and slip patterns emerging from collective dislocation interactions. This scale invariance implies that the spatial fluctuations of dislocation density and/or slip never vanish as one coarsens the observation scale. An immediate consequence is that a priori obvious concepts such as "slip bands" or dislocation density can be ill-defined. These detailed characterizations of the plastic flow heterogeneity also challenge the modelling of plasticity.

Introduction

Since the early days of the dislocation-related theory of plasticity [START_REF] Orowan | Zur Kristallplasticitat III. : Uber die Mechanismus des Gleitvorganges[END_REF][START_REF] Taylor | The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical[END_REF], the main question was to relate the microscale physics (from the crystal lattice scale, b, to the average distance between neighbouring dislocations, l d =1/ρ 1/2 ) to the macroscale behaviour. The celebrated Orowan's relation, dγ/dt = ρ m 〈v〉b which expresses the macroscopic strain-rate as a function of the density of mobile dislocations ρ m and of their averaged velocity 〈v〉, is based on two basic micro-macro concepts : (i) the mechanistic approach and (ii) mean-field and averaging procedures. E.g., in the Orowan's relation, it is implicitly assumed that averaging is meaningful for both v and the spatial distribution of dislocations. This coarse-graining can fail for three reasons: (i) spatial long-range correlations exist for ρ m , (ii) long-range correlations exist for v, or (iii) the distribution of v is "wild", i.e. such that a variance cannot be defined and the central limit theorem cannot be applied (e.g. [START_REF] Sornette | Critical phenomena in natural sciences[END_REF]). In other words, spatial and temporal fluctuations on v and ρ m are too strong (more precisely, these fluctuations do not vanish relatively to the mean as the observation scale is coarsened). Recent works [START_REF] Miguel | Intermittent dislocation flow in viscoplastic deformation[END_REF] showed that condition (iii) is often fulfilled (see below), whereas we show in this paper that conditions (i) and (ii) can be also encountered, thus questioning the applicability of such simple coarsegraining procedures in plasticity.

Since these seminal works [START_REF] Orowan | Zur Kristallplasticitat III. : Uber die Mechanismus des Gleitvorganges[END_REF][START_REF] Taylor | The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical[END_REF], the notion of dislocation density has been extensively used in the literature. A well-known example is the Taylor's relation for the flow stress [START_REF] Taylor | Plastic strain in metals[END_REF], σ c ~ Gbρ 1/2 , which expresses, through a mean-field approach, the idea that the flow stress σ c is governed by dislocation interactions. Phenomenological models formulated in terms of differential equations for the evolution of dislocation densities and using the Taylor's relation were developed from the late 70's to model work-hardening (e.g. [START_REF] Kocks | Laws for work-hardening and low-temperature creep[END_REF]). As stressed by Zaiser and Seeger [START_REF] Zaiser | Long-range internal stresses, dislocation patterning and work-hardening in crystal plasticity[END_REF], those early phenomenological density-based models were unable to account for a rather ubiquitous (at least in the case of materials with high dislocation mobility) experimental fact that became apparent at the same time: the spontaneous emergence of strong heterogeneities in the dislocation patterns. Indeed, the intrinsic heterogeneous nature of slip, which emerge from the collective behaviour of moving dislocations, is now well-established, with dislocations tending to move cooperatively in groups of spatially correlated slip planes rather than individually and independently from each other [START_REF] Neuhauser | Slip-line formation and collective dislocation motion[END_REF]. In the spatial domain, this has been formulated by the concept of "slip lines" or "slip bands" revealed by a topographic analysis of the surface of deformed samples, a subject reviewed by Neuhauser [START_REF] Neuhauser | Slip-line formation and collective dislocation motion[END_REF] almost 25 years ago. This spatial heterogeneity is accompanied by intermittency, as plastic deformation takes place through localised "bursts" or avalanches, characterized by very large instantaneous strain-rates, whereas dislocations are almost immobile everywhere else [START_REF] Miguel | Intermittent dislocation flow in viscoplastic deformation[END_REF]. Recent works showed that this collective dislocation dynamics is characterized by different scaling laws as well as space [START_REF] Weiss | Three dimensional mapping of dislocation avalanches: clustering and space/time coupling[END_REF] and time [START_REF] Weiss | Dislocation avalanche correlations[END_REF] correlations. In particular, the power law distributions of dislocation velocities [START_REF] Miguel | Intermittent dislocation flow in viscoplastic deformation[END_REF] question the averaging procedure used in the Orowan's relation, as the variance of the distribution is ill-defined in this case (see above).

