× R Filippo Morabito
email: filippo.morabito@univ-mlv.fr

A

We show the existence in the space H 2 ×R of a family of embedded minimal surfaces of genus 1 k < +∞ and finite total extrinsic curvature with two catenoidal type ends and one middle planar end. The proof is based on a gluing procedure.

Introduction

During recent years the study of the minimal surfaces in the product spaces M × R with M = H2 , S 2 has become more and more active. The development of the theory of the minimal surfaces in these spaces started with [START_REF] Rosenberg | Minimal surfaces in M × R[END_REF] by H. Rosenberg and continued with [START_REF] Meeks | The theory of minimal surfaces in M × R, Comment[END_REF] and [START_REF] Meeks | Stable minimal surfaces in M × R[END_REF] by W. H. Meeks and H. Rosenberg. In [START_REF] Nelli | Minimal surfaces in H 2 × R[END_REF] B. Nelli and H. Rosenberg showed the existence in H 2 × R of a rich family of examples including helicoids, catenoids and, solving Plateau problems, of higher topological type examples inspired by the theory of minimal surfaces in R 3 . In [START_REF] Hauswirth | Minimal surfaces of Riemann type in three-dimensional product manifolds[END_REF] L. Hauswirth constructed and classified the minimal surfaces foliated by horizontal constant curvature curves in M × R, where M is H 2 , R 2 or S 2 . Other examples of minimal surfaces of genus 0 in these product manifolds are described by R. Sa Earp and E. Toubiana in [START_REF] Sa Earp | Screw motion surfaces in H 2 × R and S 2 × R[END_REF].

C. Costa in [START_REF] Costa | Imersões minimas en R 3 de gênero un e curvatura total finita[END_REF][START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF] and D. Hoffman and W. H. Meeks in [START_REF] Hoffman | A complete embedded minimal surface in R 3 with genus one and three ends[END_REF], [START_REF] Hoffman | The asymptotic behavior of properly embedded minimal surfaces of finite topology[END_REF] and [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF] described a minimal surface in R 3 of genus 1 k < +∞ and finite total curvature with two ends asymptotic to the two ends of a catenoid and a middle end asymptotic to a plane. We will denote the Costa-Hoffman-Meeks surface of genus k by M k .

The aim of this work is to show the existence in the space H 2 × R of a family of surfaces inspired by M k . We will prove the following result. Theorem 1.1. For all 1 k < +∞ there exists in H 2 × R a one-parameter family of embedded minimal surfaces of genus k and finite total extrinsic curvature with three horizontal ends: two catenoidal type ends and a middle planar end.

We will observe that it is more convenient to construct a minimal surface enjoying the same properties mentioned in the statement of theorem in the Riemannian manifold (D 2 × R, g hyp ) where g hyp = dx 2 1 +dx 2 2

(1-x 2 1 -x 2

2 ) 2 + dx 2 3 . It is usually denoted by M 2 (-4) × R, to point out that the sectional curvature of D 2 × {0} endowed with the metric

dx 2 1 +dx 2 2 (1-x 2 1 -x 2
The main result is proved by a gluing procedure (see for example [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF]) usually adopted to construct in R 3 new examples starting from known minimal surfaces. We consider a scaled version of a compact piece of a Costa-Hoffman-Meeks type surface, which is contained in a cylindrical neighbourhood of {0, 0}×R ⊂ M 2 (-4)× R of sufficiently small radius. Actually it's possible to prove that, in the same set, the mean curvature of such a surface with respect to the metric g hyp , up to an infinitesimal term, equals the Euclidean one. We glue the surface described above along its three boundary curves to two minimal graphs that are respectively asymptotic to an upper half catenoid and a lower half catenoid defined in M 2 (-4)× R and to a minimal graph over M 2 (-4)×{0} which goes to zero in a neighbourhood of ∂ ∞ M 2 (-4) × {0}. The existence of these surfaces is proved in sections 5 and 7.

The author wishes to thank his thesis director, L. Hauswirth, for having brought this problem to his attention.

Preliminaries

In this work we will consider the unit disk model for H 2 . Let (x 1 , x 2 ) denote the coordinates in the unit disk D 2 and x 3 the coordinate in R. Then the space D 2 × R is endowed with the metric

g H 2 ×R = 4(dx 2 1 + dx 2 2 ) (1 -x 2 1 -x 2 2 ) 2 + dx 2 3 .
As mentioned in the Introduction, one of the surfaces involved in the gluing procedure is a compact piece of a scaled version of the Costa-Hoffman-Meeks surface, that is, a minimal surface in R 3 endowed with the Euclidean metric g 0 . To simplify as much as possible the proof of the main theorem, it is convenient to consider a Riemannian manifold endowed with a metric more similar to g 0 than the standard metric of H 2 × R. The best choice is

g hyp = dx 2 1 + dx 2 2 (1 -x 2 1 -x 2 2 ) 2 + dx 2 3 ,
because g hyp → g 0 if (x 1 , x 2 ) → (0, 0). This is the reason that induces us to give a proof of Theorem 1.1 working in the Riemannian manifold M 2 (-4) × R. Now we suppose to have shown the existence of a minimal surface in this Riemannian manifold. We need to show how it is possible to obtain a minimal surface in H 2 ×R. Let ḡ be the metric defined on D 2 × R by

ḡ = 4g hyp = 4(dx 2 1 + dx 2 2 ) (1 -x 2 1 -x 2 2 ) 2 + 4dx 2 3 .
We consider the map f :

(D 2 × R, g H 2 ×R ) → (D 2 × R, ḡ) defined by (1) (x 1 , x 2 , x 3 ) → x 1 , x 2 , x 3 2 .
It is easy to see that it is an isometric embedding, that is, the pull-back of the metric ḡ by f equals g H 2 ×R . So if Σ is a minimal surface in (D 2 × R, ḡ), then the image of Σ by f -1 is a minimal surface in H 2 × R. Now we turn our attention to the Riemannian manifold M 2 (-4) × R, mentioned in the Introduction. In the following we will adopt the simplified notation M 2 × R. We recall that the metric ḡ has been defined as 4g hyp , g hyp being the metric of M 2 × R. As a consequence the mean curvature of a surface Σ in M 2 × R equals the mean curvature of Σ in (D 2 × R, ḡ) multiplied by 4. So if a surface is minimal in M 2 × R, it is also minimal with respect to the metric ḡ.

We can conclude that if Σ is a minimal surface in M 2 × R, then f -1 (Σ) is a minimal surface in H 2 × R.

Remark 2.1. To prove Theorem 1.1 we will need to consider spaces of functions invariant under the actions of the isometries of R 3 which keep unchanged the Costa-Hoffman-Meeks surface (appropriate rotations about the vertical coordinate axis x 3 , the reflection with respect to the horizontal plane x 3 = 0 and the vertical plane x 2 = 0). These are isometries of M 2 × R as well. So we will continue using the same language as if we are in R 3 .

Minimal graphs in M 2 × R

We denote by H u the mean curvature of the graph of the function u over a domain in D 2 . Its expression is [START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF] 2H u = F div ∇u

1 + F |∇u| 2 ,
where F = 1x 2 1x 2 2 2 = 1r 2 2 and div denotes the divergence in R 2 . For the details of the computation, see subsection 12.3. Let Σ u be the graph of the function u. In this section we want to obtain the expression of the mean curvature of the surface Σ u+v , that is, the graph of the function u + v. It can also be considered as the vertical graph of the function v over Σ u . We will show how it follows from (2) that the linearized mean curvature, which we denote by L u , is given locally by:

(3) L u v := F div ∇v

1 + F |∇u| 2 -F ∇u ∇u • ∇v (1 + F |∇u| 2 ) 3 .
Furthermore we will give the expression of H u+v , the mean curvature of the graph of the function u + v, in terms of the mean curvature of Σ u , that is, H u . In the following we will restrict our attention to two cases: the plane (in section 5), that is, u = 0, and (in section 7) a piece of catenoid defined on the domain

{(θ, r) ∈ M 2 × {0} | r ∈ [r ε , 1]}, where r ε = ε/2.
Here we will show that:

(4) 2H u+v = 2H u + L u v + F Q u ( √ F ∇v, √ F ∇ 2 v),
where Q u has bounded coefficients if r ∈ [r ε , 1] and it satisfies Q u (0, 0) = 0 and ∇Q u (0, 0) = 0. To show this, we observe that

(5) 1 1 + F |∇(u + v)| 2 = 1 1 + F |∇u| 2 -F ∇u • ∇v (1 + F |∇u| 2 ) 3 + Q u,1 (v).
Q u,1 (v) has the following expression:

(6) -F |∇v| 2 2 (1 + F |∇(u + tv)| 2 ) 3/2 + 3F 2 ∇u • ∇v + t|∇v| 2 2 2 (1 + F |∇(u + tv)| 2 ) 5/2 ,
with t ∈ (0, 1), and it satisfies Q u,1 (0) = 0, ∇Q u,1 (0) = 0. To prove (5) it's sufficient to set

f (t) = 1 1 + F |∇(u + tv)| 2
and to write down the Taylor's series of order one of this function and to evaluate it in t = 1. That is, f (1) = f (0) + f (0) + 1 2 f ( t), with t ∈ (0, 1). We insert (5) in the expression that defines 2H u+v to get

F div ∇(u + v) 1 + F |∇u| 2 -F ∇(u + v) ∇u • ∇v (1 + F |∇u| 2 ) 3 + ∇(u + v)Q u,1 (v) = 2H u +F div ∇v 1 + F |∇u| 2 -F ∇u ∇u • ∇v (1+F |∇u| 2 ) 3 +F Q u ( √ F ∇v, √ F ∇ 2 v).
Since we assume that Σ u is a minimal surface, we will consider H u = 0.

Remark 3.1. The minimal surfaces in the families we will construct in sections 5 and 7 have finite total extrinsic curvature. These minimal surfaces are graphs over the domain

{(θ, r) ∈ M 2 | r ∈ [r ε , 1]} of functions of class C 2,α .
The total extrinsic curvature of the graph S of a function u defined on M 2 is the integral of the extrinsic curvature, that is, For the expressions of the coefficients of the first and of the second differential form see subsection 12.3. Once their expressions have been replaced in [START_REF] Hoffman | A complete embedded minimal surface in R 3 with genus one and three ends[END_REF], it is clear that, taking into account that u is a C 2,α class function, S K ext dA is bounded. This observation allows us to state that this property also holds for the surface obtained by a gluing procedure in section 11. In fact the total extrinsic curvature of this last surface equals the sum of the total extrinsic curvature of the surfaces glued together, that is, a compact piece of a Costa-Hoffman-Meeks type surface and three minimal graphs over the domain described above. Because of the compactness, the contribution to the total curvature of the piece of the Costa-Hoffman-Meeks type example is bounded. Then the result follows immediately, taking into account the observation made above concerning the graph of C 2,α class functions over M 2 .

The mapping properties of the Laplace operator

Now we restrict our attention to the case of the minimal surfaces close to M 2 × {0}, that is, the graph of the function u = 0. In this case we obtain immediately from (3) that L u=0 = F Δ 0 , where Δ 0 denotes the Laplacian in the flat metric g 0 of the unit disk D 2 .

In this section we will study the mapping properties of Δ 0 . In the sequel we will use the polar coordinates (θ, r).

In particular our aim is to solve in a unique way the problem:

Δ 0 w = f in S 1 × [r 0 , 1]
, w| r=r 0 = ϕ with r 0 ∈ (0, 1), considering a convenient normed functions space for w, f and ϕ, so that the norm of w is bounded by that of f . Now we can give the definition of the space of functions we will consider.

Definition 4.1. Given ∈ N, α ∈ (0, 1), and the closed interval

I ⊂ [0, 1], we define C ,α (S 1 × I)
to be the space of functions w := w(θ, r) in C ,α loc (S 1 × I) for which the norm w C ,α (S 1 ×I) is finite and whose graph surfaces are invariant with respect to symmetry with respect to the x 2 = 0 plane, with respect to the rotation of an angle 2π k+1 about the vertical x 3 -axis, with respect to the composition of a rotation of angle π k+1 about the x 3 -axis and the symmetry with respect to the x 3 = 0 plane.

We recall that one of the surfaces involved in the gluing procedure we will follow to prove the main theorem is a surface derived by the Costa-Hoffman-Meeks surface. This surface, as explained in subsection 9.1, enjoys many properties of symmetry that we want to be inherited by the surface obtained by the gluing procedure. This is the reason for which we have chosen the functional space described above. Proposition 4.2. Given r 0 ∈ (0, 1), there exists an operator

G r 0 : C 0,α (S 1 × [r 0 , 1]) -→ C 2,α (S 1 × [r 0 , 1]) f -→ w := G r 0 (f )
satisfying the following statements:

(i) Δ 0 w = f on S 1 × [r 0 , 1], (ii) w = 0 on S 1 × {r 0 } and S 1 × {1}, (iii) ||w|| C 2,α (S 1 ×[r 0 ,1]) c ||f || C 0,α (S 1 ×[r 0 ,1]
) , for some constant c > 0 which does not depend on r 0 , f and w.

The proof of this result is contained in subsection 12.2.

