¸ois Bacher

POINCAR É METRIC OF HOLOMORPHIC FOLIATIONS WITH NON-DEGENERATE SINGULARITIES

Keywords: Mathematics Subject Classification. Primary 37F75, Secondary 37A Singular holomorphic foliation, Leafwise Poincaré metric

Consider a Brody hyperbolic foliation F with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of F .

INTRODUCTION

The ergodic theory of laminations by Riemann surfaces has received a lot of attention during the last two decades. Much progress has been focused on the global dynamical properties of laminations by hyperbolic Riemann surfaces with singularities, and in particular singular holomorphic foliations. Indeed, most of interesting holomorphic foliations are singular and hyperbolic. In the case of P n , all foliations are singular. For d P N, let F d pP n q be the space of foliations of degree d on P n . By the results of Lins Neto and Soares [START_REF] Neto | Algebraic solutions of one-dimensional foliations[END_REF] and Jouanolou [START_REF] Jouanolou | Équations de Pfaff algébriques[END_REF], the singularities of a generic foliation F P F d pP n q, for d ě 2, are all non-degenerate. Moreover, by the results of Glutsyuk [START_REF] Glutsyuk | Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety[END_REF] and Lins Neto [START_REF] Neto | Uniformization and the Poincaré metric on the leaves of a foliation by curves[END_REF], if all the singularities of a foliation F P F d pP n q, for d ě 2, are non-degenerate, then the foliation F is hyperbolic. It is even Brody hyperbolic in the sense of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. Moreover, Loray and Rebelo build in [START_REF] Loray | Minimal, rigid foliations by curves on CP n[END_REF] a non-empty open set in F d pP n q in which every foliation has all its leaves that are dense. When n " 2, Nguyên computes the Lyapunov exponent of a generic foliation F P F d pP 2 q in [START_REF] Nguyên | Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2[END_REF][START_REF] Viêt-Anh | Singular holomorphic foliations by curves II: Negative Lyapunov exponent[END_REF]. We recall briefly some recent developments and refer the reader to the survey articles [START_REF] Fornaess | Riemann surface laminations with singularities[END_REF][START_REF] Dinh | Some open problems on holomorphic foliation theory[END_REF][START_REF] Nguyên | Ergodic theory for Riemann surface laminations: a survey[END_REF][START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF] for more detailed expositions.

By solving heat equations with respect to a positive harmonic current, Dinh, Nguyên and Sibony obtain in [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF] abstract geometric ergodic theorems for laminations and singular holomorphic foliations. As a consequence, they obtain a concrete and effective Birkhoff ergodic theorem in the case of holomorphic foliations with linearizable singularities. In a series of two articles [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF][START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF], these authors study the modulus of continuity of the leafwise Poincaré metric for a compact lamination and for a holomorphic foliation with linearizable singularities. Using this transversal regularity, they introduce the hyperbolic entropy and establish the finiteness of this entropy for a large class of foliations in dimension 2 with linearizable singularities.

Let us recall more precisely their results about the Poincaré metric. Let M be a compact complex manifold and F " pM, L , Eq be a smooth foliation by hyperbolic Riemann surfaces. That is, L is an atlas of flow boxes of M zE and E is the singular set of F . Let g P be the leafwise Poincaré metric of F and fix an ambient smooth Hermitian metric g M on M . Consider then the function η : M zE Ñ p0, `8q given by η 2 g P " 4g M , where on the right hand side g M is restricted to each leaf of F . The normalization of this function will be explained in Subsection 2.1. The two main theorems of Dinh, Nguyên and Sibony we are interested in are the following. Theorem 1.1 (Dinh-Nguyên-Sibony [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF]). Let F " pM, L q be a transversally smooth compact foliation by hyperbolic Riemann surfaces. Then, the function η is transversally Hölder continuous. Moreover, the Hölder exponent can be expressed in geometric terms. Theorem 1.2 (Dinh-Nguyên-Sibony [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]). Let F " pM, L , Eq be a Brody hyperbolic singular holomorphic foliation on a compact complex manifold M . Suppose that all singularities of F are linearizable. Then, there exist constants C ą 0 and α P p0, 1q such that |ηpxq ´ηpyq| ď C ˆmax plog ‹ d px, Eq , log ‹ d py, Eqq log ‹ dpx, yq ˙α , x, y P M zE, where log ‹ " 1 `|log| is a log-type function, d px, Eq denotes the distance from x to the singular set E and dp¨, ¨q is the distance induced by the ambient Hermitian metric g M .

So, it is natural to wonder whether these basic questions can be solved when the singularities of F are more general. The purpose of this work is to provide a positive answer to the problem of transversal regularity of the Poincaré metric when the singularities are all non-degenerate. Our main result is the following. Theorem 1.3. Let F " pM, L , Eq be a Brody hyperbolic singular holomorphic foliation on a compact complex manifold M with Hermitian metric g M . Suppose that all the singularities of F are non-degenerate. Then, there exist constants C ą 0 and α P p0, 1q such that Let us explain the method of our proof. We basically follow the approach of Dinh, Nguyên and Sibony in [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] using the Beltrami equation. However, their proof uses the peculiar geometry of a linearizable singularity as a local model in order to study the regularity of the Beltrami coefficient near the singularities. Namely, they use a sort of local invariance by homothety of a linearizable singularity. In the context of a nondegenerate singularity, such a local model is not available anymore. The novelty of our work is to settle the same kind of estimates as theirs in this more general situation.

The article is organized as follows. In Section 2, we recall some definitions about hyperbolic singular holomorphic foliations and the Poincaré metric. We specify the type of singularities we will consider, to point out our improvements to Theorem 1.2. We also state some general results, which will be useful in our proof. Next, we build in Section 3 a local orthogonal projection from a leaf onto another in singular charts. We get precise estimates on its C 2 ´norm using that the singularities are non-degenerate. More precisely, we use only the speed of expansion of a flow in a leaf L x with respect to the Hermitian metric g M . We get the same kind of estimates as in the case of linearizable singularities. Finally, we prove Theorem 1.3 in Section 4. We follow closely the proof of Theorem 1.2, with a slightly different exposition.

Notations. Throughout this paper, we will denote by D the unit disk of C, and rD (respectively rD) the open (respectively closed) disk of radius r P R ˚for the standard Euclidean metric of C. For R P R ˚, we will also denote by D R (respectively D R ) the open (respectively closed) disk of hyperbolic radius R in D, so that D R " rD with r " e R ´1 e R `1 , or if r P p0, 1q, with R " ln 1`r 1´r . More generally, for ρ P R ˚and U a subset of a vector space, ρU will denote the image of U by the homothety z Þ Ñ ρz.

Throughout this paper, we will denote by C, C 1 , C 2 , etc. . . positive constants which will not always be the same.

Acknowledgments.