In this paper, we first review recent advances in the experimental characterization of slip heterogeneity and the associated spatial correlations of dislocation densities. We show that these slip patterns are characterized by scaling laws and long-ranged correlations. Then, we discuss the pertinence (and limitations) of classical and widely-used concepts such as slip bands, dislocation densities, or strain gradient plasticity. We also discuss, in this context, the consequences of such scale-free slip and dislocation patterns to constrain future modelling developments, with a focus on the recent density-based approaches.

Heterogeneity, spatial correlations and scaling in dislocation and slip patterns

The most direct method to investigate dislocation patterns is the analysis of transmission electron microscopy (TEM) images. Although TEM investigations of dislocation structures and patterns are countless, quantitative analyses of dislocation density spatial correlations as well as of statistical and scaling properties are difficult and tedious, and therefore scarce. As this subject was reviewed by Zaiser and Seeger [START_REF] Zaiser | Long-range internal stresses, dislocation patterning and work-hardening in crystal plasticity[END_REF], we simply recall here that the emergence of fractal, scale-free dislocation arrangements has been revealed by boxcounting measurements or by the distribution of dislocation cell sizes [START_REF] Gil Sevillano | The fractal nature of gliding dislocation lines[END_REF][START_REF] Gil Sevillano | Patterns in heavily deformed metals[END_REF][START_REF] Hahner | Fractal dislocation patterning during plastic deformation[END_REF]. These fractal patterns are necessarily associated with long-ranged, non periodic spatial correlations, and differ in this respect to random arrangements (no correlations) or to patterns with periodic arrangements (periodic correlations). Periodic patterns seem to be a specific feature of cyclic deformation (fatigue) [START_REF] Zaiser | Long-range internal stresses, dislocation patterning and work-hardening in crystal plasticity[END_REF], a subject not treated here.

A complementary, but indirect method to explore the spatial heterogeneity of dislocation patterns is the X-ray line profile analysis [START_REF] Ungar | Dislocation densities, arrangements and character from X-ray diffraction experiments[END_REF][START_REF] Szekely | Statistic properties of dislocation structures investigated by X-ray diffraction[END_REF], which allows to estimate the average dislocation density, 〈ρ〉, as well as the associated fluctuations quantified by the second-order moment of the distribution, 〈ρ 2 〉. For compressed Cu single crystals, Szekely et al. [START_REF] Szekely | Statistic properties of dislocation structures investigated by X-ray diffraction[END_REF] found a good (anti)correlation between the relative dislocation density fluctuation, quantified by the ratio δ=〈ρ 2 〉 1/2 /〈ρ〉, and the fractal dimension of the dislocation patterns deduced from the distribution of cell sizes: the larger the heterogeneity, the larger the density fluctuation deduced from X-ray, and the smaller the fractal dimension deduced from TEM observations.

The most common method to study the spatial heterogeneity of slip and dislocation patterns, which was also the first developed, is the analysis of the surface of deformed samples. When dislocations emerge at the surface of a sample, slip steps are created. The heterogeneous nature of slip is then characterized by the development of roughness at the surface, which can be analyzed by shadowed microscopic images of sample surfaces or of replicas, or, more recently and directly, by atomic force microscopy (AFM) or scanning interferometry [START_REF] Zaiser | Self-affine surface morphology of plastically deformed metals[END_REF]. In his classic paper, Neuhauser [START_REF] Neuhauser | Slip-line formation and collective dislocation motion[END_REF] reviewed the observations available at that time, stressed the connection between the emergence of the so-called "slip bands" and collective dislocation motion, but did not perform any correlation or scaling analysis.