A family of minimal surfaces close to M 2 × {0}

In this section we will show the existence of minimal graphs in M 2 × R over D 2 -B r ε , having prescribed boundary and which are asymptotic to it. We recall that r ε = ε/2. We will reformulate the problem to use the Schäuder fixed point theorem. We know already that the graph of a function v, denoted with Σ v , is minimal, if and only if the function v is a solution of ( 8)

F Δ 0 v + Q 0 √ F ∇v, √ F ∇ 2 v = 0.
This equation is a simplified version (since u = 0) of ( 4). The operator

Q 0 has bounded coefficients for r ∈ [r ε , 1]. Its expression is div (∇v Q 0,1 )
, where Q 0,1 is given by [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF]. To simplify the notation, in the sequel we will write Q 0 (•) in place of

Q 0 √ F ∇•, √ F ∇ 2 • . Now let's consider a function ϕ ∈ C 2,α (S 1
) which is even with respect to θ, collinear to cos(j(k + 1)θ) (for k 1 fixed) with j 1 and odd and such that [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF] ϕ C 2,α κε 2 .

We define

w ϕ (•, •) := H r ε ,ϕ (•, •)
, where H is the operator of harmonic extension introduced in Proposition 12.1. The particular choice of ϕ assures that its harmonic extension belongs to the functional space of Definition 4.1.

In order to solve equation ( 8), we look for v of the form v = w ϕ + w, where w ∈ C 2,α (S 1 × [r ε , 1]) and v = ϕ on S 1 × {r ε }. Using Proposition 4.2, we can rephrase this problem as a fixed point problem, [START_REF] Jleli | Constant mean curvature hypersurfaces[END_REF] w = S(ϕ, w), where the nonlinear mapping S which depends on ε and ϕ is defined by

S(ϕ, w) := -G r ε (Q 0 (w ϕ + w)) ,
and where the operator G r ε is defined in Proposition 4.2. To prove the existence of a fixed point for [START_REF] Jleli | Constant mean curvature hypersurfaces[END_REF] we need the following result, which states that S(ϕ, •) is a contraction mapping:

Lemma 5.1. Let ϕ ∈ C 2,α (S 1
) be a function satisfying [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF] and enjoying the properties given above. There exist some constants c κ > 0 and ε κ > 0 such that

(11) S(ϕ, 0) C 2,α (S 1 ×[r ε ,1]) c κ ε 4
and, for all ε ∈ (0, ε κ ),

S(ϕ, v 2 ) -S(ϕ, v 1 ) C 2,α (S 1 ×[r ε ,1]) 1 2 v 2 -v 1 C 2,α (S 1 ×[r ε ,1]) , S(ϕ 2 , v) -S(ϕ 1 , v) C 2,α (S 1 ×[r ε ,1]) cε 2 ϕ 2 -ϕ 1 C 2,α (S 1 ) ,
where c is a positive constant, for all 4 and for all boundary data ϕ 1 , ϕ 2 ∈ C 2,α (S 1 ) enjoying the same properties as ϕ.

v 1 , v 2 ∈ C 2,α (S 1 × [r ε , 1]) such that v i C 2,α 2c κ ε
Proof. We know from Proposition 4.

2 that G r ε (f ) C 2,α c f C 0,α . Then S(ϕ, 0) C 2,α = G r ε (Q 0 (w ϕ )) C 2,α c Q 0 (w ϕ ) C 0,α .
To find an estimate of the norm above we recall that ϕ 2,α κε 2 and thanks to Proposition 12.1 we obtain

w ϕ C 2,α c ϕ C 2,α c κ ε 2 . Then Q 0 (w ϕ ) C 0,α c w ϕ 2 C 2,α c ϕ 2 C 2,α c κ ε 4 . So we can conclude S(ϕ, 0) C 2,α c κ ε 4 .
As for the second estimate, we observe that

S(ϕ, v 2 ) -S(ϕ, v 1 ) C 2,α c Q 0 (w ϕ + v 2 ) -Q 0 (w ϕ + v 1 ) C 0,α .
Thanks to the considerations made above it follows that

||Q 0 (w ϕ + v 2 ) -Q 0 (w ϕ + v 1 )|| C 0,α c||v 2 -v 1 || C 2,α w ϕ C 2,α c κ ε 2 ||v 2 -v 1 || C 2,α . Then S(ϕ, v 2 ) -S(ϕ, v 1 ) C 2,α c κ ε 2 ||v 2 -v 1 || C 2,α .
To show the third estimate we proceed as above:

S(ϕ 2 , v) -S(ϕ 1 , v) C 2,α c Q 0 (w ϕ 2 + v) -Q 0 (w ϕ 1 + v) C 0,α c||w ϕ 2 -w ϕ 1 || C 2,α v C 2,α cε 2 ||ϕ 2 -ϕ 1 || C 2,α . Theorem 5.2. Let B := {w ∈ C 2,α (S 1 × [r ε , 1)) | ||w|| C 2,α
2c κ ε 4 } and ϕ be as above. Then the nonlinear mapping S(ϕ, •) defined above has a unique fixed point v in B.

Proof. The previous lemma shows that, if ε is chosen small enough, the nonlinear mapping S(ϕ, •) is a contraction mapping from the ball B of radius 2c κ ε 4 in C 2,α (S 1 × [r ε , 1]) into itself. This value follows from the estimate of the norm of S(ϕ, 0). Consequently, thanks to the Schäuder fixed point theorem, S(ϕ, •) has a unique fixed point v in this ball.

We have proved the existence of a minimal surface with respect to the metric g hyp , denoted by S m (ϕ), which is close to D 2 -B r ε ⊂ M 2 × {0}, and close to its boundary is the vertical graph over the annulus B 2r ε -B r ε of a function which can be expanded as

Ūm (θ, r) = H r ε ,ϕ (θ, r) + Vm (θ, r), with || Vm || C 2,α cε 2 .
From the properties of the extension operator H r ε ,ϕ (see Proposition 12.1) and Proposition 4.2 we can see that Ūm (θ, r) tends to 0 as r → 1. In other terms S m (ϕ) is asymptotic to

M 2 × {0}. Furthermore it is clear that S m (ϕ) is embedded in M 2 × R.
The function Vm depends nonlinearly on ε, ϕ. Furthermore, as it is easy to prove thanks to the third estimate of Lemma 5.1, it satisfies

(12) Vm (ε, ϕ)(•, r ε •) -Vm (ε, ϕ )(•, r ε •) C 2,α ( B2 -B 1 ) cε ϕ -ϕ C 2,α (S 1 ) . 6. The catenoid in M 2 × R
The catenoid in the space M 2 × R can be obtained by revolution around the x 3 -axis, {0, 0} × R, of an appropriate curve γ (see [START_REF] Nelli | Minimal surfaces in H 2 × R[END_REF]). We consider a vertical geodesic plane containing the origin of M 2 and the curve γ. Let r be the Euclidean distance between the point of γ at height t and the x 3 -axis: we denote with r = r(t) a parametrization of γ.

The surface obtained by revolution of γ is minimal with respect to the metric g hyp if and only if r = r(t) satisfies the following differential equation (see subsection 12.4):

(13) r(t) ∂ 2 r ∂t 2 - ∂r ∂t 2 -(1 -r(t) 4 ) = 0.
A first integral for this equation is:

(14) ∂r ∂t 2 = Cr 2 -(1 + r 4 )
with C > 2 and constant. By the resolution of equation ∂r ∂t

2 = 0, it is easy to prove that the function r(t) has a minimum value r min given by:

r min = C - √ C 2 -4 2 = C/2 + 1 2 - C/2 -1 2 < 1.
Since we assume C = 1 ε 4 , we get

r min = C/2 + 1 2 - C/2 -1 2 = √ C 2 1 + 1 C -1 + 1 C + O 1 C 2 = 1 √ C + O 1 C 3/2 = ε 2 + O(ε 6 ).
We denote with C t and C b , respectively, the piece of the catenoid contained in

M 2 × R + and M 2 × R -.
We set t ε = -ε 2 ln ε. We need to find the parametrization of C t and C b as graphs on the horizontal plane respectively for t

∈ [t ε -ε 2 ln 2, t ε + ε 2 ln 2] and t ∈ [-t ε -ε 2 ln 2, -t ε + ε 2 ln 2]
. We start by finding the expression of r(t) for t in the interval specified before. Lemma 6.1. For ε > 0 small enough, we have

r(t) = ε 2 cosh t ε 2 + O(ε 6 e t ε 2 ) and ∂ t r(t) = sinh t ε 2 + O(ε 4 e t ε 2 ) for t ∈ [0, t ε + ε 2 ln 2]. Moreover if t ∈ [t ε -ε 2 ln 2, t ε + ε 2 ln 2], we derive r(t) ∈ [ 1 4 ε + c 1 ε 3 , ε + c 2 ε 3 ], ∂ t r(t) ∈ [ 1 4ε -c 1 ε, 1 ε -c 2 ε],
for some positive constants c 1 , c 2 , c 1 , c 2 .

Proof. We define the function v(t) in such a way that r(t) = r(0) cosh v(t), with v(0) = 0 and r(0) the minimum for r(t). It satisfies

Cr 2 (0) -(1 + r 4 (0)) = 0, from which (15) 1 = Cr 2 (0) -r 4 (0).
Plugging r(t) in ( 14) and using [START_REF] Meeks | Stable minimal surfaces in M × R[END_REF], we have

(∂ t v) 2 = C -r 2 (0)(1 + cosh 2 v(t))
and under the hypothesis

t ε 2 v (t) t ε 2 + 1 we obtain that (∂ t v) 2 = C +O(ε 4 e 2t ε 2 ) and then v(t) = √ Ct+O(ε 6 e 2t ε 2 ). We remark a posteriori that t ε 2 v(t) t ε 2 +1 holds for t ∈ [0, t ε + ε 2 ln 2], ε > 0 small enough. Since r(0) = r min = ε 2 + O(ε 6 ), we get (16) r(t) = r(0) cosh v(t) = ε 2 cosh t ε 2 + O(ε 6 e t ε 2 ). If t ∈ [t ε -ε 2 ln 2, t ε + ε 2 ln 2], then we easily obtain r(t) ∈ [ 1 4 ε + c 1 ε 3 , ε + c 2 ε 3 ], for some positive constants c 1 , c 2 . Using ∂ t r(t) = sinh t ε 2 + O(ε 4 e t ε 2 ), we find ∂ t r(t) ∈ [ 1 4ε -c 1 ε, 1 ε -c 2 ε]
for some positive constants c 1 , c 2 . Now we can prove a lemma that gives us the parametrization of the pieces of the catenoid whose height t belongs to a neighbourhood of t ε and -t ε . Lemma 6.2. For ε > 0 small enough and t ∈ [t εε 2 ln 2, t ε + ε 2 ln 2], the surface C t can be seen as the graph, over the annulus B 2r ε -B r ε /2 , of the function W t (θ, r) which satisfies

(17) W t (θ, r) = ε 2 ln 2r ε 2 + O C 2,α b (ε 3 ). Similarly if t ∈ [-t ε -ε 2 ln 2, -t ε + ε 2 ln 2]
, the surface C b can be seen as the graph over B 2r ε -B r ε /2 of the function

W b (θ, r) = -ε 2 ln 2r ε 2 + O C 2,α b (ε 3 ).
Proof. The first result easily follows from the hypothesis and equation [START_REF] Morabito | Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces[END_REF]. The second result can be shown by observing that C b is the image of C t by the reflection with respect to the x 3 = 0 plane. In other terms, W b (θ, r) = -W t (θ, r).

7.

A family of minimal surfaces close to a catenoid on

S 1 × [r , 1]
In this section we want to show the existence of minimal graphs in M 2 × R over the parts of the surfaces C t and C b (described in the previous section) defined on S 1 × [r ε , 1] ⊂ M 2 × {0} and asymptotic to them. We know that the graph of the function u + v is minimal, u being the function whose graph is the catenoid, if and only if v is a solution of the equation ( 18)

H u+v = 0
whose expression is given by ( 4). The explicit expression of

L u v is (19) F 1 √ A Δ 0 v + ∂ r 1 √ A ∂ r v - 1 A 3 2 ∂ r u ∂ r (F ∂ r u) ∂ r v -F ∂ r u ∂ r 1 A 3 2 ∂ r u ∂ r v , where F = (1 -r 2 ) 2 , A = 1 + F |∇u| 2 = (C -2)r 2 Cr 2 -1 -r 4 and ∂ r u = ± 1 √ Cr 2 -1 -r 4 ,
as is easy to obtain using ( 14). It's useful to observe that since we assume C = 1

ε 4 and r ε = ε/2, we have that, for r ∈ [r ε , 1], A = 1 + O(ε 2 ), ∂ r u = O(ε), ∂ r A = (2C -4)(-r + r 5 ) (Cr 2 -1 -r 4 ) 2 = O(ε)
and

∂ 2 rr u = ∓ (Cr -2r 3 ) (Cr 2 -1 -r 4 ) 3 = O(1).
Taking into account these estimates, we can conclude that

Lu v := √ A ∂ r 1 √ A ∂ r v - 1 A 3 2 ∂ r u ∂ r (F ∂ r u) ∂ r v -F ∂ r u ∂ r 1 A 3 2 ∂ r u ∂ r v = l 1 ∂ r v + l 2 ∂ 2 rr v, ( 20
)
where

l 1 , l 2 = O(ε). Then we can write √ AL u v = F Δ 0 v + Lu v .
We remark that we have already studied the mapping properties of the operator Δ 0 in section 4.