The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the project QuaSiDy (ANR-21-CE40-0016). We would like to thank Julio Rebelo for helping us to find Example 4.12. We are grateful to the referee for his helpful remarks.

PRELIMINARIES

2.1. Poincaré metric of a holomorphic foliation. Let F " pM, L , Eq be a singular holomorphic foliation by curves on a complex manifold M . For x P M zE, we denote by L x the leaf of F passing through x.

For two complex manifolds N and N 1 , let H pN, N 1 q be the set of holomorphic functions from N to N 1 . Define

H pD, F q " ␣ f P H pD, M zEq | f pDq Ă L f p0q ( .
Fix a Hermitian metric g M on M and define the function η : M zE Ñ p0, `8s as follows.

ηpxq " sup

! }α 1 p0q} g M | α P H pD, F q , αp0q " x ) ,
where }v} g M is the norm of v P T x M with respect to the metric g M , that is }v} g M " a g M,x pv, vq.

The function η is defined to satisfy the following facts, proven by Verjovsky [START_REF] Verjovsky | A uniformization theorem for holomorphic foliations. The Lefschetz centennial conference, Part III[END_REF].

Proposition 2.1.

(1) For x P M zE, ηpxq ă `8 if and only if the leaf L x is hyperbolic, that is, it is uniformized by the Poincaré disk D.

(2) If L x is hyperbolic, we have ηpxq " }u 1 p0q} g M , where u : D Ñ L x is any uniformization of L x such that up0q " x. (3) If L x is hyperbolic, then 4g M η 2 induces the Poincaré metric on L x . In the general case, the regularity of the function η is very weak. In fact, it is not always continuous. Fornaess and Sibony have only shown in [START_REF] Fornaess | Riemann surface laminations with singularities[END_REF]Theroem 20] that it is lower semicontinuous if M is compact. They also give sufficient conditions for η to be continuous. In our context, they are not necessarilly satisfied.

We recall from [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]

F j B Bz j , F j P H pU, Cq , 1 ď j ď n.
If in some chart centered at p, X is of the form ř n j"1 λ j z j B Bz j with all λ j ‰ 0, we say that p is a linearizable singularity.

The functions F j can be developed as a power series F j " ř αPN n c α,j z α . The 1-jet of X at p is defined in the chart pU, zq as

X 1 " n ÿ j"1 ÿ |α|ď1 c α,j z α B Bz j .
See for example [9, Chapter I] for more details. If the 1-jet of X has an isolated singularity at p, we say that p is a non-degenerate singularity. Proof. Take r " e R ´1 e R `1 the corresponding Euclidean radius such that |z| ď r and |w| ď r. Using the straight line from z to w, we get the following bound for the Poincaré distance between z and w.

d P pz, wq ď 2 |z ´w| 1 ´r2 ď e R |z ´w| , if R is sufficiently large, since 1 ´r2 " 4e R pe R `1q 2 . □
We also recall the following version of the Grönwall Lemma, which will be useful for our further work. See for example [START_REF] Palis | Welington: Geometric theory of dynamical systems. An introduction[END_REF] for a proof. For f : U Ñ V a C 1 function between two vector spaces or manifolds, and for u P U , we denote by d u f the differential of f with respect to u. Lemma 2.5 (Grönwall). Let φ be a C 1 function from an open ball B r centered at the origin of radius r in R m to C n . Let also C be a non negative constant. If we suppose that for all x P B r , }d x φ} ď C }φpxq} , then for x P B r , we have }φpxq} ď }φp0q} e C}x} .

LOCAL STUDY OF THE ORTHOGONAL PROJECTION ONTO A LEAF

We want to study the orthogonal projection from a leaf L x near x onto another leaf L y near y inside a singular chart of F . Consider then a foliation F on 3ρD n for some ρ ą 0 with a non-degenerate singularity at the origin. We suppose that F is generated by a holomorphic vector field X on 3ρD n . We endow 3ρD n with the usual Hermitian product x¨, ¨y of C n and the associated norm }¨}. Since 0 is a non-degenerate singularity, there exists a constant C 0 ą 1 such that

(3.1) C ´1 0 }z} ď }Xpzq} ď C 0 }z} , z P 3ρD n .
For z P 2ρD n , denote by φ z the flow of X starting at z. Namely, φ z : pC, 0q Ñ 3ρD n is a maximal solution of the Cauchy problem

$ & % Bφ z Bt ptq " Xpφ z ptqq φ z p0q " z.
Taking the derivative of the first line with respect to t, we find that there exists a constant C 1 ą 0 such that

(3.2) › › › › B 2 φ z Bt 2 ptq › › › › ď C 1 }Xpφ z ptqq} , › › › › B 3 φ z B 3 t ptq › › › › ď C 1 }Xpφ z ptqq} . Since we have › › Bφz Bt ptq › › ď C 0 }φ z ptq}, it follows from Grönwall Lemma that φ z is at least defined on a disk of radius r 0 " 1 C 0 ln 3 2 .
Again by Grönwall Lemma, it is also clear that for t P r 0 D, }φ z ptq} ě 2 3 }z}. In particular, there exists a constant C ą 1 such that C ´1 }Xpzq} ď }Xpφ z ptqq} ď C }Xpzq} for t P r 0 D.

We first show that the local orthogonal projection exists, is unique, and that it is C 1 . For x, y P 2ρD n zt0u, define

(3.3)
gpt, uq " B }φ x ptq ´φy puq} 2 Bu pt, uq " xφ x ptq ´φy puq, Xpφ y puqqy .

For t, u P r 0 D, if gpt, uq " 0, then the vector φ x ptq ´φy puq is orthogonal to the tangent line to L y in φ y puq. That is how we will characterize the local orthogonal projection.

Lemma 3.1.