Sprusil and Hnilica [START_REF] Sprusil | Fractal character of slip lines of Cd single crystals[END_REF] were the first to recognize the scale invariant, fractal character of slip patterns in plastically deformed Cd single crystals (figure 1), an hexagonal material in which the stage I of deformation is characterized by single-slip plasticity along the basal planes. Qualitatively, scale invariance is illustrated by the similarity of the patterns whatever the magnification. More quantitatively, these authors measured the number of dark lines, identified as "slip bands", per unit length along a line perpendicular to the basal planes, for each magnification, and obtained the following scaling (Figure 2):

) 1 ( ~1 1 - - D D b l M ρ
where ρ b is the density of slip bands (in m -1 ) along a line perpendicular to the slip planes, M the magnification, l the image resolution (l~1/M) and D is the fractal dimension characterizing the set of slip band coordinates (0 < D < 1). Sprusil and Hnilica reported a value of D≈0.5 for this set of points [START_REF] Sprusil | Fractal character of slip lines of Cd single crystals[END_REF] (Figure 2). This fractal pattern fundamentally differs from a random, uncorrelated poissonian arrangement, and is an indication of long-ranged correlations. Using different methodologies (such as the so-called gap, yardstick or box-counting methods) applied on similar sets of slip band coordinates, Kleiser and Bocek [START_REF] Kleiser | The fractal nature of slip in crystals[END_REF] concluded on the scale invariant character of slip patterns in Cu single crystals, and reported D-values between 0.35 and 0.7.

A strong limitation of these early works is that they did not resolve the elevation of the slip steps, which is related to the number of dislocations that glide along a given "slip band", or, in other words, to the local amount of plastic strain. Recent investigations based on AFM or scanning white-light interferometry allowed to quantitatively analyse this surface roughness of plastically deformed samples [START_REF] Zaiser | Self-affine surface morphology of plastically deformed metals[END_REF]. The elevation fluctuations (the roughness) can be linked to the spatial heterogeneity of strain, i.e. to local strain gradients. Therefore, by nature, the dislocation arrangements revealed by this technique are associated with strain gradients. The local elevation gradient (the local slope) ∂z/∂x is related to the local amount of plastic strain ε p (x) as follows [START_REF] Zaiser | Scale invariance in plastic flow of crystalline solids[END_REF]:

) 2 ( ) ( p p x x z ε ε - = ∂ ∂ where z(x)
is the elevation at position x along the profile.

Nadgorny et al. [START_REF] Nadgorny | Evolution of self-affine surface roughness in plastically deforming KCl single crystals[END_REF] showed that an initially smooth surface of a KCl single crystal developed self-affine roughness during deformation. This self-affine scaling, documented on 1-d profiles of the surfaces, is expressed by:

) 3 ( ) ( ) ( H x z x x z x z Δ Δ = Δ + -
where Δx is the size of the window over which the elevation difference (a proxy of the roughness) is measured, and H is the so-called Hurst's exponent characterizing the self-affine regime. Note that a more conventional estimate of roughness, the standard deviation of elevation over the window, is expected to follow the same scaling law [START_REF] Schmittbuhl | Reliability of self-affine measurements[END_REF]. For the stage I of deformation characterized by single-slip and low hardening, these authors reported a selfaffine scaling over the entire available range of observation (from about 300 nm to 200 µm) with H ≈ 0.65 [START_REF] Nadgorny | Evolution of self-affine surface roughness in plastically deforming KCl single crystals[END_REF]. For stage II characterized by multi-slip and hardening, a small scale regime (H ≈ 0.7) seems to cross over to a large scale regime (H ≈ 0.5) above few tens of µm.

When the Hurst's exponents are larger than 0.5, this self-affine regime is the hallmark of scale invariance and long-ranged spatial correlations for the strain pattern, i.e. somehow for the dislocation density profile. Indeed, it implies that the power spectrum of the elevation profile z(x) scales as (e.g. [START_REF] Schmittbuhl | Scaling invariance of crack surfaces[END_REF]):

) 4 ( ) ( µ f f E -
where f is the frequency (in m -1 ), E is the power spectrum, and the exponent µ is related to H through µ=2H+1. Consequently, the gradient ∂z/∂x, which is directly linked to the local plastic strain (see relation ( 2)), is also characterized by a power law spectrum, E(f) ~ f -α , where α = µ-2, i.e. α=0.3 for KCl single crystals during stage I. As the power spectrum is the Fourier transform of the auto-correlation function [START_REF] Schmittbuhl | Scaling invariance of crack surfaces[END_REF], this power law behaviour with α>0 is the hallmark of long-ranged spatial correlations of the plastic strain ε p (x). By comparison, an uncorrelated pattern would be characterized by a "white noise" with α=0, i.e. µ=2 and H=0.5

for the integrated profile. This might be the case for the strain pattern observed towards large scales during stage II (H≈0.5, see above).