Let Σ u be the graph of the function u. Then the graph of a function v over Σ u is minimal if and only if v is a solution of the following equation:

(21) Δ 0 v + Lu v + √ AQ u (v) = 0, where Q u (•) := Q u √ F ∇•, √ F ∇ 2 • .
Thanks to the observations on the functions A and ∂ r u, we can conclude that

Q u has bounded coefficients if r ∈ [r ε , 1].
Now we consider a function ϕ ∈ C 2,α (S 1 ) which is even with respect to θ, collinear to cos(j(k + 1)θ) (for k 1 fixed) and such that

(22) ϕ C 2,α κε 2 .
We define

w ϕ (•, •) := H r ε ,ϕ (•, •),
where the operator H r ε ,ϕ has been introduced in Proposition 12.1.

In order to solve equation ( 21), we look for v of the form v = w ϕ + w, where

w ∈ C 2,α (S 1 × [r ε , 1]) and v = ϕ on S 1 × {r ε }.
We can rephrase this problem as a fixed point problem, that is,

(23) w = S(ϕ, w),
where the nonlinear mapping S is defined by

S(ϕ, w) := -G r ε Lu (w ϕ + w) + √ AQ u (w ϕ + w) ,
and where the operator G r ε is defined in Proposition 4.2. To prove the existence of a solution for (23) we need the following result, which states that S(ϕ, •) is a contraction mapping.

Lemma 7.1. Let ϕ ∈ C 2,α (S 1 ) be a function satisfying (22) and enjoying the properties given above. There exist some constants c κ > 0 and ε κ > 0, such that

(24) S(ϕ, 0) C 2,α (S 1 ×[r ε ,1]) c κ ε 3
and, for all ε ∈ (0, ε k ),

S(ϕ, w 2 ) -S(ϕ, w 1 ) C 2,α (S 1 ×[r ε ,1]) 1 2 w 2 -w 1 C 2,α (S 1 ×[r ε ,1]) , S(ϕ 2 , w) -S(ϕ 1 , w) C 2,α (S 1 ×[r ε ,1]) cε ϕ 2 -ϕ 1 C 2,α (S 1 ) ,
where c is a positive constant, for all

w 1 , w 2 ∈ C 2,α (S 1 × [r ε , 1]) such that w i C 2,α (S 1 ×[r ε ,1]) 2c κ ε 4 and for all boundary data ϕ 1 , ϕ 2 ∈ C 2,α (S 1
) enjoying the same properties as ϕ.

Proof. We know from Proposition 4.2 that G r ε (f ) C 2,α c f C 0,α . Then S(ϕ, 0) C 2,α c Lu w ϕ + √ AQ u (w ϕ ) C 0,α c Lu w ϕ C 0,α + Q u (w ϕ ) C 0,α .
Here we have used the fact that A = 1 + O(ε 2 ). So we need to find the estimates of each summand. We recall that ϕ C 2,α κε. Thanks to Proposition 12.1 we get that

w ϕ C 2,α c ϕ C 2,α (S 1 ) c κ ε 2 .
We use [START_REF] Rosenberg | Minimal surfaces in M × R[END_REF] for finding the estimate of Lu w ϕ . We obtain

Lu w ϕ C 0,α cε w ϕ C 2,α c κ ε 3 .
The last term is estimated by observing that

Q u (w ϕ ) C 0,α c w ϕ 2 C 2,α c κ ε 4 .
Putting together all these estimates we get

S(ϕ, 0) C 2,α c κ ε 3 .
As for the second estimate, we observe that

S(ϕ, w 2 ) -S(ϕ, w 1 ) = -G r ε Lu (w ϕ + w 2 ) + √ AQ u (w ϕ + w 2 ) + G r ε Lu (w ϕ + w 1 ) + √ AQ u (w ϕ + w 1 ) . Consequently S(ϕ, w 2 ) -S(ϕ, w 1 ) C 2,α c Lu (w ϕ + w 2 ) -Lu (w ϕ + w 1 ) + √ AQ u (w ϕ + w 2 ) - √ AQ u (w ϕ + w 1 ) C 0,α = c Lu (w 2 -w 1 ) + √ A (Q u (w ϕ + w 2 ) -Q u (w ϕ + w 1 )) C 0,α c Lu (w 2 -w 1 ) C 0,α + ||Q u (w ϕ + w 2 ) -Q u (w ϕ + w 1 )|| C 0,α .
We observe that from the considerations above it follows that

Lu (w 2 -w 1 ) C 0,α cε||w 2 -w 1 || C 2,α
and

||Q u (w ϕ + w 2 ) -Q u (w ϕ + w 1 )|| C 0,α c||w 2 -w 1 || C 2,α w ϕ C 2,α c κ ε||w 2 -w 1 || C 2,α . Then S(ϕ, w 2 ) -S(ϕ, w 1 ) C 2,α cε||w 2 -w 1 || C 2,α .
To show the third estimate we observe that

S(ϕ 2 , w) -S(ϕ 1 , w) C 2,α c Lu (w ϕ 2 -w ϕ 1 ) C 0,α + ||Q u (w ϕ 2 + w) -Q u (w ϕ 1 + w)|| C 0,α cε ϕ 2 -ϕ 1 C 2,α (S 1 ) + w C 2,α ϕ 2 -ϕ 1 C 2,α (S 1 ) cε ϕ 2 -ϕ 1 C 2,α (S 1 ) . Theorem 7.2. Let B := {w ∈ C 2,α (S 1 × [r ε , 1]) | ||w|| C 2,α 2c κ ε 3 }. Then the nonlinear mapping S(ϕ, •) defined above has a unique fixed point v in B.
Proof. The previous lemma shows that, if ε is chosen small enough, the nonlinear mapping S(ϕ, •) is a contraction mapping from the ball B of radius 2c κ ε 3 in C 2,α (S 1 × [r ε , 1]) into itself. This value follows from the estimate of the norm of S(ϕ, 0). Consequently, thanks to the Schäuder fixed point theorem, S(ϕ, •) has a unique fixed point v in this ball.

We have proved the existence of a minimal surface with respect to the metric g hyp , S t (ϕ), which is close to the piece of catenoid C t introduced in section 6 and close to its boundary is a graph over the annulus

B 2r ε -B r ε of the function Ūt (θ, r) = ε 2 ln 2r ε 2 + H r ε ,ϕ (θ, r) + Vt (θ, r), with Vt C 2,α cε 2 .
From the properties of the extension operator H r ε ,ϕ (see Proposition 12.1) and Proposition 4.2 we can see that S t (ϕ) is asymptotic to C t if r tends to 1 and it is embedded in M 2 × R.

The function Vt depends nonlinearly on ε, ϕ. Furthermore it satisfies

(25) Vt (ε, ϕ)(•, r ε •) -Vt (ε, ϕ )(•, r ε •) C 2,α ( B2 -B 1 ) cε ϕ -ϕ C 2,α (S 1 ) .
This estimate follows from Lemma 7.1. Now it is easy to show the existence of a minimal surface S b (ϕ), which is close to the part of the catenoid denoted by C b introduced in section 6, and close to its boundary is a graph over the annulus B 2r ε -B r ε . We start observing that C b can be obtained by reflection of C t with respect to the x 3 = 0 plane. So we can define S b (ϕ) as the image of S t (ϕ) by the composition of a rotation by an angle π k+1 about the x 3 -axis and the reflection with respect to the horizontal plane. This choice (in particular the apparently unnecessary rotation) is indispensable to assure that the surface we will construct by the gluing procedure in section 11 has the same properties of symmetry as the Costa-Hoffman-Meeks surface. See subsection 9.1 for more information.

It is clear that S b (ϕ) is the graph over the annulus

B 2r ε -B r ε of the function Ūb (θ, r) = -Ūt θ - π k + 1
, r .

The relation between the mean curvatures of a surface in D 2 × R with respect to two different metrics

In this section we want to express the mean curvature H hyp of a surface in D 2 × R with respect to the metric g hyp in terms of the mean curvature H e of the same surface with respect to the Euclidean metric g 0 .

We recall that, if x 1 , x 2 denote the coordinates in D 2 and x 3 the coordinate in R, then

g hyp = dx 2 1 + dx 2 2 F + dx 2 3 , where F = 1 -x 2 1 -x 2 2 2 = 1 -r 2 2
and

g 0 = dx 2 1 + dx 2 2 + dx 2 3 .
If N hyp denotes the normal vector to a surface Σ with respect to the metric g hyp , then its mean curvature with respect to the same metric is given by

H hyp (Σ) := - 1 2 trace X → [ ∇X N hyp ] T ,
where [•] T denotes the projection on the tangent bunble T Σ and ∇ is the Riemannian connection relative to g hyp . The mean curvature of Σ with respect to g 0 , denoted by H e (Σ), is given by

H e (Σ) := - 1 2 trace X → [∇ X N e ] T ,
where N e denotes the normal vector to Σ with respect to the metric g 0 and ∇ is the flat Riemannian connection. The Christoffel symbols, Γ k ij , associated to the metric g hyp all vanish except

Γ 1 11 = Γ 2 21 = Γ 2 12 = -Γ 1 22 = 2x 1 √ F , Γ 1 12 = Γ 2 22 = Γ 1 21 = -Γ 2 11 = 2x 2 √ F . Let ∂ 1 = ∂ ∂x 1 , ∂ 2 = ∂ ∂x 2 , ∂ 3 = ∂ ∂x 3
be the elements of a basis of the tangent space. Now, if X = i X i ∂ i and Y = j Y j ∂ j are two tangent vector fields, the expression of the covariant derivative in (D 2 × R, g hyp ) is given by

∇X Y = k ⎛ ⎝ X(Y k ) + i,j X i Y j Γ k ij ⎞ ⎠ ∂ k . It is clear that (26) ∇X Y = ∇ X Y + 2 k=1 i,j X i Y j Γ k ij ∂ k .
We suppose that N hyp = (N 1 , N 2 , N 3 ). From (26) we get the relation

(27) ∇X N hyp = ∇ X N hyp + 2 k=1 2 i,j=1 X i N j Γ k ij ∂ k .
We start by evaluating the term ∇ X N hyp . We observe that the normal vector N e = (N 1 , N 2 , N 3 ) to Σ with respect to the metric g 0 does not coincide with N hyp . But it is clear that

N hyp = (N 1 , N 2 , N 3 ) = ( √ F N 1 , √ F N 2 , N 3 ).
We observe that

∇ X N hyp = 3 k=1 X(N k )∂ k = X( √ F N 1 )∂ 1 + X( √ F N 2 )∂ 2 + X(N 3 )∂ 3 .
We can write X(

√ F N k ) = (1 -r 2 )X(N k ) -X(r 2 )N k , for k = 1, 2. Since X(N k ) = l X l ∂ x l N k and X(r 2 ) = 2x 1 X 1 + 2x 2 X 2 , it follows that X( √ F N k ) = X(N k ) -2x 1 X 1 + 2x 2 X 2 N k -r 2 3 l=1 X l ∂ x l N k , for k = 1, 2. We can conclude that ∇ X N hyp = k X(N k )∂ k is given by 3 k=1 X(N k )∂ k -2x 1 X 1 + 2x 2 X 2 2 k=1 N k ∂ k -r 2 2 k=1 3 l=1 X l ∂ x l N k ∂ k .
Inserting this equality into (26) and observing that 3 k=1 X(N k )∂ k = ∇ X N e , we obtain

∇ X N hyp = ∇ X N e -2x 1 X 1 + 2x 2 X 2 2 k=1 N k ∂ k -r 2 2 k=1 3 l=1 X l ∂ x l N k ∂ k .
Replacing this result into (27) we find the following expression of ∇X N hyp , which we will consider to compute the trace. We will assume X to be a vector field tangent to Σ. Then (28)

∇X

N hyp = ∇ X N e -2x 1 X 1 + 2x 2 X 2 2 k=1 N k ∂ k -r 2 2 k=1 3 l=1 X l ∂ x l N k ∂ k + 2 k=1 2 i,j=1 X i N j Γ k ij ∂ k .
We start to study the second summand. 2x

1 X 1 + 2x 2 X 2 2 k=1 N k ∂ k is the vector whose components with respect to the basis (∂ 1 , ∂ 2 , ∂ 3 ) are given by ⎡ ⎣ 2x 1 N 1 2x 2 N 1 0 2x 1 N 2 2x 2 N 2 0 0 0 0 ⎤ ⎦ ⎡ ⎣ X 1 X 2 X 3 ⎤ ⎦ .
So the trace of the mapping

3 i=1 X i ∂ i → 2x 1 X 1 + 2x 2 X 2 2 k=1 N k ∂ k T equals 2O(x 1 N 1 + x 2 N 2 ).
The components of the vector

2 k=1 3 l=1 X l ∂ x l N k ∂ k with respect to the basis (∂ 1 , ∂ 2 , ∂ 3 ) are given by ⎡ ⎣ ∂ x 1 N 1 ∂ x 2 N 1 ∂ x 3 N 1 ∂ x 1 N 2 ∂ x 2 N 2 ∂ x 3 N 2 0 0 0 ⎤ ⎦ ⎡ ⎣ X 1 X 2 X 3 ⎤ ⎦ .
So the trace of the mapping

3 i=1 X i ∂ i → r 2 2 k=1 3 l=1 X l ∂ x l N k ∂ k T equals O r 2 (∂ x 1 N 1 + ∂ x 2 N 2 ) .
As for the last term of (28), we can state that

2 k=1 i,j X i N j Γ k ij ∂ k is the vector whose components with respect to the basis (∂ 1 , ∂ 2 , ∂ 3 ) are given by (29) ⎡ ⎢ ⎣ 2x 1 N 1 √ F + 2x 2 N 2 √ F 2x 2 N 1 √ F -2x 1 N 2 √ F 0 -2x 2 N 1 √ F + 2x 1 N 2 √ F 2x 1 N 1 √ F + 2x 2 N 2 √ F 0 0 0 0 ⎤ ⎥ ⎦ ⎡ ⎣ X 1 X 2 X 3 ⎤ ⎦ .
Taking into account the equalities

N 1 = √ F N 1 and N 2 = √ F N 2
, it is easy to conclude that the trace of the mapping

i X i ∂ i → ⎡ ⎣ 2 k=1 i,j X i N j Γ k ij ∂ k ⎤ ⎦ T equals 4O (x 1 N 1 + x 2 N 2 )
. From the definition of the mean curvatures it is easy to obtain the following relation:

(30) H hyp (Σ) = H e (Σ) -O(x 1 N 1 + x 2 N 2 ) + r 2 O (∂ x 1 N 1 + ∂ x 2 N 2 ) .
We have proved the following result.