There exist a radius r 1 ă r 0 and a positive number ε 0 satisfying the following conditions. Let x, y be two points in 2ρD n zt0u such that }x ´y} ď ε }Xpxq} for ε ă ε 0 . For t P r 1 2 D, there exists a unique u P r 1 D satisfying gpt, uq " 0. Furthermore, |t ´u| " Opεq. Moreover, Jac u pgqpt, uq ‰ 0. In particular, by the implicit function theorem, there exists a C 1 function f : r 1 2 D Ñ r 1 D such that gpt, f ptqq " 0 for t P r 1 2 D. Proof. Let us note first that for |t| ă r 0 , there exists a constant C ą 1 such that C ´1 }x ´y} ď }φ x ptq ´φy ptq} ď C }x ´y}. Indeed, by Grönwall Lemma, C " e cr 0 , for c such that }Xpx 1 q ´Xpy 1 q} ď c }x 1 ´y1 }. Define N puq " }φ x ptq ´φy puq} 2 . By compactness of r 1 D, N attains its minimum somewhere in u 0 P r 1 D. N is of class C 1 , so either u 0 satisfies BN Bu pt 0 q " gpt, u 0 q " 0, or |u 0 | " r 1 . We show that for u too far from t, N puq ą }φ x ptq ´φy ptq} 2 " N ptq. Namely, }φ x ptq ´φy puq} ě |t ´u| }Xpφ y ptqq} ´}φ x ptq ´φy ptq} ´}φ y ptq ´φy puq ´pt ´uqXpφ y ptqq} , ě `|t ´u| `Opεq `Op|t ´u| 2 q ˘}Xpφ y ptqq} , Note that }Xpφ y ptqq} ě C }Xpxq}. It follows that for a sufficiently small r 1 , ε 0 sufficiently small chosen in consequence and |t ´u| ą Opεq, }φ x ptq ´φy puq} ą Cε }Xpxq} ě }φ x ptq ´φy ptq}. Therefore, t ´u0 " Opεq, u 0 does not belong to the boundary t|u| " r 1 u and gpt, u 0 q " 0. We give in Figure 1 a schematic view of the quantities involved in our computation. Now, we come to the uniqueness of u 0 . Take another u such that gpt, uq " 0 and |u| ă r 1 , and compute gpt, uq ´gpt, u 0 q " 0 " xφ x ptq ´φy pu 0 q, Xpφ y puqq ´Xpφ y pu 0 qqy `xφ y pu 0 q ´φy puq, Xpφ y puqqy , 0 " `pu 0 ´uq `Op|u ´u0 | 2 q `Opε |u ´u0 |q ˘}Xpφ y puqq} 2 . L x L y Xpx 1 q

x 1 " φ x ptq Xpy 1 q y 1 " φ y ptq pt ´uqXpy 1 q φ y puq Op|t ´u| 2 q Opε}Xpxq}q FIGURE 1. Proof of Lemma 3.1. Quantities involved in the proof that u cannot be too far from t.

For r 1 and ε small enough, this is possible only if u " u 0 and hence u 0 is unique. It remains to show that the Jacobian with respect to u does not vanish in pt, u 0 q. Derivating (3.3) with respect to u and ū, we get

(3.4) Bg Bu pt, uq " ´}Xpφ y puqq} 2 , Bg Bu pt, uq " B φ x ptq ´φy puq, B 2 φ y Bu 2 puq F . Using (3.2), it follows that (3.5) Jac u pgqpt, u 0 q " Bg Bu Bg Bu Bg Bu Bg Bu pt, u 0 q ě p1 ´Cεq }Xpφ y pu 0 qq} 4 ě C 2 }Xpxq} 4 ą 0.
The existence and regularity of the function f is a consequence of the implicit function theorem and the uniqueness of u 0 to glue together the local implicit functions. □

We will need in Section 4 a C 2 ´estimate for the orthogonal projection. We begin by controlling the C 2 ´norm of f . Lemma 3.2. With the notations of Lemma The estimate on the C 1 ´norm follows. We continue with the derivatives of order 2 of f ´id. Recall that

d A det ¨H " Tr ´pcompAqq T H ¯,
where compAq denotes the comatrix of A, Tr is the trace operator, and d A det is the differential of the determinant with respect to the matrix A. Denote by Jptq " Jac u pgqpt, f ptqq and differentiate the holomorphic part of (3.6). We will also need some estimates on the distances induced by the standard Hermitian metric restricted to L x and L y . Denote by d L the distance induced by the restriction of the standard Hermitian metric on a leaf L. Let x 1 and x 2 be two points in a sufficiently small neighbourhood U of x in L x such that φ x is bijective from a neighbourhood of 0 onto 2U . This neighbourhood can be chosen of radius Op}x}q by (3.1) and (3.2). Suppose that x 1 " φ x pt 1 q and x 2 " φ x pt 2 q and denote by γ a path of minimal length in L x from x 1 to x 2 . The path γ stays in 2U . Then, φ y ˝f ˝φ´1

d
x ˝γ is a path in L y from Φ xy px 1 q to Φ xy px 2 q. It follows easily that there exists a constant C 6 such that (3.9)

d Ly pΦ xy px 1 q, Φ xy px 2 qq ď C 6 d Lx px 1 , x 2 q.

PROOF OF THE MAIN THEOREM

We are now ready to prove Theorem 1.3. Throughout this section, we will have to consider three different distances and will be very careful to distinguish them by our notations. We denote by d the distance induced by g M on M . For L a leaf of F , we consider two distances: ' d P the distance induced by the Poincaré metric on L ; ' d L , as in Section 3, the distance induced by the restriction of g M on L. For a set K and two functions u, v from K to L or M , we define the distances between u and v. By compactness of M , it is sufficient to show the result (1.1) only for x and y close to each other. Hence, we will suppose that

max plog ‹ d px, Eq , log ‹ d py, Eqq log ‹ dpx, yq ď e ´α´1 R ,
for R big enough and some α that we will specify later.

4.1. Geometric setup. Note that the existence of C in (1.1) is independent of the metric g M . Then, we may build g M by a partition of the unity satisfying some peculiar hypothesis that will help us in our proof. By compactness of M , we can find a finite open covering U " pU p , U i q pPE,iPI , where

(1) U p is a neighbourhood of p and we have a holomorphic chart U p » 4ρD n centered at p in which g M is the standard Hermitain metric }dz} 2 on a smaller polydisk 3ρD n . We suppose that F is generated on U p by some holomorphic vector field X p . (2) pU i q iPI is an open covering of M z Ť pPE 1 4 U p by flow boxes. We also suppose that F is generated on U i by a holomorphic vector field X i . To simplify the computations, we suppose that the diameter of M for g M is lower than 1. That way, for δ a distance on M , log ‹ δ " 1´log δ. Denote by d ą 0 some small distance such that any point of M admits a holomorphic neighbourhood of radius at least d. We suppose that F is Brody hyperbolic and we will denote by A the constant appearing in Definition 2.2.

The following important local estimate on η is needed.

Theorem 4.1 (Lins Neto-Canille Martins [START_REF] Neto | Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves[END_REF]Theorem 3]). Let F be a holomorphic foliation by Riemann surfaces on a ball rB with non-degenerate singularity at the origin, where B is the unit ball of C n . Then there exists a smaller radius 0 ă ρ ă r such that

C ´1 }z} log ‹ }z} ă ηpzq ă C }z} log ‹ }z} , z P ρBzt0u.
We obtain the following global estimate on η.

Proposition 4.2.

There exists a constant C 1 ą 1 such that C ´1 1 d px, Eq log ‹ d px, Eq ď ηpxq ď C 1 d px, Eq log ‹ d px, Eq , x P M zE. Proof. Suppose first that x P M zE is far from E. By the embedding of the plaque passing through x in a flow box, we get ηpxq ą C. Moreover, by Brody hyperbolicity of F , we have ηpxq ă A. Since x is far from E, the conclusion of Proposition 4.2 follows.