In single crystals, at least during stage I, the upper bound of scaling is set by the sample size. Similar roughness analyses performed on polycrystalline Cu [START_REF] Zaiser | Self-affine surface morphology of plastically deformed metals[END_REF] an Al-Mg alloys [START_REF] Wouters | On the evolution of surface roughness during deformation of polycrystalline aluminium alloys[END_REF] revealed an auto-correlation function of the roughness profile z(x) that saturates at scales larger than the average grain size 〈d〉. This defines 〈d〉 as the correlation length ξ of the slip patterns. Towards small scales, scaling was observed down to the resolution scale of the analysis, i.e. down to 50 nm in the case of atomic force microscopy [START_REF] Zaiser | Self-affine surface morphology of plastically deformed metals[END_REF], with 0.8≤H≤ 0.92, i.e. 0.6≤α ≤0.84 for the power spectrum of the local strain pattern ε p (x).

The main drawback of these roughness analyses is that, as already stressed by Neuhauser [START_REF] Neuhauser | Slip-line formation and collective dislocation motion[END_REF], the presence of a free surface is expected to influence the operation of dislocation sources and dislocations patterns near the surface. Consequently, they might not be representative of bulk plasticity. Recently, we performed at ESRF post-mortem synchrotron X-ray topography analyses of ice single crystals deformed under pure torsion [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF]. The torsion axis was parallel to the c-axis of the samples. Hexagonal ice Ih is characterized by single slip plasticity along the basal planes (i.e. stage II of deformation is never observed), which were therefore parallel to the shear plane in these experiments.

Although a macroscopic strain gradient is geometrically imposed from the centre of the cylindrical samples to the surface, no gradient is imposed along the direction perpendicular to the basal planes. During synchrotron X-ray topography, the diffracted intensity records the distortion field, in our case related to orientation contrast. For our deformed samples, it is related to the dislocation density at the origin of the lattice distortion, i.e. only "excess dislocations" [START_REF] Varadhan | Coupling the dynamics of statistically distributed and excess dislocations[END_REF] responsible for lattice distortion are accessible by this technique. Diffracted intensity variations provide a map of dislocation density fluctuations within the bulk. Figure 3a shows an example of a diffraction pattern on the prismatic plane, with the long dimension corresponding to the height of the specimen, along the c-axis, while the width corresponds to the width of the specimen illuminated by the X-ray beam. Darker zones correspond to high dislocation density regions. The pattern shown on figure 3a reveals a strong heterogeneity of the dislocation density along a direction perpendicular to the slip planes. We analyzed this heterogeneity along 1-d intensity profiles of the topograph perpendicular to the basal plane (figure 3b), by means of a spectral analysis. We observed a power law regime for the power spectrum of the intensity record over the entire available scale range (bounded by the resolution of the analysis, 10 µm, and the sample size, ≈ 7 mm), E(f) ~ f -α , with α=1.3 for all the samples analyzed [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF](Figure 4). As for the elevation gradients ∂z/∂x of the surfaces of deformed samples (see above), these power laws reveal the scale invariance of the intensity records, i.e. of the dislocation density arrangement along a line perpendicular to the basal planes, as well as the associated long-range correlations. With this X-ray topography analysis, as the measured intensity is a proxy of the local dislocation density, the intensity gradients ∂I/∂x are a proxy of dislocation density gradients. The observed value of α=1.3<2 identifies the intensity profile as a so-called anti-persistent pattern [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF], i.e. a local increase of the density (positive gradient) is expected, in the probabilistic sense, to be surrounded by negative gradients. In other words, the density gradients are anti-correlated, although the absolute values of the gradients remain positively correlated [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF].

Therefore, surface and bulk analyses of slip and dislocation patterns are both characterized by scale invariance and spatial long-range correlations, although the exponent α of the power spectrum is larger for the dislocation density patterns revealed by X-ray topography than those obtained for strain patterns from roughness analyses. In simpler words, bulk density patterns appear more correlated (less noisy) than surface strain patterns. At this stage, we don't know if this difference is due to the materials analyzed, to the difference between bulk and surface properties, or to the fact that a dislocation density cannot be directly translated into a local strain, even for the single-slip plasticity under torsion studied for the ice samples [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF].