H hyp (S) = H e (S) -O(x 1 N 1 + x 2 N 2 ) + r 2 O (∂ x 1 N 1 + ∂ x 2 N 2 ) ,
where x 1 , x 2 are the Euclidean coordinates on D 2 and r 2 = x 2 1 + x 2 2 .

A scaled Costa-Hoffman-Meeks type surface

In this section we will describe the surface obtained by scaling of M k , the Costa-Hoffmann-Meeks surface of genus k 1 (see C. Costa [START_REF] Costa | Imersões minimas en R 3 de gênero un e curvatura total finita[END_REF], [START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF] and D. Hoffman and W. H. Meeks [START_REF] Hoffman | The asymptotic behavior of properly embedded minimal surfaces of finite topology[END_REF], [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF]) and we will study the mapping properties of its Jacobi operator. We denote by M k,ε the image of M k by a homothety of ratio ε 2 . We will adapt to our situation some of the analytical tools used in [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF] to show the existence of a family of minimal surfaces in R 3 close to M k with one planar end and two slightly bent catenoidal ends by an angle ξ ∈ (-ξ 0 , ξ 0 ), ξ 0 > 0 and small enough. We denote an element of this family by

M k (ξ). Then M k (ξ)| ξ=0 = M k .
9.1. The Costa-Hoffman-Meeks surface. We start by giving a brief description of the surface M k . After suitable rotation and translation, M k enjoys the following properties.

(1) It has one planar end E m asymptotic to the x 3 = 0 plane, one top end E t and one bottom end E b that are respectively asymptotic to the upper end and to the lower end of a catenoid with x 3 -axis of revolution. The planar end E m is located between the two catenoidal ends. (2) It is invariant under the action of the rotation of angle 2π k+1 about the x 3axis, under the action of the symmetry with respect to the x 2 = 0 plane and under the action of the composition of a rotation of angle π k+1 about the x 3 -axis and the symmetry with respect to the x 3 = 0 plane.

(3) It intersects the x 3 = 0 plane in k + 1 straight lines, which intersect themselves at the origin with angles equal to π k+1 . The intersection of M k with the plane x 3 = const ( = 0) is a single Jordan curve. The intersection of M k with the upper half space x 3 > 0 (resp. with the lower half space x 3 < 0) is topologically an open annulus. We denote by X i , with i = t, b, m, the parametrization of the end E i and with X i,ε the parametrization of the corresponding end E i,ε of M k,ε . Now we give a local description of the surface M k,ε near its ends and we introduce coordinates that we will use. The planar end. The planar end E m,ε of the surface M k,ε can be parametrized by

(32) X m,ε (x) := ε 2 x |x| 2 , ε 2 u m (x) ∈ R 3 , where x ∈ Bρ 0 (0)-{0} ⊂ R 2 .
Here ρ 0 > 0 is fixed small enough. In the sequel, where necessary, we will consider on B ρ 0 (0) also the polar coordinates (θ, ρ)

∈ S 1 × [0, ρ 0 ].
The function u m satisfies the minimal surface equation, which has the following form:

(33) 2H u = |x| 4 ε 2 div ∇u (1 + |x| 4 |∇u| 2 ) 1/2 = 0.
It can be shown (see [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF]) that the function u m can be extended at the origin continuously using the Weierstrass representation. In particular we can prove that

u m ∈ C 2,α ( Bρ 0 ) and u m = O C 2,α b (|x| k+1
), where the expression O C n,α b (g) denotes a function that, together with its partial derivatives of order less than or equal to n + α, is bounded by a constant times g. Furthermore, taking into account the symmetries of the surface, it is possible to show that the function u m , in polar coordinates, has to be collinear to cos(j(k + 1)θ), with j 1 and odd.

If we linearize in u = 0 the nonlinear equation (33), we obtain the expression of an operator which is, up to a multiplication by ε 4 , the Jacobi operator about the plane, that is, L R 2 = |x| 4 Δ 0 . To be more precise, the linearization of (33) gives (34)

L u v = |x| 4 ε 2 div ∇v 1 + |x| 4 |∇u| 2 -|x| 4 ∇u ∇u • ∇v (1 + |x| 4 |∇u| 2 ) 3 .
We will give the expression of H u+v , the mean curvature of the graph of the function u + v, in terms of the mean curvature of Σ u , that is, H u .

Here we will show that

(35) 2H u+v = 2H u + L u v + |x| 4 ε 2 Q u (|x| 2 ∇v, |x| 2 ∇ 2 v),
where Q u satisfies Q u (0, 0) = 0, ∇Q u (0, 0) = 0.

To show (35), we start by observing that (36) 1

1 + |x| 4 |∇(u + v)| 2 = 1 1 + |x| 4 |∇u| 2 -|x| 4 ∇u • ∇v (1 + |x| 4 |∇u| 2 ) 3 + Q u,1 (v),
where the function

Q u,1 satisfies Q u,1 (0) = 0, ∇Q u,1 (0) = 0.
The proof of that is very close to the one that appears in section 3: it's necessary only to replace F by |x| 4 . So we can omit some details. Secondly we observe that 2H u+v is given by

|x| 4 ε 2 div ∇(u + v) 1 + |x| 4 |∇u| 2 -|x| 4 ∇(u + v) ∇u • ∇v (1 + |x| 4 |∇u| 2 ) 3 + ∇(u + v)Q u,1 (v) = 2H u + |x| 4 ε 2 div ∇v 1 + |x| 4 |∇u| 2 -|x| 4 ∇u ∇u • ∇v (1 + |x| 4 |∇u| 2 ) 3 + |x| 4 ε 2 Q u (|x| 2 ∇v, |x| 2 ∇ 2 v).
From this the wanted expression follows.

Since we assume that Σ u is a minimal surface, we will consider

H u = 0.
Following what we have done in section 7 replacing F by |x| 4 we get:

(37)

|x| 4 ε 2 Δ 0 v + 1 + |x| 4 |∇u| 2 Lu v + Q u (|x| 2 ∇v, |x| 2 ∇ 2 v) = 0, where Lu v is a second order linear operator with coefficients in O C 2,α b (|x| k+1 ).
It is important to remark that if the function v satisfies equation (37) with u = u m , then the graph of the function ε 2 (u m + v) is minimal. Now we are interested in finding the equation which a function w must satisfy in such a way that the surface parametrized by X m,ε +we 3 , that is, the graph of w over the middle end E m,ε , is minimal, which is equivalent to requiring that the graph of ε 2 u m + w is minimal. Then we can obtain the wanted equation by replacing v by w/ε 2 in (37). So we get (38)

|x| 4 ε 2 1 ε 2 Δ 0 w + 1 + |x| 4 |∇u| 2 1 ε 2 Lu w + Q u |x| 2 ε 2 ∇w, |x| 2 ε 2 ∇ 2 w = 0. If we set Q ε,u (•) := |x| 4 ε 2 1 + |x| 4 |∇u| 2 Q u |x| 2 ε 2 ∇•, |x| 2 ε 2 ∇ 2
• to simplify the notation, we can write this equation in the following way:

(39) |x| 4 ε 4 Δ 0 w + |x| 4 ε 4 1 + |x| 4 |∇u| 2 Lu w + Q ε,u (w) = 0.
The catenoidal ends. We denote by X c the parametrization of the standard catenoid C whose axis of revolution is the x 3 -axis. Its expression is

X c (s, θ) := (cosh s cos θ, cosh s sin θ, s) ∈ R 3 ,
where (s, θ) ∈ R × S 1 . The unit normal vector field about C is given by n c (s, θ) := 1 cosh s (cos θ, sin θ,sinh s).

The catenoid C may be divided into two pieces, denoted by C ± , which are defined as the image by X c of R ± × S 1 . For any ε > 0, we define the catenoid C ε as the image of C by a homothety of ratio ε 2 . We denote by X c,ε := ε 2 X c its parametrization. Of course, by this transformation, two surfaces correspond to C ± . We denote them by C ε,± .

Up to some dilation, we can assume that the two ends E t,ε and E b,ε of M k,ε are asymptotic to some translated copy of the catenoid parametrized by X c,ε in the vertical direction. Therefore, E t,ε and E b,ε can be parametrized, respectively, by

(40) X t,ε := X c,ε + w t n c + σ t,ε e 3 for (s, θ) ∈ (s 0 , ∞) × S 1 , (41) X b,ε := X c,ε -w b n c -σ b,ε e 3 for (s, θ) ∈ (-∞, -s 0 ) × S 1
, where σ t,ε , σ b,ε ∈ R and the functions w t , w b tend exponentially fast to 0 as s goes to ±∞, reflecting the fact that the ends are asymptotic to a catenoidal end. Furthermore, taking into account the symmetries of the surface, it is easy to show that the functions w t , w b , in terms of the (s, θ) coordinates, have to be collinear to cos(j(k + 1)θ), with j ∈ N and must satisfy w b (s, θ) = -w t (-s, θ -π k+1 ). Furthermore we have σ t,ε = σ b,ε . In section 3 of [START_REF] Mazzeo | Connected sums of constant mean curvature surfaces in Euclidean 3 space[END_REF] the expression of the mean curvature operator of a surface close to a scaled standard catenoid is given. We can adapt this result to our situation.

We obtain that the surface parametrized by X c,ε + w n c is minimal if and only if the function w satisfies the minimal surface equation

(42) 1 ε 4 L C w + Q ε (w) = 0, L C
being the Jacobi operator about the catenoid, i.e.

L C w = 1 cosh 2 s ∂ 2 ss w + ∂ 2 θθ w + 2w cosh 2 s , and (43)
Q ε (w) = 1 ε 2 cosh 2 s Q 2,ε w ε 2 cosh s + 1 ε 2 cosh s Q 3,ε w ε 2 cosh s .
Here Q 2 , Q 3 are nonlinear second order differential operators which are bounded in C k (R × S 1 ), for every k, and satisfy

Q 2 (0) = Q 3 (0) = 0, ∇Q 2 (0) = ∇Q 3 (0) = 0, ∇ 2 Q 3 (0) = 0 together with: Q j (v 2 ) -Q j (v 1 ) C 0,α ([s,s+1]×S 1 ) c sup i=1,2 v i C 2,α ([s,s+1]×S 1 ) j-1 v 2 -v 1 C 2,α ([s,s+1]×S 1 ) (44) for all s ∈ R and all v 1 , v 2 such that v i C 2,α ([s,s+1]×S 1 )
1. The constant c > 0 does not depend on s.

For all ρ < ρ 0 and s > s 0 , we define (45)

M k,ε (s, ρ) := M k,ε -X t,ε ((s, ∞) × S 1 ) ∪ X b,ε ((-∞, -s) × S 1 ) ∪ X m,ε (B ρ (0)) .
The parametrizations of the three ends of M k,ε induce a decomposition of M k,ε into slightly overlapping components: a compact piece M k,ε (s 0 + 1, ρ 0 /2) and three noncompact pieces X t,ε ((s 0 , ∞) × S 1 ), X b,ε ((-∞, -s 0 ) × S 1 ) and X m,ε ( Bρ 0 (0)).

We define a weighted space of functions on M k,ε .