Suppose next that x P 3 4 U p for some p P E. Considering the restriction of F to U p and applying Theorem 4.1, we get a holomorphic function u : D Ñ L x X U p such that up0q " x and ηpxq ě }u 1 p0q} g M ą C ´1d px, Eq log ‹ d px, Eq. For the upper estimate, recall that F is Brody hyperbolic. Hence, there exists a radius R 0 , independant of x P 3 4 U p , such that the disk in L x of radius R 0 with respect to the Poincaré metric is contained in U p . Denote by u x a uniformization of L x such that u x p0q " x. For the corresponding Euclidean radius r 0 " e R 0 ´1 e R 0 `1 , we have u x pr 0 Dq Ă U p . By the extremal property of Poincaré metric and Theorem 4.1, we get ηpxq ă r ´1 0 Cd px, Eq log ‹ d px, Eq. □

We are going to prove our Hölder estimate by building a nearly holomorphic mapping ψ : D R Ñ L y , which is close to a uniformization of L x . Next, we solve a Beltrami equation to have a holomorphic function close to ψ which will make ηpxq and ηpyq close. To produce this nearly holomorphic mapping, we shall use the orhogonal projection we studied earlier. Note that in [3, Lemmas 2.11 and 2.12], the three authors use the invariance by homothety of F in the case of linearizable singularities. Here, we had to establish in Section 3 the same estimates for orthogonal projections, since this trick is not available anymore in our context.

Denote by L x rrs the ball of center x and radius r for the distance d Lx . What we have proven in Section 3 (see (3.8), (3.9) and the fact that Φ xy is defined on a neighourhood of radius Op}x}q), together with the classical estimates of orthogonal projection on nonsingular flow boxes, can be summarized by the following lemma in our geometric setup.

Lemma 4.3.

There exist constants ε 0 , ε 1 , k and K such that for x, y P M zE, if dpx, yq ď ε 1 d px, Eq, then there exists a local orthogonal projection (4) If x 1 P L x rε 0 d px, Eqs, y 1 P L y rkε 0 d px, Eqs and dpx 1 , y 1 q ď ε 1 d px 1 , Eq, then Φ x 1 y 1 " Φ xy on the intersection of their domains of definition.

Φ xy : L x rε 0 d px, Eqs Ñ L y rkε 0 d py,

4.2.

Proof of the Hölder estimate. We begin our proof with the second step. Take x, y P M zE and take u x : D Ñ L x a uniformization of L x such that u x p0q " x. Such a covering is unique up to a rotation. Take also u y : D Ñ L y a uniformization of L y such that u y p0q " y. The following lemma is very similar to [3, Proposition 3.6], the proof of which is partially referred to [2, Proposition 2.2].

Lemma 4.4. Let R ą 0 and suppose that

(1) There exists a smooth map ψ : D R Ñ L y without critical point such that ψp0q " y and d D R pψ, u x q ď e ´2R .

(2) }dψ} 8 ď 2A for the constant A appearing in Definition 2.2. We consider the norm of dψ with respect to the Poincaré metric on the source D R and the restriction of g M on the goal L y . (3) There exists a smooth map Ψ : D R Ñ D such that Ψp0q " 0, u y ˝Ψ " ψ and the Beltrami coefficient µ Ψ of Ψ satisfies }µ Ψ } C 1 ď e ´2R . Here, we consider the norm of µ Ψ with respect to the standard Euclidean metric on D at the goal and the source.

If R is sufficiently large, then there exists a constant C 2 such that

ηpxq ´ηpyq ď C 2 e ´R.
We recall that the Beltrami coefficient µ Ψ is defined by BΨ Bt " µ Ψ BΨ Bt . Remark 4.5. Note that the existence of Ψ in (3) such that Ψp0q " 0 and u y ˝Ψ " ψ is a consequence of ( 1) and the fact that D R is simply connected. The smooth map Ψ is just the unique lifting of ψ via u y such that Ψp0q " 0.

Proof. Let us first translate our conditions in a commutative diagram. The mention H (resp. C 8 ) on an arrow will denote a holomorphic (resp. smooth) function.

D uy H D R Ψ C 8 9 9 C 8 ψ / / _ L y D H ux / / L x .
We want to find v : D R Ñ D holomorphic and q : D R Ñ D R smooth and close to the identity such that Ψ " v ˝q. To make v holomorphic, we should have Bq Bt " µ Ψ

Bq

Bt . It was proven by Schatz and Earle in [START_REF] Earle | Teichmüller theory for surfaces with boundary[END_REF] that this Beltrami equation can be solved with qp0q " 0, and }q ´id} 8 ď κ }µ Ψ } C 1 ď κe ´2R , for some constant κ ą 0. We also have }q ´1 ´id} 8 ď κe ´2R for another constant κ. By construction, v " Ψ ˝q´1 is holomorphic. Let us complete our commutative diagram.

D R v H / / D uy H D R Ψ C 8 9 9 C 8 ψ / / _ q C 8 O O L y D H ux / / L x
The estimate for η will come from a comparison between the two holomorphic mappings u y ˝v and u x . Denote by v y : D Ñ L y defined by v y ptq " u y ˝vprtq with r " e R ´1 e R `1 the Euclidean radius associated to R. Since v y is holomorphic and v y p0q " ψ ˝q´1 p0q " y, by definition of η, we have ηpyq ě

› › v 1 y p0q › › g M
. Consider a radius r 0 such that u x `r0 D ˘and v y `r0 D ˘are both included in a common holomorphic chart of M . It is clear that this radius can be chosen at least equal to d´e ´2R 2A ě d 4A if R is large enough. We apply the Cauchy formula in this chart. We get (4.1)

› › u 1 x p0q ´v1 y p0q › › g M,x ď C › › u 1 x p0q ´v1 y p0q › › ď 4AC d sup tPr 0 D }u x ptq ´vy ptq} ď 4CC 1 A d sup tPr 0 D dpu x ptq, v y ptqq,
where the norm without index is the Euclidean norm in the chart, and the constants C, C 1 are given by the equivalence of Hermitian metrics. Note also that we have

(4.2) ηpxq ´ηpyq ď }u 1 x p0q} g M,x ´› › v 1 y p0q › › g M,y ď › › u 1 x p0q ´v1 y p0q › › g M,x `C2 Adpx, yq,
where the constant C 2 comes from the difference between

› › v 1 y p0q › › g M,x and › › v 1 y p0q › › g M,y
, that is, from the spatial variation of g M . Since dpx, yq ď e ´2R , the second term is well controlled. By (4.2) and (4.1), it is sufficient to estimate d r 0 D pu x , v y q. Take t P r 0 D and remember that u y ˝v " ψ ˝q´1 . We have

(4.3) dpu x ptq, v y ptqq ď dpu x ptq, u x prtqq `dpu x prtq, ψprtqq `d `ψprtq, ψ `q´1 prtq ˘ď p1 ´rqr 0 sup τ Pr 0 D }u 1 x pτ q} g M `e´2R `}dψ} 8 d P `rt, q ´1prtq ˘.
Using an automorphism of D, we show that for |τ | ď r 0 , }u

1 x pτ q} g M " }u 1 uxpτ q p0q} g M 1´|τ | 2 ď A 1´r 2 0 . Moreover, 1 ´r " 2 e R `1 ď 2e ´R.
Since d pq ´1prtq, rtq ď Ce ´2R , by Lemma 2.4, d P prt, q ´1prtqq ď Ce ´R. Back to (4.3), we get some constant C 1 such that

dpu x ptq, v y ptqq ď C 1 e ´R.
Finally, by (4.1) and (4.2), we obtain ηpxq ´ηpyq ď C 2 e ´R.

