A COSTA-HOFFMAN-MEEKS TYPE SURFACE IN $\mathbb{H}^2 \times \mathbb{R}$

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A COSTA-HOFFMAN-MEEKS TYPE SURFACE IN H ² × R

Filippo Morabito

To cite this version:

Filippo Morabito. A COSTA-HOFFMAN-MEEKS TYPE SURFACE IN H

²

× R. Transactions of

the American Mathematical Society, American Mathematical Society, 2011, 363 (1), pp.1-36. �hal-

00693003�

A COSTA-HOFFMAN-MEEKS TYPE SURFACE IN H

²

× R

FILIPPO MORABITO

Abstract. We show the existence in the spaceH²×Rof a family of embedded minimal surfaces of genus 1k <+∞and finite total extrinsic curvature with two catenoidal type ends and one middle planar end. The proof is based on a gluing procedure.

1. Introduction

During recent years the study of the minimal surfaces in the product spaces M × R with M = H

²

, S

²

has become more and more active. The development of the theory of the minimal surfaces in these spaces started with [20] by H. Rosenberg and continued with [14] and [15] by W. H. Meeks and H. Rosenberg. In [17] 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 [5] L. Hauswirth constructed and classified the minimal surfaces foliated by horizontal constant cur- vature 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 [21].

C. Costa in [1, 2] and D. Hoffman and W. H. Meeks in [7], [8] and [9] 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 enjoy- ing the same properties mentioned in the statement of theorem in the Riemannian manifold (D

²

× R, g

hyp

) where g

hyp

=

₍₁^dx_−x²¹2^+dx22

1−x²₂)²

+ dx

²₃

. It is usually denoted by M

²

(−4) × R, to point out that the sectional curvature of D

²

× {0} endowed with the metric

₍₁^dx_−x²¹2^+dx22

1−x²₂)²

equals − 4. We observe that H

²

= M

²

( − 1). Once having con- structed this surface, it is easy to obtain by a diffeomorphism the wanted minimal surface in H

²

× R.

1

The main result is proved by a gluing procedure (see for example [6]) 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.

2. 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

_H²_×R

= 4(dx

²₁

+ dx

²₂

)

(1 − x

²₁

− x

²₂

)

²

+ dx

²₃

.

As mentioned in the Introduction, one of the surfaces involved in the gluing pro- cedure 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

²₁

+ dx

²₂

(1 − x

²₁

− x

²₂

)

²

+ dx

²3

,

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 ¯ g be the metric defined on D

²

× R by

¯

g = 4g

hyp

= 4(dx

²₁

+ dx

²₂

)

(1 − x

²₁

− x

²₂

)

²

+ 4dx

²₃

.

We consider the map f : (D

²

× R, g

_H²_×R

) → (D

²

× R, g) 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 ¯ g by f equals g

_H²_×R

. So if Σ is a minimal surface in (D

²

× R, g), ¯ 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 ¯ g 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

2

mean curvature of Σ in (D

²

× R, ¯ g) multiplied by 4. So if a surface is minimal in M

²

× R, it is also minimal with respect to the metric ¯ g.

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

.

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

(2) 2H

u

= F div

∇u 1 + F |∇ u |

²

, where F =

1 − x

²₁

− x

²₂

2

=

1 − r

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 := Fdiv

∇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 3

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) +

¹₂

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,

(7)

S

K

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

22

− b

²₁₂

, I = g

11

g

22

− g

²₁₂

, dA = √

I. 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 (7), 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

.

4. 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

= FΔ

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=r₀

= 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.

4

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

loc^,α

(S

¹

× I) for which the norm w

_C^,α(S¹×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

_k+1^2π

about the vertical x

3

-axis, with respect to the composition of a rotation of angle

_k+1^π

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

r0

: C

^0,α

(S

¹

× [r

0

, 1]) −→ C

^2,α

(S

¹

× [r

0

, 1])

f −→ w := G

r0

(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(S¹×[r₀,1])

c ||f ||

_C^0,α(S¹×[r₀,1])

, 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.