Definition 9.1. Given ∈ N, α ∈ (0, 1) and δ ∈ R, the space C ,α δ (M k,ε ) is defined to be the space of functions in C ,α loc (M k,ε ) for which the following norm is finite:

w C ,α δ (M k,ε ) := w C ,α (M k,ε (s 0 +1,ρ 0 /2)) + w • X m,ε C ,α ( Bρ 0 (0)) + w • X t,ε C ,α δ ([s 0 ,+∞)×S 1 ) + w • X b,ε C ,α δ ((-∞,-s 0 ]×S 1 )
, where

f C ,α δ ([s 0 ,+∞)×S 1 ) = sup s s 0 e -δs f C ,α ([s,s+1]×S 1 ) , f C ,α δ ((-∞,-s 0 ]×S 1 ) = sup s -s 0 e δs f C ,α ([s-1,s]×S 1 )
and which are invariant under the action of the symmetry with respect to the x 2 = 0 plane, that is, w(p) = w(p) for all p ∈ M k,ε , where p := (x 1 , -x 2 , x 3 ) if p = (x 1 , x 2 , x 3 ), invariant with respect to a rotation of angle 2π k+1 about the x 3axis and to the composition of a rotation of angle π k+1 about the x 3 -axis and the symmetry with respect to the x 3 = 0 plane. We remark that there is no weight on the middle end. In fact we compactify this end and we consider a weighted space of functions defined on a two-ended surface. We will perturb the surface M k,ε by the normal graph of a function u ∈ C 2,α δ (M k,ε ). 9.2. The Jacobi operator. The Jacobi operator about M k,ε is

L M k,ε := Δ M k,ε + |A M k,ε | 2 ,
where |A M k,ε | is the norm of the second fundamental form on M k,ε .

In the parametrization of the ends introduced above, the volume forms dvol M k,ε can be written as γ t ds dθ and γ b ds dθ near the catenoidal type ends and as γ m dx 1 dx 2 near the middle end. Now we can define globally on M k,ε a smooth function

(46) γ : M k,ε -→ [0, ∞)
that is identically equal to ε 4 on M k,ε (s 0 -1, 2ρ 0 ) and equal to γ t (resp. γ b , γ m ) on the end E t,ε (resp. E b,ε , E m, ). They are defined in such a way that on X t,ε ((s 0 , ∞) × S 1 ) and on X b,ε ((-∞, -s 0 ) × S 1 ) we have

γ • X t,ε (s, θ) ∼ ε 4 cosh 2 s and γ • X b,ε (s, θ) ∼ ε 4 cosh 2 s.
Finally on X m,ε (B ρ 0 ), we have

γ • X m,ε (x) ∼ ε 4 |x| 4 .
It is possible to check that

L ε,δ : C 2,α δ (M k,ε ) -→ C 0,α δ (M k,ε ) w -→ γ L M k,ε (w)
is a bounded linear operator. The subscript δ is meant to keep track of the weighted space over which the Jacobi operator is acting. Observe that the function γ is here to counterbalance the effect of the conformal factor

1 √ |g M k,ε |
in the expression of the Laplacian in the coordinates we use to parametrize the ends of the surface M k,ε . This is precisely what is needed to have the operator defined from the space C 2,α δ (M k,ε ) into the target space C 0,α δ (M k,ε ). To have a better grasp of what is going on, let us linearize the nonlinear equation (42) at w = 0. We get the expression of the Jacobi operator about the scaled catenoid C ε :

L C ε := 1 ε 4 cosh 2 s ∂ 2 s + ∂ 2 θ + 2 cosh 2 s .
We can observe that the operator cosh 2 s L C ε maps the space (cosh s) δ C 2,α ((s 0 ,+∞) ×S 1 ) into the space (cosh s) δ C 0,α ((s 0 , +∞) × S 1 ).

Similarly, if we linearize the nonlinear equation (33) at u = 0, we obtain (see (3) with u = 0), up to a multiplication by 1/ε 4 , the expression of the Jacobi operator about the plane:

1 ε 4 L R 2 := |x| 4 ε 4 Δ 0 .
Again, the operator γ 1 ε 4 L R 2 = Δ 0 clearly maps the space C 2,α ( Bρ 0 ) into the space C 0,α ( Bρ 0 ). Now, the function γ plays, for the ends of the surface M k,ε , the role played by the function cosh 2 s for the ends of the standard catenoid and the role played by the function |x| -4 for the plane. Since the Jacobi operator about M k,ε is asymptotic to

1 ε 4 L R 2 at E m,ε and is asymptotic to L C ε at E t,ε and E b,ε , we conclude that the operator L ε,δ maps C 2,α δ (M k,ε ) into C 0,α δ (M k,ε
). Now we recall the notion of nondegeneracy introduced in [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF].

Definition 9.2. The surface M k,ε is said to be nondegenerate if L ε,δ is injective for all δ < -1.

It is useful to observe that a duality argument in the weighted Lebesgue spaces implies that

(L ε,δ is injective) ⇔ (L ε,-δ is surjective) if δ / ∈ Z.
See [START_REF] Melrose | The Atiyah-Patodi-Singer index theorem[END_REF] and [START_REF] Jleli | Constant mean curvature hypersurfaces[END_REF] for more details.

The nondegeneracy of M k,ε is related to the mapping properties of L ε,δ and to the kernel of this operator. From the observations made above, it follows that at the catenoidal type ends the Jacobi operators of M k,ε and M k are respectively asymptotic to L C ε and L C which coincide up to a multiplication by ε 4 ; at the middle end they are respectively asymptotic to 1 ε 4 L R 2 and L R 2 . So we could transpose some of the results about the surface M k (0) contained in [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF] related to the study of its mean curvature operator, to the surface M k,ε , including nondegeneracy. The only difference is that here we work with spaces of functions invariant with respect to all of the symmetries of M k . The Jacobi fields. It is known that a smooth one-parameter group of isometries containing the identity generates a Jacobi field, that is, a solution of the equation L M k,ε u = 0. The Jacobi fields of this type, which are invariant with respect to the mirror symmetry by the x 2 = 0 plane, the rotation by 2π k+1 about the x 3 -axis, the composition of the rotation by π k+1 about the x 3 -axis and the mirror symmetry with respect to the x 3 = 0 plane, are generated by dilations. Of course the Jacobi equation has other solutions which are not taken into account because they are not invariant under the action of the symmetries listed above. See [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF] for details.

The Killing vector field Ξ(p) = p, which is associated to the one-parameter group of dilations, generates the Jacobi field

Φ(p) := n(p) • p.
It is clear that Φ(p) grows linearly and so it is not bounded.

With this notation, we define the deficiency space

D := Span{χ t Φ, χ b Φ },
where χ t is a cutoff function that is identically equal to 1 on X t,ε ((s 0 +1, +∞)×S 1 ), identically equal to 0 on M k,ε -X t,ε ((s 0 , +∞) × S 1 ) and that is invariant under the action of the symmetries listed above. The cutoff function χ b is obtained from χ t by using the symmetries. Clearly, if δ < 0, then Lε,δ :

C 2,α δ (M k,ε ) ⊕ D -→ C 0,α δ (M k,ε ) w -→ γ L M k,ε w
is a bounded linear operator. Thanks to a result of S. Nayatani about the dimension of the kernel of the Jacobi operator of M k , shown in [START_REF] Nayatani | Morse index of complete minimal surfaces, The problem of Plateau[END_REF][START_REF] Nayatani | Morse index and Gauss maps of complete minimal surfaces in Euclidean 3space[END_REF] and extended in [START_REF] Morabito | Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces[END_REF], we can state that there is not any bounded Jacobi field which is invariant with respect to the symmetries of M k,ε . So we get the following result about the operator L ε,δ .

Proposition 9.3. We choose δ ∈ (1, 2). Then the operator L ε,δ is surjective. Moreover, there exists G ε,δ , a right inverse for L ε,δ whose norm is bounded.

This fact, together with an adaptation to our setting of the linear decomposition lemma proved in [START_REF] Kusner | The moduli space of complete embedded constant mean curvature surfaces[END_REF] for constant mean curvature surfaces (see also [START_REF] Jleli | Constant mean curvature hypersurfaces[END_REF] for minimal hypersurfaces), allows us to prove the following result. Proposition 9.4. We choose δ ∈ (-2, -1). Then the operator Lε,δ is surjective.

10. An infinite dimensional family of minimal surfaces which are close to a compact part of a scaled Costa-Hoffman-Meeks type surface in M 2 × R

We recall that in section 8 we found that the mean curvature with respect to the metric g hyp of a surface S in M 2 × R can be expressed in terms of the Euclidean mean curvature of S and the components of the normal vector to the same surface with respect to the flat metric g 0 .

In this section we will apply this result to prove the existence of a family of embedded minimal surfaces with respect to the metric g hyp which are close to the piece of the surface M k,ε contained a cylindrical neighbourhood of radius

r ε = ε/2 of {0, 0} × R.
We start by giving the statement of a result that can be easily obtained by [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF], Lemma 2.2. It describes the region of the surface M k,ε which can be parametrized as a graph on an annular neighbourhood of r ε contained in the x 3 = 0 plane.

Lemma 10.1.

There exists ε 0 > 0 such that, for all ε ∈ (0, ε 0 ), an annular part of the ends E t,ε , E b,ε and E m,ε of M k,ε can be written as vertical graphs over the annulus B 2r ε -B r ε /2 of the functions

Z t (θ, r) = σ t,ε + ε 2 ln 2r ε 2 + O C 2,α b (ε 3 ), (47) Z b (θ, r) = -Z t θ - π k + 1 , r . (48)
As for the parametrization of the planar end, it satisfies

(49) Z m (θ, r) = O C 2,α b ε 2 r ε 2 -(k+1)
.

Here (θ, r) are the polar coordinates in the

x 3 = 0 plane. The functions O C 2,α b (f )
are defined in the annulus B 2r ε -B r ε /2 and are bounded in the C 2,α b topology by a constant (independent by f ) multiplied by f , where the partial derivatives are computed with respect to the vector fields r ∂ r and ∂ θ .

We set s ε =ln ε, ρ ε := 2ε and we define M T k,ε to be equal to M k,ε , from which we have removed the image of (s ε , +∞) × S 1 by X t,ε , the image of (-∞, -s ε ) × S 1 by X b,ε and the image of B ρ ε (0) by X m,ε . The values of s ε and ρ ε have been chosen in such a way that the surface M T k,ε is contained in a neighbourhood of radius r ε = ε/2 of {0, 0} × R. In this section we will prove the existence of a family of minimal surfaces close to M T k,ε . To this aim we will use Proposition 8.1 and we will follow the work [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF]. First, we modify the parametrization of the ends E t,ε , E b,ε and E m,ε , for appropriate values of s and x, so that the images of r = r ε by

(50) θ → (r cos θ, r sin θ, Z b (θ, r)), θ → (r cos θ, r sin θ, Z t (θ, r))
correspond, respectively, up to a vertical translation, to the horizontal curves at heights ±ε 2 ln(2r ε /ε 2 ). The curve θ → (r cos θ, r sin θ, Z m (θ, r)), if r = r ε , corresponds, up to a vertical translation, to a horizontal curve at height ε 2 r ε /ε 2 -(k+1) .

The second step is the modification of the unit normal vector field on M k,ε into a transverse unit vector field ñε in such a way that it coincides with the normal vector field n ε on M k,ε , is equal to e 3 on the graph over B 2r ε -B r ε /2 of the functions U t and U b and interpolates smoothly between the different definitions of ñε in different subsets of M T k,ε . Finally we observe that close to E t,ε , we can give the following estimate:

(51) ε 4 cosh 2 s L M k,ε v -(ε 4 cosh 2 s) -1 (∂ ss v + ∂ θθ v) c (cosh 2 s) -1 v .
This follows easily from (42) together with the fact that w t decays at least like (cosh 2 s) -1 on E t,ε . Similar considerations hold close to the bottom end E b,ε . Near the middle planar end E m,ε , we observe that the following estimate holds:

(52) ε 4 |x| -4 L M k,ε v -|x| 4 ε -4 Δ 0 v c |x| 2k+3 ∇v .
This follows easily from (34) together with the fact that u m decays at least like

|x| k+1 on E m,ε .
The graph of a function u, Σ u , using the vector field ñε , is a minimal surface (with respect to the metric g 0 ) if and only if u is a solution of a second order nonlinear elliptic equation of the form This operator is supported in a neighbourhood of {±s ε }×S 1 and of S 1 ×{ρ ε }. It is possible to show that the coefficients of ε 4 Lε are uniformly bounded by a constant times ε 4 . We start by noticing that the conformal factor (cosh 2 s) -1 contributes with a term equal to ε 2 . Furthermore the fact that ñε

H e (Σ u ) = L M T k,ε u -Lε u -Q ε (u) = 0, where L M T k,ε is the Jacobi operator about M T k,ε , Q ε is
, n ε = 1 + O C 2,α b (ε 2 )
in a neighbourhood of {±s ε }×S 1 and the result of [START_REF] Hauswirth | Higher genus Riemann minimal surfaces[END_REF], appendix B, show that the change of vector field induces a linear operator whose coefficients are bounded by a constant times ε 2 . The contribution of the change of parametrization can be estimated as follows. As for the catenoidal ends, we determine the difference between the value of functions ( 47) and ( 48), for r in a neighbourhood of r ε , and the height of the horizontal boundary curve (see (50)). We obtain it is bounded by a constant times ε 3 . Using (49) we can show that a similar estimate holds for the planar end. Now, we consider three functions ϕ t , ϕ b , ϕ m ∈ C 2,α (S 1 ) which are even, with respect to θ, ϕ t is collinear to cos(j(k + 1)θ) (k 1 fixed) with j 1, ϕ b (θ) = -ϕ t (θ -π k+1 ), while ϕ m is collinear to cos(l(k + 1)θ), with l 1 and odd. Assume that they satisfy (53)

ϕ t C 2,α + ϕ b C 2,α + ϕ m C 2,α κ ε 2 .
We set Φ := (ϕ t , ϕ b , ϕ m ) and we define w Φ to be the function equal to (1) χ + H ϕ t (s εs, •) on the image of X t,ε , where χ + is a cutoff function equal to 0 for s s 0 + 1 and identically equal to 1 for s ∈ [s 0 + 2, s ε ];

(2) χ -H ϕ b (s + s ε , •) on the image of X b,ε , where χ -is a cutoff function equal to 0 for s -s 0 -1 and identically equal to

1 for s ∈ [-s ε , -s 0 -2]; (3) χ m Hρ ε ,ϕ m (•, •) on the image of X m,ε
, where χ m is a cutoff function equal to 0 for ρ ρ 0 and identically equal to 1 for ρ ∈ [ρ ε , ρ 0 /2]; (4) zero on the remaining part of the surface M T k,ε . The cutoff functions just introduced must have the same symmetry properties as the functions in C 2,α δ (M k,ε ). H and H are harmonic extension operators introduced respectively in Propositions 12.3 and 12.2.