□

We continue the proof of Theorem 1.3 similarly to [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Proposition 3.7]. We show that if x and y are sufficiently close, then there exist ỹ P L y close to y and x P L x close to x such that x, ỹ and y, x satisfy the hypothesis of Lemma 4.4. We suppose more precisely that (4.4) log ‹ dpx, yq ě e α ´1R max plog ‹ d px, Eq , log ‹ d py, Eqq , for some fixed α we will specify later. We begin by some work on L x which will be helpful to build the function ψ by orthogonal projection. Consider ξ P D such that d P p0, ξq " R. We want to subdivide the geodesic r0, ξs in subsegments rξ j , ξ j`1 s, for j P 0, N ´1 . Set ξ 0 " 0 and for j P 0, N , x j " u x pξ j q. Choose at each step ξ j`1 P rξ j , ξs such that for all ζ P rξ j , ξ j`1 s, d Lx pu x pζq, x j q ď ε 0 d px j , Eq and d Lx px j`1 , x j q " ε 0 d px j , Eq. At last step, we set ξ N " ξ and suppose that for all ζ P rξ N ´1, ξ N s, d Lx pu x pζq, x N ´1q ď ε 0 d px N ´1, Eq. Denote by R j " d P p0, ξ j q and r j " |ξ j |. This setup will enable us to consider the local orthogonal projections Φ x j y j by Lemma 4.3, where y 0 " y and y j`1 " Φ x j y j px j`1 q (see Lemma 4.9 below).

At this stage, it is not certain that this process stops and that N is finite, but the lemmas below will give us a bound for N . Lemma 4.6. We have the following estimate comparing d Lx px j , x j`1 q and d P pξ j , ξ j`1 q. d Lx px j , x j`1 q ď 1 2 sup rξ j ,ξ j`1 s pη ˝ux q d P pξ j , ξ j`1 q " 1 2 sup rξ j ,ξ j`1 s pη ˝ux q pR j`1 ´Rj q.

Proof. Denote by γ a parametrization of the geodesic rξ j , ξ j`1 s. We have

d Lx px j , x j`1 q ď ż 1 0 › › pu x ˝γq 1 ptq › › g M dt ď sup rξ j ,ξ j`1 s pη ˝ux q ż 1 0 |γ 1 ptq| }u 1 x pγptqq} g M ηpu x pγptqqq dt. Now, }u 1 x pγptqq} g M " }u 1 uxpγptqq p0q} g M 1´|γptq| 2
" ηpuxpγptqqq 1´|γptq| 2 . It follows that d Lx px j , x j`1 q ď sup rξ j ,ξ j`1 s pη ˝ux q

ż 1 0 |γ 1 ptq| 1 ´|γptq| 2 dt ď 1 2 sup rξ j ,ξ j`1 s
pη ˝ux q d P pξ j , ξ j`1 q.

Since d P pξ j , ξ j`1 q " R j`1 ´Rj , the result of the lemma follows. □

The following lemma is an analog of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Lemma 3.4] Let γ : r0, 1s Ñ D be a parametrization of the geodesic from σ to ζ. Consider τ " sup tt P r0, 1s | @t 1 P r0, ts , d pu x pγpt 1 qq, Eq ď min pd pz, Eq , 3ρqu, with by convention τ " 0 if d pw, Eq ą 3ρ. Denote by ζ 1 " γpτ q and w 1 " u x pζ 1 q. For t P r0, τ s, u x pγptqq is contained in a singular chart 3 4 U p on which g M is the standard Hermitian metric of C n . Therefore d pu x pγptqq, Eq " }u x pγptqq} and the second inequality of Proposition 4.2 will apply on it. We split our integral in two. pη ˝ux q .

Denote by Σ " sup rξ j ,ξ j`1 s pd pu x , Eqq. The function t Þ Ñ t log ‹ t is increasing on p0, 1q, thus R j`1 ´Rj ě 2ε 0 C ´1 1 d px j , Eq Σ log ‹ Σ .

Since for ζ P rξ j , ξ j`1 s, d Lx pu x pζq, x j q ď ε 0 d px j , Eq, we have Σ ď p1 `ε0 qd px j , Eq. It follows that

R j`1 ´Rj ě 2ε 0 C ´1 1 1 `ε0 1 log ‹ Σ .
Moreover, by Lemma 4.7, log ‹ Σ ď e C 3 `C4 R log ‹ d px, Eq. Hence,

R j`1 ´Rj ě 2ε 0 C ´1 1 e ´C3 ´C4 R p1 `ε0 q log ‹ d px, Eq .
This bound is uniform in j, and it is clear that

N ď R inf j pR j`1 ´Rj q ď C 1 e C 3 p1 `ε0 q 2ε 0 log ‹ d px, Eq Re C 4 R .

□

We are now able to construct the function ψ that we need to apply Lemma 4.4.

Lemma 4.9. Suppose that x, y P M zE are such that (4.4) for a sufficiently large R. Then, there exists a smooth function ψ : D R Ñ L y without critical point, such that d D R pψ, u x q ď e ´2R and }dψ} 8 ď 2A. Here, we consider the norm of dψ with respect to the Poincaré metric on the source D R and the restriction of g M on the goal L y .