To end with this section about experimental characterization of slip patterns, one should mention that a 3-d analysis of dislocation avalanche locations monitored by acoustic emission during the creep of an ice single crystal revealed a scale-free pattern with a fractal dimension D=2.5, in agreement with the analyses detailed above [START_REF] Weiss | Three dimensional mapping of dislocation avalanches: clustering and space/time coupling[END_REF].

Slip bands, dislocation densities, strain-gradient plasticity and modelling frameworks

The observations reported above argue for the rather ubiquitous character of spatial long-range correlations and scaling in dislocation and slip patterns. In single crystals, scaling holds from the smallest scales that can be explored by modern investigation methods (few tens of nm with AFM) to the scale of the sample. In polycrystalline materials, the correlation length ξ, i.e. the upper bound of scaling, seems to be set by the average grain size.

These observations question the pertinence of classical and widely-used concepts such as slip bands or dislocation densities, and should constrain future modelling developments of plasticity.

Slip bands and dislocations densities

The concept of "slip band" emerged naturally when the spatial heterogeneity of plastic flow was recognized [START_REF] Neuhauser | Slip-line formation and collective dislocation motion[END_REF]. However, this a priori simple concept is misleading when scaling is present. Indeed, relation [START_REF] Orowan | Zur Kristallplasticitat III. : Uber die Mechanismus des Gleitvorganges[END_REF] implies that the density of slip bands ρ b is scale dependent: more and more slip bands per unit length appear as one increases the resolution. Any parameter related to ρ b , such as, e.g., an average spacing between bands, has therefore to be taken with caution.

Similarly, the definition of a volumic statistical dislocation density ρ implies that such averaging is meaningful at the scale considered, that is, the associated spatial fluctuations at that scale are small compared to the mean. In other words, the inevitable small scale heterogeneity (at the crystal lattice scale, dislocations are discrete defects) can be homogenised at some scale if the fluctuations vanish relatively to the mean above this scale.

The scale invariance of the slip and dislocation patterns reported above might actually imply exactly the reverse. In the case of a so-called monofractal pattern, the ratio δ=〈ρ 2 〉 1/2 /〈ρ〉 (as well as any higher order ratio 〈ρ q 〉 1/q /〈ρ〉 with q>2) remains constant whatever the scale of observation, i.e. fluctuations never vanish and dislocation density is ill-defined. For the dislocation patterns revealed by X-ray topography on ice samples, a detailed analysis might suggest a small departure from this monofractal scaling, with a possible decrease of δ with increasing scale as δ ~ Δx - β , with β in the range 0.05-0.1 (figure 5). These estimated β-values are an upper bound, as δ-values are biased (underestimated) by a finite size effect towards large scales. Therefore, this decrease of fluctuations towards large scale, if any, is very slow.

In the case of strain patterns revealed by surface roughness, Zaiser [START_REF] Zaiser | Scale invariance in plastic flow of crystalline solids[END_REF] checked on KCl single crystals that the surface roughness patterns were monofractal, i.e.:

) 5 ( 4 1 , ) ( ) ( / 1 / 1 ≤ ≤ ∀ Δ Δ = Δ + - q for q x z x x z x z H q q q q
As the elevation gradient ∂z/∂x is related to the local amount of plastic strain ε p (x) (see above), relation ( 5) implies:

) 6 ( 4 1 , ~1 / 1 / 1 ≤ ≤ ∀ Δ Δ Δ - q for q x x z H q q q q p ε
, which shows that the strain fluctuations remain constant whatever the scale of observation, in good agreement with our analysis of dislocation density profiles. This situation is true up to the correlation length ξ, meaning that a statistical dislocation density is ill-defined whatever the scale in the case of single crystal plasticity. On the other hand, in polycrystals, the average grain size 〈d〉 appears as an upper bound of scaling. Consequently, the fluctuations vanish as one coarsens the scale above 〈d〉. Mean-field approaches and homogenisation procedures might therefore be adequate if the representative volume element includes several grains (here, we do not consider other sources of heterogeneity such as textures and fabrics, which could be important at scales larger than 〈d〉).