We would like to prove that, under appropriate hypotheses, the graph Σ u over M T k,ε of the function u = w Φ + v is a minimal surface with respect to the metric g hyp . We want to point out that to construct the graph of the function u, here we consider the normal vector field with respect to the Euclidean metric g 0 . The equation to solve is:

H hyp (Σ u ) = 0.
If we denote by N u = (N 1 (u), N 2 (u), N 3 (u)) the unit normal vector to Σ u , by equation (31) we can express H hyp (Σ u ) in terms of the Euclidean mean curvature and write the equation to solve as

H e (Σ u ) -O(x 1 N 1 (u) + x 2 N 2 (u)) + r 2 O(∂ x 1 N 1 (u) + ∂ x 2 N 2 (u)) = 0, where x 1 , x 2 are the coordinates on D 2 and r 2 = x 2 1 + x 2 2 .
To simplify the notation we set P (u

) := O(x 1 N 1 (u) + x 2 N 2 (u)) -r 2 O(∂ x 1 N 1 (u) + ∂ x 2 N 2 (u)).
Taking into account that u = w Φ + v, now the expression of the equation to solve is given by

L M T k,ε (w Φ + v) -Lε (w Φ + v) -Q ε (w Φ + v) -P (w Φ + v) = 0.
The resolution of the previous equation is obtained by the following fixed point problem:

(54) v = T (Φ, v) with T (Φ, v) = G ε,δ • E ε γ Lε (w Φ + v) + P (w Φ + v) -L M T k,ε w Φ + Q ε (w Φ + v)
, where δ ∈ (1, 2), the operator G ε,δ is defined in Proposition 9.3 and E ε is a linear extension operator such that

E ε : C 0,α δ (M T k,ε ) -→ C 0,α δ (M k,ε ), where C 0,α δ (M T k,ε ) denotes the space of functions of C 0,α δ (M k,ε ) restricted to M T k,ε . It is defined by E ε v = v in M T k,ε , E ε v = 0 in the image of [s ε + 1, +∞) × S 1 by X t,ε , in the image of (-∞, -s ε -1] × S 1 by X b,ε and in the image of B ρ ε /2 by X m,ε . Finally E ε v is an interpolation of these values in the remaining part of M k,ε such that (E ε v) • X t,ε (s, θ)=(1 + s ε -s)(v • X t,ε (s ε , θ)), for (s, θ) ∈ [s ε , s ε + 1] × S 1 , (E ε v) • X b,ε (s, θ)=(1 + s ε + s)(v • X b,ε (s ε , θ)), for (s, θ) ∈ [-s ε -1, -s ε ] × S 1 , (E ε v) • X m,ε (θ, ρ) = 2 ρ ε ρ -1 (v • X m,ε (θ, ρ ε )) for (θ, ρ) ∈ S 1 × [ρ ε /2, ρ ε ]. Remark 10.2. From the definition of E ε , if supp v ∩ (B ρ ε -B ρ ε /2 ) = ∅, then (E ε v) • X m,ε C 0,α ( Bρ 0 ) cρ -α ε v • X m,ε C 0,α (B ρ 0 -B ρ ε ) .
This phenomenon of explosion of the norm does not occur near the catenoidal type ends:

(E ε v) • X t,ε C 0,α ([s 0 ,+∞)×S 1 ) c v • X t,ε C 0,α ([s 0 ,s ε ]×S 1 ) .
A similar equation holds for the bottom end. In the following we will assume α > 0 and close to zero.

The existence of a solution v ∈ C 2,α δ (M T k,ε ) for equation ( 54) is a consequence of the following result, which proves that T (Φ, •) is a contraction mapping.

Lemma 10.3. Let δ ∈ (1, 2) and Φ = (ϕ t , ϕ b , ϕ m ) ∈ [C 2,α (S 1
)] 3 satisfy (53) and enjoy the properties given above. There exist constants c κ > 0 and ε κ > 0, such that

(55) T (Φ, 0) C 2,α δ (M k,ε ) c κ ε 5/2
and, for all ε ∈ (0, ε κ ), α ∈ (0, 1/2),

T (Φ, v 2 ) -T (Φ, v 1 ) C 2,α δ (M k,ε ) 1 2 v 2 -v 1 C 2,α δ (M k,ε ) , T (Φ 2 , v) -T (Φ 1 , v) C 2,α δ (M k,ε ) cε 3/2 Φ 2 -Φ 1 C 2,α (S 1 ) ,
where c is a positive constant, with

Φ 2 -Φ 1 C 2,α (S 1 ) = ϕ t,2 -ϕ t,1 C 2,α (S 1 ) + ϕ b,2 -ϕ b,1 C 2,α (S 1 ) + ϕ m,2 -ϕ m,1 C 2,α (S 1 ) for all v, v 1 , v 2 ∈ C 2,α δ (M k,ε ) and satisfying v C 2,α δ 2 c κ ε 5/2
and for all boundary data Φ i = (ϕ t,i , ϕ b,i , ϕ m,i ) ∈ [C 2,α (S 1 )] 3 , i = 1, 2, enjoying the same properties as Φ.

Proof. We recall that the Jacobi operator associated to M k,ε , is asymptotic (up to a multiplication by 1/ε 4 ) to the operator of the catenoid near the catenoidal ends, and it is asymptotic to the Laplacian near the planar end. The function w Φ is identically zero far from the ends where the explicit expression of L M k,ε is not known: this is the reason for our particular choice in the definition of w Φ . Then from the definition of w Φ and thanks to Proposition 9.3 we obtain the estimate

E ε γL M k,ε w Φ C 0,α δ (M k,ε ) = γL M T k,ε -∂ 2 s + ∂ 2 θ (w Φ • X t,ε ) C 0,α δ ([s 0 +1,s ε ]×S 1 ) + γL M T k,ε -∂ 2 s + ∂ 2 θ (w Φ • X b,ε ) C 0,α δ ([-s ε ,-s 0 -1]×S 1 ) + ρ -α ε γL M T k,ε -Δ 0 (w Φ • X m,ε ) C 0,α (S 1 ×[ρ ε ,ρ 0 ]) c cosh -2 s (w Φ • X t,ε ) C 0,α δ ([s 0 +1,s ε ]×S 1 ) + c cosh -2 s (w Φ • X b,ε ) C 0,α δ ([-s ε ,-s 0 -1]×S 1 ) + cε -α ρ 2k+3 ∇(w Φ • X m,ε ) C 0,α (S 1 ×[ρ ε ,ρ 0 ]) c κ ε 4 + c κ ε 5/2 c κ ε 5/2 .
To obtain this estimate we used the following:

sup

[s 0 +1,s ε ]×S 1 e -δs cosh -2 s(w Φ • X t,ε/2 ) C 0,α ([s,s+1]×S 1 ) c sup [s 0 +1,s ε ]×S 1 e -δs e -2(s ε -s) e -2s φ t C 2,α (S 1 ) c e -2s ε φ t C 2,α (S 1 ) c κ ε 4
(a similar estimate holds for the bottom end) and

ρ -α ε ρ 2k+3 ∇(w Φ • X m,ε ) C 0,α (S 1 ×[ρ ε ,ρ 0 ]) cε -α ρ ε φ m C 2,α (S 1 ) c κ ε 5/2
together with the fact that s ε =ln ε and ρ ε = 2ε, from which e -2s ε = ε 2 and ρ -α ε = (2ε) -α . Using the properties of Lε and the definition of γ (see ( 46)), we obtain

E ε γ Lε w Φ C 0,α δ (M k,ε ) cε 2 w Φ • X t,ε C 0,α δ ([s 0 +1,s ε ]×S 1 ) + cε 2 w Φ • X b,ε C 0,α δ ([-s ε ,-s 0 -1]×S 1 ) + cε 2-α w Φ • X m,ε C 0,α (S 1 ×[ρ ε ,ρ 0 /2]) c κ ε 5/2 . The estimate of E ε (γP (w Φ )) C 0,α δ (M k,ε
) is related to the estimate of the horizontal components and their derivatives of order one of the normal vector to the surface and to the definition of the function

γ on M T k,ε . It is convenient to recall that the operator E ε smoothly extends a function g ∈ C 0,α δ (M T k,ε ) to a function in C 0,α δ (M k,ε
), substantially leaving it unchanged on M T k,ε and setting it equal to zero on the remaining part of M k,ε . P keeps track of the difference of the mean curvatures of Σ u computed with respect to two different metrics. It is sufficient to estimate the norm of γP (w Φ ) only on M T k,ε . The function γ equals ε 4 cosh 2 s at the catenoidal ends of M T k,ε , equals ε 4 /|x| 4 at the middle end, and equals ε 4 far from the ends. We recall that |s| ∈ [s 0 , s ε ] and |x| = ρ ∈ [ρ ε , ρ 0 ]. Furthermore it is easy to prove that the horizontal components N 1 , N 2 of the normal vector (with respect to the metric g 0 ) to the graph of w Φ over the middle end of M T k,ε , are, in absolute value, smaller than a constant times ε 2 . Their derivatives ∂ x 1 N 1 , ∂ x 2 N 2 are bounded by a constant times ε. As for the values of the coordinates x 1 and x 2 , we recall that we are working inside a cylindrical neighbourhood of radius r ε = ε/2. We get

E ε (γP (w Φ )) C 0,α δ (M k,ε ) c ε 5/2 .
As for the last term, we recall that the expression of the operator Q ε depends on the type of end we are considering (see equations ( 39) and ( 43)). It follows that

E ε (γQ ε (w Φ )) C 0,α δ (M k,ε ) c κ ε 5/2 .
In fact

E ε (γQ ε (w Φ )) C 0,α δ (M k,ε ) cε 2 w Φ ε 2 cosh s • X t,ε 2 C 2,α δ/2 ([s 0 +1,s ε ]×S 1 ) + cε 2 w Φ ε 2 cosh s • X b,ε 2 C 2,α δ/2 ([-s ε ,-s 0 -1]×S 1 ) + cε 2(1-α) |x| 2 ε 2 w Φ • X m,ε 2 C 2,α (S 1 ×[ρ ε ,ρ 0 /2]) c κ ε 5/2 .
As for the second estimate, we recall that

T (Φ, v) := G ε,δ • E ε γ P (w Φ + v) + Lε (w Φ + v) -L M k,ε w Φ + Q ε (w Φ + v) .
Then

T (Φ, v 2 ) -T (Φ, v 1 ) C 2,α δ (M k,ε ) E ε (γ (P (w Φ + v 2 ) -P (w Φ + v 1 ))) C 0,α δ (M k,ε ) + E ε γ Lε (v 2 -v 1 ) C 0,α δ (M k,ε ) + E ε (γ (Q ε (w Φ + v 1 ) -Q ε (w Φ + v 2 ))) C 0,α δ (M k,ε
) . We observe that from the considerations above it follows that

E ε (γ (P (w Φ + v 2 ) -P (w Φ + v 1 ))) C 0,α δ (M k,ε ) cε 3/2 ||v 2 -v 1 || C 2,α δ (M k,ε ) , E ε γ Lε (v 2 -v 1 ) C 0,α δ (M k,ε ) cε 2 ||v 2 -v 1 || C 2,α δ (M k,ε )
and

E ε (γ (Q ε (w Φ + v 1 ) -Q ε (w Φ + v 2 ))) C 0,α δ (M k,ε ) c||v 2 -v 1 || C 2,α δ (M k,ε ) ε 2 w Φ ε 2 cosh s • X t,ε C 0,α (E t,ε ) + ε 2 w Φ ε 2 cosh s • X b,ε C 0,α (E b,ε ) +ε 2-α |x| 2 ε 2 w Φ • X m,ε C 0,α (E m,ε ) c κ ε 2 ||v 2 -v 1 || C 2,α δ (M k,ε ) . Then T (Φ, v 2 ) -T (Φ, v 1 ) C 2,α δ (M k,ε ) cε 3/2 ||v 2 -v 1 || C 2,α δ (M k,ε
) . To show the last estimate it is sufficient to observe that

T (Φ 2 , v) -T (Φ 1 , v) C 2,α δ (M k,ε ) E ε (γ (P (w Φ 2 + v) -P (w Φ 1 + v))) C 0,α δ (M k,ε ) + E ε γ Lε (w Φ 2 -w Φ 1 ) C 0,α δ (M k,ε ) + E ε (γ (Q ε (w Φ 2 + v) -Q ε (w Φ 1 + v))) C 0,α δ (M k,ε ) cε 3/2 Φ 2 -Φ 1 C 2,α (S 1 ) + c v C 0,α δ (M k,ε ) Φ 2 -Φ 1 C 2,α (S 1 ) cε 3/2 Φ 2 -Φ 1 C 2,α (S 1 ) . Theorem 10.4. Let B := {w ∈ C 2,α δ (M k,ε ) | ||w|| C 2,α 2c κ ε 5/2 }.
Then the nonlinear mapping T (Φ, •) defined above has a unique fixed point v in B.