Proof. Note that d Lz pz, u z ptqq ď Ad P p0, tq. Hence, d P p0, tq ď R 0 " d A implies that u z ptq is contained in a holomorphic chart of M . Consider r 0 " e R 0 ´1 e R 0 `1 , |t| ď r 0 2 and apply Cauchy formula. We get

}u 2 x ptq} g M ď C }u 2 x ptq} ď C πr 0 ż 2π 0 › › u 1 x `r0 e iθ ˘› › dθ ď CC 1 p1 ´r2 0 q πr 0 ż 2π 0 › › u 1 x `r0 e iθ ˘› › g M 1 ´r2 0 dθ ď 2CC 1 Ap1 ´r2 0 q r 0 ,
where the constants C and C 1 come from the equivalence of the standard Hermitian metric and g M in the holomorphic chart. □

Proof of Lemma 4.10. Recall that locally, Ψ " u ´1 ỹ ˝Φj ˝ux , where Φ j " Φ x j y j . An explicit computation and Lemma 4.3 give that (4.8)

|µ Ψ | `ˇˇˇB µ Ψ Bt ˇˇˇ`ˇˇˇB µ Ψ Bt ˇˇˇď ˜1 `› › `BΦ j ˘˝u x › › }pBΦ j q ˝ux } ¸ˆ}u 1 x } }pBΦ j q ˝ux } › › D 2 Φ j ˝ux › › `2 › › u 2 y ˝Ψ› › }u 1 x } › › u 1 y ˝Ψ› › 2 › › `BΦ j ˘˝u x › › ¸`ˆ1 `2}u 2 x } }u 1 x } ˙› › `BΦ j ˘˝u x › › }pBΦ j q ˝ux } ď 3e K d D R pu x , ψq d px j , Eq ˜1 `}u 1 x } d px j , Eq `› › u 2 ỹ ˝Ψ› › }u 1 x } › › u 1 ỹ ˝Ψ› › 2 `}u 2 x } }u 1 x } ¸.
For t P D, consider the automorphism f t : z Þ Ñ z`t 1`tz of D, and define v z " u z ˝ft " u uzptq . We get

u 1 z ptq " v 1 z p0q 1 ´|t| 2 , u 2 z ptq " v 2 z p0q `1 ´|t| 2 ˘2 `2tv 1 z p0q `1 ´|t| 2 ˘2 .
Applying this to z " x and z " ỹ, using Brody hyperbolicity, Lemma 4.11 and Proposition 4.2 in (4.8), we obtain

|µ Ψ ptq| `ˇˇˇB µ Ψ Bt ptq ˇˇˇ`ˇˇˇB µ Ψ Bt ptq ˇˇˇď 3e K 1 ´|t| 2 d D R pu x , ψq d px j , Eq ˆ1 `A d px j , Eq `C6 A `2A 2 C 2 1 pd pψ, Eq log ‹ d pψ, Eqq 2 `C6 `2A C 1 d pu x , Eq log ‹ d pu x , Eq ˙.
Using Lemma 4.7 on d px j , Eq, d pu x , Eq and d pψ, Eq, together with (4.7) and 1 ´|t| 2 ě e ´R, we obtain This, with the symmetric statement |ηpyq ´ηpỹq| ď e ´R, together with (4.9), give us the result of Theorem 1.3. □ We finish our paper with an example that illustrates our enhancement to Theorem 1.2. It is a foliation on P 2 that satisfies the hypothesis of Theroem 1.3, but at least one of its singularities is non-linearizable. We leave the details of the easy computations to the reader.

}µ Ψ } C 1 ď C exp ´log ‹ d px, Eq ´C1 Re C 4 R ´eα ´1R ¯¯ď e ´2R , if
Example 4.12. Consider in C 2 the polynomial vector field Xpz, wq " `2z `w2 ´z3 ˘B Bz ``w ´z2 w ˘B Bw .

The field X defines a singular holomorphic foliation by Riemann surfaces on C 2 that extends to a foliation F " pP 2 , L , Eq of degree 2 on P 2 . Computing the vector fields that define F in the two other affine charts of P 2 , one can check that F does not have any singularity on the line at the infinity. Moreover, solving X " 0, one gets that E " ! p0, 0q; p ? 2, 0q; p´?2, 0q; p1, iq; p1, ´iq; p´1, 1q; p´1, ´1q

) .

Computing the Jacobian matrix of X in each of these singularities, one can check easily that every singularity of F is non-degenerate. By the results of Glutsyuk [START_REF] Glutsyuk | Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety[END_REF] and Lins Neto [START_REF] Neto | Uniformization and the Poincaré metric on the leaves of a foliation by curves[END_REF], it follows that F is a Brody hyperbolic foliation on P 2 and Theorem 1.3 applies.

We claim that X is not linearizable near 0. In fact, X is essentially the most basic example of resonance (see [START_REF] Martinet | Classification analytique des équations différentielles non linéaires résonnantes du premier ordre[END_REF] or [START_REF] Dulac | Recherches sur les points singuliers des équations différentielles[END_REF] for more details about this phenomenon). In order to prove our claim, we blow up the origin and compare the forms of the liftings of X and its linear part X ℓ " 2z B Bz `w B Bz . Consider one of the usual projections πpt, wq " ptw, wq from the total space of the blowing-up to C 2 . On the one hand, π ˚Xℓ " t B Bt `w B Bw ;

and on the other hand, π ˚X " pt `wq B Bt `pw ´t2 w 3 q B Bw .

These two vector fields cannot define the same foliation. Indeed, their linear parts are not equivalent up to multiplication by λ P C ˚. Hence the singularity of F at 0 is not linearizable.

( 1 . 1 )

 11 |ηpxq ´ηpyq| ď C ˆmax plog ‹ d px, Eq , log ‹ d py, Eqq log ‹ dpx, yq ˙α , x, y P M zE, where we use the same notations as in Theorem 1.2.

d

  K pu, vq " sup xPK dpupxq, vpxqq, d P,K pu, vq " sup xPK d P pupxq, vpxqq, d L,K pu, vq " sup xPK d L pupxq, vpxqq.

  the following.

	vector field	n
		ÿ
	X "
		j"1
	Definition 2.2. We say that F is Brody hyperbolic if there exists a positive constant A
	such that η ă A on M zE.
	Definition 2.3. Near a singularity p, there exists an open chart U Ă M with coordinates
	z P C n , such that the leaves of F are locally defined as the complex flow of a holomorphic

Some useful results. Let

  us prove the following lemma, which gives a bound to the Poincaré distance on a cut-off disk if the Euclidean distance is very small. For z, w P D, we denote by d P pz, wq the Poincaré distance between z and w.

	2.2.

Lemma 2.4. Let

  z, w P D R with R P R ˚. If |z ´w| ď e ´R and if R is sufficiently large, then d P pz, wq ď e R |z ´w|.

  3.1, the C 2 ´norm of f satisfies }f ´id} C 2 " Opεq. Proof. Lemma 3.1 implies that }f ´id} 8 " Opεq. Let us now differentiate f ´id. The implicit function theorem relates the derivatives of g and f by Bu ˇˇpt, f ptqq is also Op}φ x ptq ´φy pf ptqq} }Xpxq}q. Denote by δ xy ptq " φ x ptq ´φy pf ptqq. By the previous estimates and (3.5), we have that }dpf ´idq} 8 is a Op}δ xy } }Xpxq} ´1q. Since }d t δ xy } ď }d t pφ x ´φy q} `}dφ y } 8 }d t pf ´idq} ď C }δ xy ptq} , Grönwall Lemma implies that }δ xy ptq} ď }x ´φy pf p0qq} e C|t| ď C 3 }x ´y} , t P

	(3.6) other hand, (3.7)	Bg Bt	Bf Bt pt, uq ptq ´1 " ´ Bf Bt ptq " ´ `Bg Bu pt, uq " xXpφ r 1 Bg Bu Bg Bu Bg Bu Bg Bu ´1 Bg Bt `Bg Bu Bg Bu Bg Bu Bg Bu pt, f ptqq, Bg Bu Bg Bu Bg Bu Bg Bu ´1 ˆBg Bt ¨Bg Bu ˙pt, f ptqq, 2 D.	On the

since

Bg

Bt " 0. By (3.4) and (3.2), ˇˇBg Bu pt, f ptqq ˇˇ" Op}φ x ptq ´φy pf ptqq} }Xpxq}q. x ptqq ´Xpφ y puqq, Xpφ y puqqy .