However, a recent analysis of dislocation avalanches in polycrystalline ice [START_REF] Richeton | Dislocation avalanches: role of temperature, grain size and strain hardening[END_REF] showed that spatial correlations and fractal scaling extended towards scales much larger than the grain size. The role of grain boundaries as barriers to dislocation and slip correlations might not be so straightforward [START_REF] Richeton | Breakdown of avalanche critical behaviour in polycrystalline plasticity[END_REF]. We note also that in single crystals, extended scaling was essentially observed during stage I (in ice and KCl, see above), i.e. in a single slip configuration. During stage II in KCl, a crossover towards a "white noise" regime (H ≈ 0.5) was observed at scales around few tens of µm [START_REF] Nadgorny | Evolution of self-affine surface roughness in plastically deforming KCl single crystals[END_REF]. This could indicate a rapid decrease of the dislocation density fluctuations above these scales, maybe as the result of forest-type interactions in different planes.

Strain-gradient plasticity

In order to account for a size dependence on strength in plasticity laws, the strain gradient theory introduced the concept of geometrically necessary dislocations (GND's), to be distinguished from the statistically stored dislocations (SSD's). From Ashby [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF] the later are supposed to accumulate in pure crystal during straining and to be responsible for the forest hardening, while the first are required for the compatible deformation of various parts of the specimen. Ashby assumed that the deformation associated with the SSD's is spatially uniform, whereas GND's accommodate local strain-gradients, i.e. non-uniform deformation.

In other words, the fluctuations of the SSD dislocation density, ρ S , are assumed to vanish rapidly above the average spacing l d . Within this framework, ρ S can be considered as a variable characterizing the state of deformation of the very material, while the GND density, ρ G , is associated to "externally" imposed strain gradients. Here, we employ the term "external" with respect to the dislocation system itself, i.e. strain gradients imposed by microstructural features such as second-phase particles or grain boundaries, or imposed by the deformation geometry (e.g. bending tests). Such assumptions allowed to introduce a dependence on strain gradient, and then on a length scale, in phenomenological plasticity laws [START_REF] Fleck | Strain gradient plasticity: theory and experiments[END_REF][START_REF] Fleck | Strain gradient plasticity[END_REF][START_REF] Gao | Modeling plasticity at the micrometer scale[END_REF].

In this strain-gradient theory of plasticity, it is therefore assumed that the deformation of a single-phase single crystal will be spatially homogeneous and accommodated by SSD's only. This approximation about the existence of a plastically homogeneous deforming crystal can now be strongly questioned regarding the experimental results about dislocation patterning reported above. Surface measurements of dislocation induced slip steps in deformed single and polycrystals as well as X-ray diffraction analyses of dislocation density patterns revealed the emergence of strain (or dislocation density) gradients self-induced by the collective behaviour of dislocations. The dislocation and slip patterns observed in single crystals as well as in grains inside polycrystals are scale invariant, meaning that these selfinduced strain gradients exist whatever the scale of observation, within the limits of the scaling regime. The fact that fluctuations of dislocation density never vanish as one coarsen the scale (see previous section) implies that the concept of SSD's becomes meaningless, at least up to the upper bound of scaling. Note that the dislocation population which accommodates the externally imposed gradients, i.e. the so-called GND's in the Ashby's strain gradient plasticity theory, has, by nature, no long-range stresses associated with it [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF].

On the other hand, the dislocation pattern which spontaneously emerge from collective effects is associated with long-range stresses, up to the correlation length ξ.

A geometrically necessary dislocation population can also be defined as the one accommodating the incompatibility of the lattice in the sense of the Nye's dislocation density tensor [START_REF] Nye | Some geometrical relations in dislocated crystals[END_REF]. This tensor provides the net Burgers vector of all dislocation lines threading a given surface. This definition is more general than the original Ashby's definition of GND's [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF]. Indeed, such dislocation population, also referred as "excess" dislocations [START_REF] Varadhan | Coupling the dynamics of statistically distributed and excess dislocations[END_REF], is, by nature, associated to long-ranged stresses and all kind of strain gradients, i.e. self-induced and externally imposed gradients. However, the associated densities remain a scale dependant concept, as they are defined for a Burgers circuit of a given size.

In the spirit of the presented experimental results, we propose to distinguish between "externally" imposed (see above for the definition of "external") and self-induced strain gradients, and the associated dislocation populations. These self-induced gradients compete with the gradients externally imposed by the microstructure or by the deformation geometry.