Proof. The previous lemma shows that, if ε is chosen small enough, the nonlinear mapping T (Φ, •) is a contraction mapping from the ball B of radius 2c κ ε 5/2 in C 2,α δ (M k,ε ) into itself. This value follows from the estimate of the norm of T (Φ, 0). Consequently, thanks to the Schäuder fixed point theorem, T (Φ, •) has a unique fixed point v in this ball.

This argument provides an embedded minimal surface with respect to the metric g hyp , M T k,ε (Φ), which is close to M T k,ε and has three boundaries. This surface close to its upper and lower boundary is a vertical graph over the annulus B r ε -B r ε /2 , with r ε = ε/2, whose parametrization is, respectively, given by

U t (θ, r) = σ t,ε + ε 2 ln 2r ε 2 + H ϕ t s ε -ln 2r ε 2 , θ + V t (θ, r), U b (θ, r) = -U t θ - π k + 1 , r ,
where s ε =ln ε. Near the middle boundary, the surface is a vertical graph whose parametrization is

U m (θ, r) = Hρ ε ,ϕ m θ, ε 2 r + V m (θ, r).
The boundaries of the surface correspond to r = r ε . All the functions V i , i = t, m, depend nonlinearly on ε, ϕ.

Lemma 10.5. The function V i (ε, ϕ i ), for i = t, b, satisfies

V i (ε, ϕ i )(•, r ε •) C 2,α ( B1 -B 1/2 ) cε 2 and (56) V i (ε, ϕ i,2 )(•, r ε •) -V i (ε, ϕ i,1 )(•, r ε •) C 2,α ( B1 -B 1/2 ) cε 3/2-δ ϕ i,2 -ϕ i,1 C 2,α (S 1 ) . The function V m (ε, ϕ) satisfies V m (ε, ϕ)(•, ρ ε •) C 2,α ( B1 -B 1/2 ) cε 2 and (57) V m (ε, ϕ m,2 )(•, ρ ε •)-V m (ε, ϕ m,1 )(•, ρ ε •) C 2,α ( B1 -B 1/2 ) cε 3/2 ϕ m,2 -ϕ m,1 C 2,α (S 1 ) .
Proof. We start by observing that the functions V t , V b , V m are the restrictions to E t,ε , E b,ε , E m,ε of a fixed point for the operator T (Φ, •). Then the second estimate follows from

V i (ε, ϕ i,2 )(•, •) -V i (ε, ϕ i,2 )(•, •) C 2,α ( Br ε -B r ε /2 ) ce δs ε (T (Φ 2 , V i ) -T (Φ 1 , V i )) • X i,ε C 2,α δ (Ω i ×S 1 ) , for i = t, b, with Ω t = [s 0 , s ε ] and Ω b = [-s ε , -s 0 ]. The third estimate comes from V m (ε, ϕ m,2 )(•, •) -V m (ε, ϕ m,2 )(•, •) C 2,α ( Br ε -B r ε /2 ) c (T (Φ 2 , V m ) -T (Φ 1 , V m )) • X m,ε C 2,α (S 1 ×[ρ ε ,ρ 0 ]) .

The matching of Cauchy data

In this section we will complete the proof of Theorem 1.1. Using the result of section 7, we obtain two minimal surfaces that are perturbations of two parts of the catenoid defined in M 2 × R. The first surface, which we denote by S t,d t (ϕ t ), after a translation by d t along the x 3 -axis, can be parameterized in B 2r ε -B r ε as the vertical graph of

Ūt (θ, r) = ε 2 ln 2r ε 2 + d t + H r ε ,ϕ t (θ, r) + O C 2,α b (ε 2 ).
The second surface, which we denote by S b,d b (ϕ b ), where ϕ b (θ) = ϕ t (θ -π k+1 ) and

d b = -d t , can be parameterized in B 2r ε -B r ε as the vertical graph of (58) Ūb (θ, r) = -Ūt θ - π k + 1 , r .
Using the result of section 5, we can construct the minimal graph S m (ϕ m ). It can be parameterized in B 2r ε -B r ε as the vertical graph of

Ūm (θ, r) = H r ε ,ϕ m (θ, r) + O C 2,α b (ε 2 ).
By the result of section 10, we can obtain a minimal surface M T k,ε (Ψ), with Ψ = (ψ t , ψ b , ψ m ), where ψ b (θ) = ψ t (θ -π k+1 ), which is close to a truncated and scaled genus k Costa-Hoffman-Meeks surface and has three boundaries. This surface is close to its upper and lower boundary, and is a vertical graph over the annulus B r ε -B r ε /2 , whose parametrization is, respectively, given by

U t (θ, r) = σ t,ε + ε 2 ln 2r ε 2 + H ψ t s ε -ln 2r ε 2 , θ + O C 2,α b (ε 2 ), U b (θ, r) = -U t θ - π k + 1 , r , (59)
where s ε =ln ε. Near the middle boundary, the surface is a vertical graph whose parametrization is

U m (θ, r) = Hρ ε ,ψ m θ, ε 2 r + O C 2,α b (ε 2 ).
We assume that the parameters and the boundary functions are chosen so that

|η t | + |η b | + ϕ t C 2,α (S 1 ) + ϕ m C 2,α (S 1 ) + ψ t C 2,α (S 1 ) + ψ m C 2,α (S 1 ) κε 2 ,
where η t = d tσ t,ε , and the constant κ > 0 is fixed large enough. The functions O C 2,α b (ε 2 ) replace the functions V t , V m , Vt , Vm that appear at the end of sections 5, 7 and 10. They depend nonlinearly on the different parameters and boundary data functions but they are bounded by a constant (independent of κ and ε) times ε 2 in the C 2,α b topology, where partial derivatives are taken with respect to the vector fields r∂ r and ∂ θ . It remains to show that, for all ε small enough, it is possible to choose the parameters and boundary functions in such a way that the surface

S t,d t (ϕ t ) ∪ S b,d b (ϕ b ) ∪ S m (ϕ m ) ∪ M T k,ε (Ψ)
is a C 1 surface across the boundaries of the different summands. Regularity theory will then ensure that this surface is in fact smooth and by construction it has the desired properties. We point out that, in this way, we will have constructed a one-parameter family of minimal surfaces: each one of them is determined by a different value of the parameter ε. Therefore this will complete the proof of the main theorem.

It is necessary to fulfill the following system of equations:

(60)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U t (•, r ε ) = Ūt (•, r ε ) U b (•, r ε ) = Ūb (•, r ε ) U m (•, r ε ) = Ūm (•, r ε ) ∂ r U b (•, r ε ) = ∂ r Ūb (•, r ε ) ∂ r U t (•, r ε ) = ∂ r Ūt (•, r ε ) ∂ r U m (•, r ε ) = ∂ r Ūm (•, r ε )
on S 1 . The first three equations together with the following identities,

H ψ t s ε -ln(2r)/ε 2 , θ | r=r ε = ψ t (θ), Hρ ε ,ψ m θ, ε 2 /r | r=r ε = ψ m (θ), H r ε ,ϕ t (θ, r) | r=r ε = ϕ t (θ), H r ε ,ϕ m (θ, r) | r=r ε = ϕ m (θ), lead to the system (61) η t + ϕ t -ψ t = O C 2,α b (ε 2 ) ϕ m -ψ m = O C 2,α b (ε 2 ),
where η t = d tσ t,ε . The relations (58) and (59) allowed us to omit one equation. The last three equations give the system (we applied Lemmas 12.4 and 12.5)

(62) ∂ * θ (ϕ t + ψ t ) = O C 1,α b (ε 2 ) ∂ * θ (ϕ m + ψ m ) = O C 1,α b (ε 2 ).
Here ∂ * θ denotes the operator which associates to φ = i 1 φ i cos(iθ) the function ∂ * θ φ = i 1 iφ i cos(iθ). The functions O C l,α (ε 2 ) in the above expansions depend nonlinearly on the different parameters and boundary data functions, but they are bounded by a constant (independent of κ and ε) times ε 2 in the C l,α topology. Projecting every equation of this system over the L 2 -orthogonal complement of Span{1}, we obtain the system (63)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ϕ t -ψ t = O C 2,α b (ε 2 ) ϕ m -ψ m = O C 2,α b (ε 2 ) ∂ * θ ϕ t + ∂ * θ ψ t = O C 1,α b (ε 2 ) ∂ * θ ϕ m + ∂ * θ ψ m = O C 1,α b (ε 2 ).
Lemma 11.1. The operator h, defined by C 2,α (S 1 ) → C 1,α (S 1 ) ϕ → ∂ * θ ϕ, acting on functions that are orthogonal to the constant function in the L 2 -sense and are even, is invertible.

Proof. We observe that if we decompose ϕ = j 1 ϕ j cos(jθ), then

h(ϕ) = j 1 ϕ j cos(jθ),
which is clearly invertible from H 1 (S 1 ) into L 2 (S 1 ). Now elliptic regularity theory implies that this is also the case when this operator is defined between Hölder spaces.

Using this result, the system (63) can be rewritten as

(64) (ϕ t , ϕ m , ψ t , ψ m ) = O C 2,α (ε 2 ).
Recall that the right hand side depends nonlinearly on ϕ t , ϕ b , ψ t , ψ m and also on the parameter η t . We look at this equation as a fixed point problem and fix κ large enough. Thanks to the estimates ( 12), ( 25), (56) with δ ∈ (1, 3/2), and (57), we can use a fixed point theorem for contraction mappings in the ball of radius κε 2 in (C 2,α (S 1 )) 4 to obtain, for all ε small enough, a solution (ϕ t , ϕ m , ψ t , ψ m ) of (64). Since the right hand side of (64) is continuous with respect to all data, we see that the fixed point (ϕ t , ϕ m , ψ t , ψ m ) depends continuously (and in fact smoothly) on the parameter η t . Inserting the found solution into (61) and (62), we see that it remains to solve an equation that can be rewritten as (65)

η t = O(ε 2 ),
where this time, the right hand side depends nonlinearly on η t . Now, provided κ has been fixed large enough, we can use the Schäuder fixed point theorem in the ball of radius κε 2 in R to solve (65), for all ε small enough. This provides a set of parameters and boundary data such that (61) and (62) hold. Equivalently we have proved the existence of a solution of system (60). So the proof of Theorem 1.1 is complete.

12. Appendix 12.1. Harmonic extension operators. The results contained in this section are about the existence of some harmonic extension operators. The first one gives the harmonic extension of a function defined on ∂B r 0 to D 2 \ B r 0 .

Proposition 12.1. There exists an operator

H r 0 : C 2,α (S 1 ) -→ C 2,α (S 1 × [r 0 , 1]),
such that for every even function ϕ(θ) ∈ C 2,α (S 1 ), the function

w ϕ = H r 0 ,ϕ solves ⎧ ⎨ ⎩ Δ 0 w ϕ = 0 on S 1 × [r 0 , 1] w ϕ = ϕ on S 1 × {r 0 } w ϕ = 0 on S 1 × {1}. Moreover, ( 66
) ||H r 0 ,ϕ || C 2,α (S 1 ×[r 0 ,1]) c ||ϕ|| C 2,α (S 1 ) ,
for some constant c > 0.

S 1 × [r 0 , 1] being a compact domain, the existence of the solution of the Dirichlet problem above is a classical result; for example, see Theorem 2.14 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

The uniqueness of the solution follows from the maximum principle. To get the wanted estimate of the norm of the solution, we observe that the maximum principle also implies that sup |w ϕ | c sup |ϕ| and that we can obtain similar estimates (involving the derivatives of ϕ) for the norm of the derivatives of w ϕ from their harmonicity. Now we give the statement of a result whose proof is contained in [START_REF] Fakhi | Existence result for minimal hypersurfaces with a prescribed finite number of planar ends[END_REF].