It follows that ˇˇBg

Bt `Bg

  Op}x ´y} }Xpxq} ´1q. Next, we differentiate the anti-holomorphic part of(3.6).Recall that }g} C 2 is a Op}Xpxq} 2 q by (3.2), ˇˇBfBt ˇˇis a Op}x ´y} }Xpxq} ´1q, ˇˇBg Bu ˇˇpt, f ptqq is a Op}x ´y} }Xpxq}q and ˇˇB Bu 2 ˇˇpt, f ptqq " ˇˇxφ x ptq ´φy pf ptqq, B 3 φy Bu 3 pf ptqqy ˇˇis a Op}x ´y} }Xpxq}q. Similarly, the Jacobian on the left hand side does not change the estimate if it is inside or outside the derivative. Hence ˇˇB Define then locally Φ xy " φ y ˝f ˝φ´1x : pL x , xq Ñ pL y , yq. By definition of f , Φ xy is the local orthogonal projection from L x in the neighbourhood of x to L y in the neighbourhood of y. Figure2gives a schematic view of this definition and its link with the function f . The radii of these neighbourhoods are O p}x}q by(3.1) and Lemma 3.1. Considering the local definition of Φ xy , (3.1) and Lemma 3.2, we get easily that (3.8) }Φ xy ´id} 8 ď C 5 }x ´y} , }Φ xy ´id} C 1 ď C 5 }x ´y} }x} , }Φ xy ´id} C 2 ď C 5

												2 g
													2 f Bt 2 ˇˇď C }x ´y} }Xpxq} ´1. This and the same inequalities
	on	› › d `Bf Bt	´1˘› › 8 give us finally
											}f ´id} C 2 ď C 4 }x ´y} }Xpxq} ´1 " Opεq.
													□
		t ˆJ	¨ˆBf Bt	´1˙˙"	´Tr	¨¨B Bu g ´Bg Bu	´Bg Bu Bg Bt Bu `Bg	‹ 'dpt,fptqq	¨Bg Bt	Bu `Bg Bu Bg	Bu Bg Bu Bg	‹ ' ‹ '˝pdt, df q. }x ´y} }x} 2 .
													˜Bg
	Here, we have denoted by d pt,f ptqq L x	Bt	`Bg Bu Bg	Bg Bu Bg	x
	O p}x ´y} }Xpxq}q, ˇˇBf		Bu	Bu	Bt and }g} C 2 ď C }Xpxq} 2 by (3.2). `Bg Bu ˇˇare y Φ xy pxq ´1˘, φ
	Differentiating (3.7), we get that L y
								ˆB Bt `B Bu	˙ˆBg Bt	`Bg Bu	˙" B	B 2 φ x Bt 2	´B2 φ y Bu 2 , X ˝φy	F	,
								ˆB Bt `B Bu	˙ˆBg Bt	`Bg Bu	˙" B X ˝φx ´X ˝φy ,	B 2 φ y Bu 2	F	.
	Hence,	ˇˇ`B Bt `B Bu	˘`Bg Bt	`Bg Bu	˘ˇa nd also that	ˇˇ`B Bt `B Bu	˘`Bg	It
	follows that
											› › › › d ˆJ	¨ˆBf Bt	› 8 ´1˙˙› › ›	ď C }Xpxq} 2 F px, yq,
	where F is given by
			F px, yq " Tr	ˆˆ}Xpxq} }x ´y} }x ´y} }x ´y} ˙ˆ}x ´y} }Xpxq} }Xpxq} }Xpxq} ˙˙`}x ´y} }Xpxq}
						" 5 }x ´y} }Xpxq} .
	By (3.5), and using estimates on ˇˇBf Bt	´1ˇˇa nd }g} C 2 , we get that	› › d `Bf Bt	´1˘› › 8 is a
				B Bt	ˆJptq	Bf Bt	ptq ˙" Bg Bu ¨B Bt	ˆBg Bt	pt, f ptqq	˙`Bg Bt	ˆBf Bt	¨B2 g BtBu	`Bf Bt	Bu 2 ¨B2 g	˙.

¸the differential of the obvious matrix application with respect to the point pt, f ptqq. Note again that ˇˇBg Bu ˇˇand ˇˇBg Bt ´1ˇˇi s a O `}x ´y} Xpxq Bt `Bg Bu ˘ˇa re Op}x ´y} }Xpxq}q. y pf ptqq " Φ xy px 1 q x 1 " φ x ptq 0 FIGURE 2. Orthogonal projection from L x near x onto L y near y (outside the singular point 0)

  Ly py, Φ xy pxqq ď kdpx, yq, (2) for x 1 , x 2 P L x rε 0 d px, Eqs, d Ly pΦ xy px 1 q, Φ xy px 2 qq ď kd Lx px 1 , x 2 q. (3) Φ xy is smooth and we have the following estimates for Φ xy ´id and its derivatives in the charts U i and U p . }Φ xy ´id} 8 ď e K dpx, Φ xy pxqq, }Φ xy ´id} C 1 ď e K dpx, Φ xy pxqq d px, Eq , }Φ xy ´id} C 2 ď e K dpx, Φ xy pxqq d px, Eq 2 .

	Eqs,
	satisfying
	(1) d

  in our context. Let ζ, σ be points of D. Denote by z " u x pζq, w " u x pσq. There exist constants C 3 , C 4 P R ˚, such that we have ˇˇˇl n log ‹ d pz, Eq log ‹ d pw, Eq ˇˇˇď C 3 `C4 d P pζ, σq.Proof. Since ζ and σ play a symmetric role, assume that d pz, Eq ě d pw, Eq.

	Lemma 4.7. We have
	(4.5)	ˇˇˇl n	log ‹ d pz, Eq log ‹ d pw, Eq	ˇˇˇ" ln " C	log ‹ d pw, Eq log ‹ d pz, Eq ż dpz,Eq dpw,Eq dt t log ‹ t " ´ln plog ‹ d pz, Eqq `ln plog ‹ d pw, Eqq .

  Now, the second integral is either on the empty set if d pz, Eq ď 3ρ, or we have d pw 1 , Eq " 3ρ and d pz, Eq ď 1. In both cases, the second integral is bounded above by ş 1 3ρ dt t log ‹ t . We treat the first integral by using the standard Hermitian metric on C n and a change of variable. In the following, ℜz denotes the real part of a complex number z.