Below the scale where externally imposed gradients dominate, set by an average particle spacing, an average grain size, or the sample size, the scale invariance of dislocation patterning prevents from any homogenisation hypotheses, and consequently from the distinction between the two dislocation populations, SSD's and GND's (in the original Ashby's framework), and the use of the strain-gradient plasticity theory. To keep using the concept of SSD's and GND's implies that a scale can be found where self-induced gradients become negligible compared with externally imposed gradients, as it seems to be the case for the average grain size in polycrystals (see above and [START_REF] Zaiser | Self-affine surface morphology of plastically deformed metals[END_REF][START_REF] Wouters | On the evolution of surface roughness during deformation of polycrystalline aluminium alloys[END_REF]). In that case, at scales well above ξ, the self-induced gradients average out, and the concept of SSD's remains possible.

The strain gradient plasticity theory is then strongly dependent on the scale at which homogenisation can be performed.

Modelling issues

As perfectly illustrated by the contributions of this special issue, the density-based modelling of dislocation systems and plasticity is a very active and promising field of research. This might appear, at first glance, somehow in contradiction with the fact that statistical dislocation density is ill-defined (at least scale-dependent) when scaling is present (see above). However, we stress again that scaling is observed up to the correlation length which is set by the grain size in polycrystals. The case of single crystals, where scaling holds up to the sample size for the observations reported above, might appear as an "ideal" case without obvious practical application. On the other hand, the development of nanotechnologies, in shrinking the structural dimensions, could raise the problem of plastic heterogeneity in these cases.

Nevertheless, this emergence of scale-invariant patterns from the collective dislocation interactions necessarily challenges the modelling of plasticity. The key issue is, as stressed by Zaiser and Hochrainer [START_REF] Zaiser | Some steps towards a continuum representation of 3D dislocation systems[END_REF], to achieve the transition between descriptions of plastic flow in terms of the dynamics of discrete dislocations (which have been successful in modelling the scale invariant properties of plasticity [START_REF] Miguel | Intermittent dislocation flow in viscoplastic deformation[END_REF]), and continuum equations formulated in terms of partial differential equations and boundary value problems. The key challenge is to adapt the classical continuum modelling framework to take into account heterogeneity and fluctuations, while solving boundary value problems at the macro-scale.

A recent advance in this respect is based on the concept of "excess dislocations" briefly described in the previous section. It is an elegant way to propose a continuum framework that is able to (actually designed for) take into account internal stress fields and strain gradients resulting from dislocation interactions [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF]. This framework, called Field Dislocation Mechanics (FDM) does not introduce phenomenological, ad hoc internal length scales to account for the observed gradients, instead incorporates physically-based length scales and spatial correlations due to dislocation interactions. In addition, FDM includes the dislocation transport equation [START_REF] Mura | Continuous distribution of moving dislocations[END_REF], which is used as an evolution equation for the dislocation densities. It provides a dynamical setting for the solution of boundary value problems with stress and dislocation densities as state variables. When the resolution is small (of the order of the mean dislocation spacing), FDM is in practice only applicable to small size systems. However, coarse-graining is possible [START_REF] Varadhan | Coupling the dynamics of statistically distributed and excess dislocations[END_REF][START_REF] Archarya | Size effects and idealized dislocation microstructure at small scales: Predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part I[END_REF]: short-range details of the internal stress field are smeared off but samples can be dealt with at a larger scale. It would be interesting in the future to test the ability of FDM to reproduce the complexity of self-induced strain gradients described in this paper.