5. 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 Sch¨ auder 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 (6). 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

(9) 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.

5

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,

(10) 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

rε

is defined in Proposition 4.2. To prove the existence of a fixed point for (10) 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 (9) and enjoying the prop- erties given above. There exist some constants c

κ

> 0 and ε

κ

> 0 such that (11) S(ϕ, 0)

_C^2,α(S¹×[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(S¹×[rε,1])

1

2 v

2

− v

1

_C^2,α(S¹×[rε,1])

, S(ϕ

2

, v) − S(ϕ

1

, v)

_C^2,α(S¹×[rε,1])

cε

²

ϕ

2

− ϕ

1

_C^2,α(S¹)

,

where c is a positive constant, for all v

1

, v

2

∈ C

^2,α

(S

¹

× [r

ε

, 1]) such that v

i

_C^2,α

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

cf

_C^0,α

. Then S(ϕ, 0)

_C^2,α

= G

r_ε

(Q

0

(w

ϕ

))

_C^2,α

cQ

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

cϕ

_C^2,α

c

κ

ε

²

. Then

Q

0

(w

ϕ

)

_C^0,α

cw

ϕ

²_C^2,α

cϕ

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

cQ

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

)||

_C0,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

ϕ2

+ v) − Q

0

(w

ϕ1

+ v)

_C^0,α

c||w

ϕ₂

− w

ϕ₁

||

_C^2,α

v

_C^2,α

cε

²

||ϕ

2

− ϕ

1

||

_C^2,α

.

6

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 nonlin- ear 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 Sch¨ auder 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

rε

⊂ 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

U ¯

m

(θ, r) = H

r_ε,ϕ

(θ, r) + ¯ V

m

(θ, r), with || V ¯

m

||

_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 ¯ U

m

(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 ¯ V

m

depends nonlinearly on ε, ϕ. Furthermore, as it is easy to prove thanks to the third estimate of Lemma 5.1, it satisfies

(12) V ¯

m

(ε, ϕ)(·, r

ε

·) − V ¯

m

(ε, ϕ

)(·, r

ε

·)

_C^2,α( ¯B2−B1)

cεϕ − ϕ

_C^2,α(S¹)

.

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 [17]). 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.

7

Since we assume C =

_ε¹4

, 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

^εt2

) and ∂

t

r(t) = sinh t

ε

²

+ O(ε

⁴

e

^εt2

) 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 (15), 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

^ε2t2

) and then v(t) = √

Ct + O (ε

⁶

e

^ε2t2

). We remark a posteriori that

_ε^t2

v(t)

_ε^t2

+ 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

^εt2

).

If t ∈ [t

ε

− ε

²

ln 2, t

ε

+ ε

²

ln 2], then we easily obtain r(t) ∈ [

¹₄

ε + c

1

ε

³

, ε + c

2

ε

³

], for some positive constants c

1

, c

2

. Using ∂

t

r(t) = sinh

_t

ε²

+ O(ε

⁴

e

^εt2

), we find

∂

t

r(t) ∈ [

_4ε¹

− 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

.

8

Lemma 6.2. For ε > 0 small enough and t ∈ [t

ε

− ε

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

rε/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 (16). 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

³²

∂

r

u ∂

r

(F ∂

r

u) ∂

r

v − F ∂

r

u ∂

r

1

A

³²

∂

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 =

_ε¹4

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

∂

_rr²

u = ∓ (Cr − 2r

³

)

(Cr

²

− 1 − r

⁴

)

³

= O(1).

Taking into account these estimates, we can conclude that L ¯

u

v := √

A

∂

r

√ 1 A

∂

r

v − 1

A

³²

∂

r

u ∂

r

(F ∂

r

u) ∂

r

v − F ∂

r

u ∂

r

1

A

³²

∂

r

u ∂

r

v

= l

1

∂

r

v + l

2

∂

_rr²

v, (20)

where l

1

, l

2

= O (ε). Then we can write √

AL

u

v = F

Δ

0

v + ¯ L

u

v

.