Proposition 12.2. There exists an operator

H : C 2,α (S 1 ) -→ C 2,α -2 ([0, +∞) × S 1
), such that for all ϕ ∈ C 2,α (S 1 ), even functions and orthogonal to e i , i = 0, 1, in the L 2 -sense, the function w = H ϕ solves

(∂ 2 s + ∂ 2 θ ) w = 0 in S 1 × [0, +∞) w = ϕ on S 1 × {0}. Moreover H ϕ C 2,α -2 (S 1 ×[0,+∞)) c ϕ C 2,α (S 1 ) ,
for some constant c > 0.

The following result gives a harmonic extension of a function on R 2 \ D ρ.

Proposition 12.3. There exists an operator

Hρ : C 2,α (S 1 ) -→ C 2,α (S 1 × [ρ, +∞)),
such that for each even function ϕ(θ) ∈ C 2,α (S 1 ), which is L 2 -orthogonal to the constant function, then w ϕ = Hρ,ϕ solves

Δw ϕ = 0 on S 1 × [ρ, +∞) w ϕ = ϕ on S 1 × {ρ}. Moreover, (67) || Hρ,ϕ || C 2,α (S 1 ×[ρ,+∞)) c ||ϕ|| C 2,α (S 1 ) ,
for some constant c > 0.

Proof. We consider the decomposition of the function ϕ with respect to the basis {cos(iθ)}, that is,

ϕ = ∞ i=1 ϕ i cos(iθ).
Then the solution w ϕ is given by

w ϕ (θ, ρ) = ∞ i=1 ρ ρ i ϕ i cos(iθ). Since ρ ρ 1, then ρ ρ i ρ ρ , we can conclude that |w(θ, ρ)| c|ϕ(θ)| and then ||w ϕ || C 2,α c||ϕ|| C 2,α .
Lemma 12.4. Let u(θ, r) be the harmonic extension defined on S 1 × [r 0 , +∞) of the even function ϕ = i 0 ϕ i cos(iθ) ∈ C 2,α (S 1 ) and such that u(θ, r 0 ) = ϕ(θ).

Then ∂ * θ ϕ(θ) = r 0 ∂ r u(θ, r) |r=r 0 . Proof. If ϕ(θ) = i 0 ϕ i cos(iθ), then the function u is given by u(θ, r) = i 0 ϕ i r r 0 i cos(iθ). Then ∂ r u(θ, r) = i 1 ϕ i r r 0 i i cos(iθ) r . Consequently ∂ * θ ϕ(θ) = r 0 ∂ r u(θ, r) |r=r 0 .
Lemma 12.5. Let u(θ, r) be the harmonic extension defined on S 1 × [0, r 0 ] of the even function ϕ ∈ C 2,α (S 1 ) and such that u(θ, r 0 ) = ϕ(θ). Then 12.2. The proof of Proposition 4.2. We start by giving the statement of a classical result about the injectivity of Δ 0 .

Lemma 12.6. Given 0 < r 0 < r 1 1, let w be a solution of Δ 0 w = 0 on S 1 × [r 0 , r 1 ] such that w(• , r 0 ) = w(• , r 1 ) = 0. Then w = 0.

As a consequence of Lemma 12.6, the operator Δ 0 is injective. Hence, the Fredholm alternative assures that there exists a unique w ∈ C 2,α (S 1 × [r 0 , 1]), with w(θ, r) satisfying:

(68)

Δ 0 w = f on S 1 × [r 0 , 1] w(•, r 0 ) = w(•, 1) = 0.

We want to prove the following assertion.

Assertion 12.7. For every 0 < r 0 < 1, f ∈ C 0,α (S 1 × [r 0 , 1]) and w ∈ C 2,α (S 1 × [r 0 , 1]) satisfying (68) there exists a constant c such that

w C 0,α (S 1 ×[r 0 ,1]) c f C 0,α (S 1 ×[r 0 ,1]) .
We suppose by contradiction that the assertion 12.7 is false; that is, there does not exist a universal constant for which the previous estimate holds. Then, for each n ∈ N, there exist r 0,n and f n , w n satisfying (68) (with r 0,n , f n , w n instead of r 0 , f, w) such that sup |w n | → +∞ as n → ∞.

Since S 1 × [r 0,n , 1] is a compact set, A n is achieved at a point (θ n , r n ) ∈ S 1 × [r 0,n , 1].

The sequence of sets I n = [ r 0,n r n , 1 r n ] converges (up to some subsequence) to a set that we denote by I ∞ . We will show that it is nonempty and contains 1. If r 0,n < r < r < 1, elliptic estimates allow us to conclude that sup

S 1 ×[r 0,n ,r ] |∇w n | c( sup S 1 ×[r 0,n ,r ] |f n | + sup S 1 ×[r 0,n ,r ] |w n |) c(1 + A n ),
where c is a constant independent of n. Then, if n → +∞, r 0,n r n → R 1 < 1 and 1 r n → R 2 > 1. The fact that R 1 < 1 follows from the above estimate for the gradient of w n near r = r 0,n . That implies that the supremum A n cannot be achieved at a point which is too close to r 0,n , that is, the point where w n vanishes. In other terms, the quotient r 0,n r n remains bounded away from 1. Using similar arguments it is possible to show that ∇w n is bounded near 1 and consequently 1 r n also remains bounded away from 1. Then we can conclude that I ∞ is not empty. We set

I ∞ = [R 1 , R 2 ], where 0 R 1 < 1 < R 2 < +∞ ∈ R.
We define From the definition of w n , we obtain that

∇ w n = 1 A n ∇w n (θ, rr n ).
Then

|∇ w n | c 1 + A n A n < 2c.
Since the sequences ( wn ) n and (∇ wn ) n are uniformly bounded, the Ascoli-Arzelà theorem assures that a subsequence of ( wn ) n converges on compact sets of S 1 × I ∞ to a nonzero function w ∞ that vanishes on S 1 × ∂I ∞ . The function w ∞ inherits the properties of wn . In particular, (69) sup

S 1 ×I ∞ |w ∞ | = 1.
In the same way it's possible to prove that a subsequence of ( fn ) n converges on compact sets of S 1 × I ∞ to the function f ∞ ≡ 0 since, if n → ∞, sup

S 1 ×I n | f n | → 0.
Then the limit function w ∞ must satisfy the differential equation Δ 0 w ∞ = 0 on S 1 ×I ∞ with null boundary conditions on ∂I ∞ . So we can conclude that ∀r ∈ I ∞ , w ∞ (θ, r) = 0. This function does not satisfy (69), a contradiction. This proves the assertion 12.7.

The elliptic estimate, |∇w| c sup

S 1 ×[r 0 ,1] |f | + sup S 1 ×[r 0 ,1]
|w| , allows us to get a uniform estimate of ∇w. This proves the existence of a solution of Δ 0 w = f defined on S 1 × [r 0 , 1] for which

w C 0,α (S 1 ×[r 0 ,1]) c f C 0,α (S 1 ×[r 0 ,1]) .
Now it is sufficient again to use elliptic estimates to obtain the estimates for the derivatives.

12.3. Minimal graphs in (D 2 × R, g hyp ). In this section, following [START_REF] Nelli | Minimal surfaces in H 2 × R[END_REF], we will find the condition to be satisfied such that the graph Σ of a function defined on D 2 is minimal with respect to the metric g hyp = dx 2 1 +dx 2 2 F + dx 2 3 , where F = (1-x 2 1x 2 2 ) 2 . We assume that the immersion of Σ in D 2 × R is given by (x 1 , x 2 ) → (x 1 , x 2 , u(x 1 , x 2 )).

The Christoffel symbols, Γ k ij , associated to g hyp all vanish except

Γ 1 11 = Γ 2 21 = Γ 2 12 = -Γ 1 22 = 2x 1 √ F , Γ 1 12 = Γ 2 22 = Γ 1 21 = -Γ 2 11 = 2x 2 √ F .
Let e 1 , e 2 , e 3 be the canonical basis of R 3 . Then ε 1 = √ F e 1 , ε 2 = √ F e 2 , ε 3 = e 3 is an orthonormal basis for M 2 × R. The coordinate vector fields on Σ are

X 1 = ε 1 √ F +u x 1 ε 3 , X 2 = ε 2 √ F +u x 2 ε 3 , N = 1 W -u x 1 √ F ε 1 -u x 2 √ F ε 2 + ε 3 , with W = 1 + F |∇u| 2 .
The induced metric on Σ is defined by

g 11 = 1 F + (u x 1 ) 2 , g 22 = 1 F + (u x 2 ) 2 , g 12 = u x 1 x 2 .
If ∇ denotes the Riemannian connection of the metric g hyp , then the coefficients of the second fundamental form are

b 11 = ∇ X 1 X 1 , N = 1 W - 2x 1 √ F u x 1 + 2x 2 √ F u x 2 + u x 1 x 1 ε 3 , b 22 = ∇ X 2 X 2 , N = 1 W 2x 1 √ F u x 1 - 2x 2 √ F u x 2 + u x 2 x 2 ε 3 , b 12 = ∇ X 1 X 2 , N = 1 W - 2x 2 √ F u x 1 - 2x 1 √ F u x 2 + u x 1 x 2 ε 3 ,
where we used the following identities:

∇ X 1 X 1 = 2x 1 ε 1 -2x 2 ε 2 + u x 1 x 1 ε 3 , ∇ X 2 X 2 = -2x 1 ε 1 + 2x 2 ε 2 + u x 2 x 2 ε 3 , ∇ X 1 X 2 = 2x 2 ε 1 + 2x 1 ε 2 + u x 1 x 2 ε 3 .
The mean curvature of Σ with respect to g hyp is given by .

H(Σ) = 1
Using the expressions of the coefficients of the first and second fundamental form, we find that

H(Σ) = F 2 div ∇u 1 + F |∇u| 2 .
12.4. Minimal surfaces of rotation in (D 2 × R, g hyp ). In this section, following [START_REF] Nelli | Minimal surfaces in H 2 × R[END_REF], we will find the condition to be satisfied such that a surface of revolution Σ in D 2 × R is minimal with respect to the metric g hyp = dx 2 1 +dx 2 2 F + dx 2 3 , where F = (1x 2 1x 2 2 ) 2 . We will assume that the immersion of Σ in D 2 × R is given, in terms of the cylindrical coordinates (r, θ, z), by: (θ, z) → (r(z), θ, z), where r(z) is a function of z.

It is convenient to express the metric g hyp in terms of the new coordinates. We find

g hyp = dr 2 + r 2 dθ 2 F + dz 2 ,
with F = (1r 2 ) 2 . The Christoffel symbols, Γ k ij , associated to g hyp all vanish except

Γ 1 11 = 2r 1 -r 2 , Γ 1 22 = - r(1 + r 2 ) 1 -r 2 , Γ 2 12 = Γ 2 21 = 1 + r 2 r(1 -r 2 )
.

SK

  ext dA = S II I dA, where I, II denote the determinants of the first and of the second fundamental form. It follows that II = b 11 b 22b 2 12 , I = g 11 g 22g 2 12 , dA = √ I.

  a nonlinear second order differential operator and Lε is a linear operator which takes into account the change of the normal vector field (only for the top and bottom ends) n ε into ñε and of the change of the parametrization.

  ∂ * θ ϕ(θ) = -r 0 ∂ r u(θ, r) |r=r 0 . Proof. If ϕ(θ) = i 0 ϕ i cos(iθ), then the function u is given by u(θ, r) = Consequently ∂ θ ϕ * (θ) = -r 0 ∂ r u(θ, r) |r=r 0 .

S 1 ×

 1 [r 0,n ,1] |f n | = 1 and A n := sup S 1 ×[r 0,n ,1]

  w n (θ, r) := 1 A n w n (θ, rr n ) and f n (θ, r) := 1 A n f n (θ, rr n ),for all (θ, r) ∈ S 1 × I n , with I n = [r 0,n /r n , 1/r n ]. These functions satisfy r 2 n Δ w n = f n .

  Proposition 8.1. Let S be a surface in D 2 × R endowed with the metric g hyp . If H hyp (•) denotes the mean curvature with respect to the metric g hyp , and if H e (•) and (N 1 , N 2 , N 3 ) denote respectively the mean curvature and the normal vector to S with respect to g 0 , then

	(31)

  2 b 11 g 22 + b 22 g 11 -2b 12 g 12 g 11 g 22g 2 12

) 2 equals -4. We observe that H 2 = M 2 (-1). Once having constructed this surface, it is easy to obtain by a diffeomorphism the wanted minimal surface in H 2 × R.

Let e 1 , e 2 , e 3 be the canonical basis of R 3 . Then the coordinate vector fields on Σ are

The induced metric on Σ is defined by

If ∇ denotes the Riemannian connection of the metric g hyp , then the coefficients of the second fundamental form are

where we used the following identities:

The mean curvature of Σ with respect to g hyp is given by .

Using the expressions of the coefficients of the first and second fundamental form, we find that H(Σ) = 0 if the function r(z) satisfies the following differential equation: r(z)r (z) -(r (z)) 2 -(1r(z) 4 ) = 0.