Other recent developments (e.g. [START_REF] Zaiser | Fluctuation phenomena in crystal plasticity -a continuum model[END_REF]) are based on mesoscopic field variables (stress, strain, dislocation densities). The phenomenological laws describing the evolution of these mesoscopic variables, such as dislocation density-based hardening laws, are inspired from the classical macroscopic modelling framework. Here, mesoscale means above the average dislocation spacing l d =1/ρ 1/2 , such that field variables can be defined, but well below the macroscopic scale of the deforming body [START_REF] Zaiser | Scale invariance in plastic flow of crystalline solids[END_REF]. In addition to these phenomenological laws, some randomness is introduced in the model, e.g. on the local (mesoscopic) yield stress, to represent the sub-mesoscopic variability arising from the fine scale dynamics of interacting dislocations. Although this introduction of randomness is ad hoc, its statistical characteristics, such as variance or correlation properties, are deduced from experiments [START_REF] Szekely | Statistic properties of dislocation structures investigated by X-ray diffraction[END_REF] or from discrete dislocation simulations. This association of (i) non-linear dynamics (a local yield stress), (ii) long-ranged elastic stresses, and (iii) randomness, is similar to statistical models of fracture of disordered media (e.g. [START_REF] Herrmann | Statistical models for the fracture of disordered media[END_REF][START_REF] Cowie | Multifractal scaling properties of a growing fault population[END_REF]) that successfully model the heterogeneity and scaling properties of fracture patterns, although in this last case the randomness introduced is uncorrelated (white noise). These stochastic models of plasticity successfully reproduce fluctuations, heterogeneity and scaling properties, such as the intermittency of plastic flow, or the spatial slip patterns described above. As an example, in the model of Zaiser and Moretti [START_REF] Zaiser | Fluctuation phenomena in crystal plasticity -a continuum model[END_REF], a "surface profile" is obtained, following relation [START_REF] Taylor | The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical[END_REF], by a direct integration of the strain pattern along a direction perpendicular to the single-slip direction. This simulated profile is self-affine (relation ( 3)) with an exponent H>0.5, i.e. the hallmark of long-ranged spatial correlations, in agreement with experiments. This suggests that the phenomenological laws (e.g. local hardening laws) introduced in the model capture the essential features of collective dislocation interactions, although such modelling framework introduces a characteristic length scale in the system (the mesoscopic scale over which field variables are defined, which defines a lower cut-off) whereas observations reported above argue for scale invariance from the nm scale to the macroscopic scale. Therefore, these approaches appear as promising compromises between discrete dislocation dynamics models, which computational costs are still much too high (especially for 3D simulations) to explore meso-to macro-scopic scales, and classical phenomenological models which are unable to account for heterogeneity of plastic flow and associated scaling laws.

Conclusion

Recent detailed analyses of slip and dislocation density patterns in various deformed materials, either performed from a roughness analysis on the samples surface, or from X-ray topography within the bulk, revealed the emergence during deformation of scale invariant patterns characterized by long-range correlations, and resulting from the collective behaviour of dislocations. In single crystals, scale invariance holds up to the sample size, whereas in polycrystals, the upper bound of scaling, i.e. the correlation length, seems to be set by the average grain size. This scale invariance implies that the spatial fluctuations of dislocation density and/or slip never vanish as one coarsens the observation scale up to the correlation length. An immediate consequence is that a priori obvious concepts such as "slip bands" or dislocation density can be ill-defined. Similarly, the concept of "statistically stored dislocations" introduced by the strain gradient plasticity theory has to be taken with caution in these cases. Instead of the separation between GND's and SSD's as originally defined by Ashby [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF], we propose to distinguish between "externally imposed" and "self-induced" strain gradients, and the associated dislocation populations. This scale invariant heterogeneity of slip emerging from the population of dislocations and their elastic mutual interactions also challenges the dislocation density-based modelling of plasticity. Recent modelling developments associating phenomenological laws on mesoscopic variables, long-ranged elastic stresses, and the introduction of randomness with statistical characteristics, such as variance or correlation properties, deduced from experiments or from discrete dislocation simulations, appear as promising directions.

Figure captions : Figure 1 .Figure 2 .

 :12 Figure captions:

Figure 4 .

 4 Figure 4. Fourier power spectrum of the intensity record of figure 3b. The slope α=1.3 is represented. The white dots are averages over bin regularly separated in logarithmic scale.

Figure 5 .Figure 1 .Figure 2 .

 512 Figure 5. Evolution of the spatial fluctuations of the dislocation density ρ, measured by the ratio δ, with the scale of observation, for the intensity profile of figure 3b. The intensity I is a proxy of ρ, so δ is determined from δ=〈ρ 2 〉Δ x 1/2 /〈ρ〉Δ x

Figure 4 .

 4 Figure 4. Fourier power spectrum of the intensity record of figure 3b. The slope α=1.3 is represented. The white dots are averages over bin regularly separated in logarithmic scale.

Figure 5 .

 5 Figure 5. Evolution of the spatial fluctuations of the dislocation density ρ, measured by the ratio δ, with the scale of observation, for the intensity profile of figure 3b. The intensity I is a proxy of ρ, so δ is determined from δ=〈ρ 2 〉Δ x 1/2 /〈ρ〉Δ x

Acknowledgments

We would like to thank M. Zaiser and C. Fressengeas for fruitful discussions, as well as Jurgen Hartwig and José Baruchel from the ID 19 beamline of ESRF. The comments of another anonymous reviewer are also acknowledged.