9

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 + ¯ L

u

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

rε,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

L ¯

u

(w

ϕ

+ w) + √

AQ

u

(w

ϕ

+ w)

,

and where the operator G

rε

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(S¹×[rε,1])

c

κ

ε

³

and, for all ε ∈ (0, ε

k

),

S(ϕ, w

2

) − S(ϕ, w

1

)

_C^2,α(S¹×[rε,1])

1

2 w

2

− w

1

_C^2,α(S¹×[rε,1])

, S(ϕ

2

, w) − S(ϕ

1

, w)

_C^2,α(S¹×[r_ε,1])

cε ϕ

2

− ϕ

1

_C^2,α(S¹)

,

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(S¹×[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

cf

_C^0,α

. Then S(ϕ, 0)

_C^2,α

c L ¯

u

w

ϕ

+ √

AQ

u

(w

ϕ

)

_C^0,α

c

L ¯

u

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(S¹)

c

κ

ε

²

.

10

We use (20) for finding the estimate of ¯ L

u

w

ϕ

. We obtain L ¯

u

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

cw

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

rε

L ¯

u

(w

ϕ

+ w

2

) + √

AQ

u

(w

ϕ

+ w

2

) + G

rε

L ¯

u

(w

ϕ

+ w

1

) + √

AQ

u

(w

ϕ

+ w

1

)

. Consequently

S(ϕ, w

2

) − S(ϕ, w

1

)

_C^2,α

c L ¯

u

(w

ϕ

+ w

2

) − L ¯

u

(w

ϕ

+ w

1

) + √

AQ

u

(w

ϕ

+ w

2

) − √

AQ

u

(w

ϕ

+ w

1

)

_C^0,α

= c L ¯

u

(w

2

− w

1

) + √

A (Q

u

(w

ϕ

+ w

2

) − Q

u

(w

ϕ

+ w

1

))

_C^0,α

c

L ¯

u

(w

2

− w

1

)

_C^0,α

+ ||Q

u

(w

ϕ

+ w

2

) − Q

u

(w

ϕ

+ w

1

)||

_C0,α

. We observe that from the considerations above it follows that

L ¯

u

(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

)||

_C0,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

L ¯

u

(w

ϕ₂

− w

ϕ₁

)

_C^0,α

+ ||Q

u

(w

ϕ₂

+ w) − Q

u

(w

ϕ₁

+ w)||

_C0,α

cε ϕ

2

− ϕ

1

_C^2,α(S¹)

+ w

_C^2,α

ϕ

2

− ϕ

1

_C^2,α(S¹)

cεϕ

2

− ϕ

1

_C^2,α(S¹)

.

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 nonlin- ear 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 Sch¨ auder fixed point theorem, S(ϕ, ·) has a

unique fixed point v in this ball.

11

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

U ¯

t

(θ, r) = ε

²

ln 2r

ε

²

+ H

r_ε,ϕ

(θ, r) + ¯ V

t

(θ, r),

with V ¯

t

_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 ¯ V

t

depends nonlinearly on ε, ϕ. Furthermore it satisfies (25) V ¯

t

(ε, ϕ)(·, r

ε

·) − V ¯

t

(ε, ϕ

)(·, r

ε

·)

_C^2,α( ¯B₂−B₁)

cεϕ − ϕ

_C^2,α(S¹)

. 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

2rε

− B

rε

. 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 U ¯

b

(θ, r) = − U ¯

t

θ − π

k + 1 , r

.

8. 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

²₁

+ dx

²₂

F + dx

²₃

, where F =

1 − x

²₁

− x

²₂

2

=

1 − r

²

2

and

g

0

= dx

²₁

+ dx

²₂

+ dx

²₃

.

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 Rie- mannian 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

,

12

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

¹₁₁

= Γ

²₂₁

= Γ

²₁₂

= −Γ

¹22

= 2x

1

√ F , Γ

¹12

= Γ

²22

= Γ

¹21

= −Γ

²11

= 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

.

13