	ż dpw 1 ,Eq dpw,Eq	dt t log ‹ t	"	ż }uxpγpτ qq} }uxpγp0qq}	dt t log ‹ t	"	ż τ 0	ℜ }u x pγptqq} 2 log ‹ }u x pγptqq} @ D pu x ˝γq 1 ptq, u x ˝γptq	dt
				ď	ż τ 0	}u 1 x pγptqq} |γ 1 ptq| }u x pγptqq} log ‹ }u x pγptqq}	dt
				ď C 1	ż τ 0	}u 1 x pγptqq} |γ 1 ptq| ηpu x pγptqqq	dt " C 1	ż τ 0	|γ 1 ptq| 1 ´|γptq| 2 dt
				ď	C 1 2	d P pζ, ζ 1 q ď	C 1 2	d P pζ, σq.
	Coming back to (4.5), and using our work on the second integral, we obtain
		ˇˇˇl n	log ‹ d pz, Eq log ‹ d pw, Eq	ˇˇˇď C	ż 1 3ρ	dt t log ‹ t	`CC 1 2	d P pζ, σq.
		ż dpz,Eq dpw,Eq	dt t log ‹ t	"	ż dpw 1 ,Eq dpw,Eq	dt t log ‹ t	`ż dpz,Eq dpw 1 ,Eq	dt t log ‹ t	.

□

We are now ready to control N with the following lemma.

Lemma 4.8.

There exists a constant C 5 such that

N ď C 5 log ‹ d px, Eq Re C 4 R ,

Proof. By Lemma 4.6, if j P 0, N ´2 , we have R j`1 ´Rj ě 2ε 0 d px j , Eq sup rξ j ,ξ j`1 s

  R is sufficently large. □ End of proof of Theorem 1.3. Lemmas 4.9 and 4.10 ensure us that we are in the setup to use Lemma 4.4. Since x and y play a symmetric role, if we denote by ỹ " Φ xy pxq and x " Φ yx pyq, we get (4.9) ηpxq ´ηpỹq ď C 2 e ´R, ηpyq ´ηpxq ď C 2 e ´R. It remains only to compare η in two close points in the same leaf. That is, showing that ηpxq (resp. ηpyq) and ηpxq (resp. ηpỹq) are close to each other. By symmetry of x and y, we only show it for x and x. Let t P D be a time such that u x ptq " x and d P p0, tq ď 2d P px, xq. By Lemma 4.3 and Proposition 4.2, we have (4.10) |t| ď d P p0, tq ď 2d P px, xq ď C d Lx px, xq d px, Eq log ‹ d px, Eq if R is sufficiently large. Hence, x and x belong to the same Hermitian chart and by Lemma 4.11, }u 2x } g M ď C on r0, ts. We have|ηpxq ´ηpxq| " ˇˇˇ} u 1 x p0q} g M ,x ´}u 1 x ptq} g M ,x 1 ´|t| 2 M ,x ´}u 1 x ptq} g M ,x ˇˇ`4 `ˇˇ} u 1 x p0q} g M ,x ´}u 1 x ptq} g M ,x ˇď C 1 dpx, yq d px, Eq `C2 }u 1 x p0q ´u1 x ptq} ď C 1 dpx, yq d px, Eq `C2 C 6 |t| ď C dpx, yq d px, Eq , where we used successively the Cauchy formula (see just after (4.3)), the Lipschitz spatial variation of g M , the equivalence of Hermitian metrics, (4.10), Lemma 4.3 and Lemma 4.11. It follows that if R is sufficiently large, |ηpxq ´ηpxq| ď e ´R.

			ď Ck	dpx, yq d px, Eq	ď	r 0 2	,
			ˇˇď
	4 4 ´r2 0	ˇˇ}u 1 x p0q} g A |t| 2 4 ´r2 0
	ď CAdpx, xq	`4A |t| 2 4 ´r2 0

Proof. We keep the notations that we introduced before Lemma 4.6. The condition that d Lx pu x pζq, x j q ď ε 0 d px j , Eq for all ζ P rξ j , ξ j`1 s allows us to use Lemma 4.3. This will ensure the existence of successive orthogonal projection Φ x j`1 y j`1 : L x j`1 rε 0 d px j`1 , Eqs Ñ L y j`1 rkε 0 d px j`1 , Eqs, with by induction y j`1 " Φ x j y j px j`1 q and y 0 " Φ xy pxq. We just have to check that at each step, x j and y j are close enough. As long as dpx j , y j q ď ε 1 d px j , Eq, we have dpx j , y j q ď e jK dpx, y 0 q ď e jK dpx, yq. Hence (4.6) dpx j , y j q d px j , Eq ď e N K dpx, yq d px j , Eq ď exp ´log ‹ d px, Eq ´C3 `pKC 5 R `1qe C 4 R ´eα ´1R ¯¯, using our hypothesis (4.4), Lemmas 4.7 and 4.8. Let us suppose that α ă C ´1 4 . If R is sufficiently large, we have dpx j , y j q ď ε 1 d px j , Eq at each step. If follows that we can define ψ : r0, ξs Ñ L y by gluing the successive Φ x j y j ˝ux on rξ j , ξ j`1 s. We can do the same work for each ξ P D such that d P p0, ξq " R and obtain a function ψ : D R Ñ L y . We could imagine that this does not give a smooth function and not even a continuous function. Take ξ 1 , ξ 2 P D R with d P p0, ξ 1 q " d P p0, ξ 2 q " R. Define ξ 1 j , ξ 2 j to have the above conditions and ˇˇξ 1 j ˇˇ" ˇˇξ 2 j ˇˇfor both segments r0, ξ 1 s and r0, ξ 2 s. If we suppose that |ξ 1 ´ξ2 | ď e ´RA ´1ε 0 exp `´e C 4 R log ‹ d px, Eq ˘, denote by x k j " u x pξ k j q and y k j " ψpξ k j q for j P 0, N and k P t1, 2u. By definition, at first step

1 on the intersection of their domains of definition. We continue by induction. By the same computation, we have at each step that d Ly py 1 j , y 2 j q ď kε 0 d `x1 j , E ˘.

Hence, ψ is locally equal to some Φ x j y j ˝ux . Then, it is smooth. Moreover, by (4.6), if R is sufficiently large, (4.7)

Since ψ is locally equal to Φ x j y j ˝ux , it is clear that ψ has no critical point. The condition on }dψ} 8 will be quite easy to obtain. Observe that locally, ψ " Φ x j y j ˝ux . By (4.6) In order to prove this result, we will need first the following estimate on the second derivative of a uniformization.

Lemma 4.11.

There exist a radius r 0 P R ˚and a constant C 6 P R ˚such that for all z P M zE, u z a uniformization of L z and |t| ď r 0 2 , }u 2 z ptq} ď C 6 .