FREE BOUNDARY MINIMAL SURFACES IN THE UNIT 3-BALL

HAL Id: hal-01119962

https://hal.archives-ouvertes.fr/hal-01119962

Preprint submitted on 24 Feb 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

FREE BOUNDARY MINIMAL SURFACES IN THE UNIT 3-BALL

Abigail Folha, Frank Pacard, Tatiana Zolotareva

To cite this version:

Abigail Folha, Frank Pacard, Tatiana Zolotareva. FREE BOUNDARY MINIMAL SURFACES IN THE UNIT 3-BALL. 2015. �hal-01119962�

ABIGAIL FOLHA, FRANK PACARD, AND TATIANA ZOLOTAREVA

Abstract. In a recent paper A. Fraser and R. Schoen have proved the ex- istence of free boundary minimal surfaces Σⁿ inB³ which have genus 0 and nboundary components, for alln≥3. For largen, we give an independent construction of Σⁿand prove the existence of free boundary minimal surfaces Σ˜ⁿ inB³ which have genus 1 andnboundary components. As ntends to infinity, the sequence Σⁿ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts ofB³ while the sequence ˜Σⁿ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts ofB³− {0}.

1. Introduction and statement of the result.

In this paper, we are interested in minimal surfaces which are embedded in the Euclidean 3-dimensional unit open ball B³ and which meet S², the boundary of B³, orthogonally. Following [2], we refer to such minimal surfaces asfree boundary minimal surfaces.

Obviously, the horizontal unit disk, which is the intersection of the horizontal plane passing through the origin with the unit 3-ball, is an example of such free boundary minimal surface. Moreover, it is the only free boundary solution of topo- logical disk type, [7]. Lets∗>0 be the solution of

s∗tanhs∗= 1.

The so calledcritical catenoidparameterized by (s, θ)7→ 1

s∗coshs∗

(coshscosθ,coshssinθ, s),

is another example of such afree boundary minimal surface. A. Fraser and M. Li conjectured that it is the only free boundary minimal surface of topological annulus type [1].

Free boundary minimal surfaces arise as critical points of the area among surfaces embedded in the unit 3-ball whose boundaries lie onS²but are free to vary onS². The fact that the area is critical for variations of the boundary of the surface which are tangent toS² translates into the fact that the minimal surface meetsS² orthogonally.

In a recent paper [3], A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces Σn in B³ which have genus 0 and n boundary components, for all n ≥ 3. For large n, these surfaces can be understood as the connected sum of two nearby parallel horizontal disks joined bynboundary bridges

F. Pacard and T. Zolotareva are partially supported by the ANR-2011-IS01-002 grant.

T. Zolotareva is partially supported by the FMJH through the ANR-10-CAMP-0151-02 grant.

1

which are close to scaled down copies of half catenoids obtained by diving a catenoid which vertical axis with a plane containing it, which are arranged periodically along the unit horizontal great circle ofS². Furthermore, asntends to infinity, these free boundary minimal surfaces converge on compact subsets of B³ to the horizontal unit disk taken with multiplicity two.

We give here another independent construction of Σn, for nlarge enough. Our proof is very different from the proof of A. Fraser and R. Schoen and is more in the spirit of the proof of the existence of minimal surfaces inS³by doubling the Clifford torus by N. Kapouleas and S.-D. Yang [4]. We also prove the existence of free boundary minimal surfaces inB³which have genus 1 andnboundary components, for allnlarge enough.

To state our result precisely, we definePnto be the regular polygon withn-sides, which is included in the horizontal planeR²× {0}and whose vertices are given by

cos

2πj n

,sin

2πj n

,0

∈R³, for j= 1, . . . , n.

We defineSn⊂O(3) to be the subgroup of isometries ofR³which is generated by the orthogonal symmetry with respect to the horizontal planex3= 0, the symmetry with respect to coordinate axisOx1and the rotations around the vertical axisOx3

which leavePn globally invariant.

Our main result reads :

Theorem 1.1. There existsn0≥0such that, for eachn≥n0, there exists a genus 0 free boundary minimal surface Σn and a genus 1 free boundary minimal surface Σ˜n which are both embedded inB³and meetS²orthogonally alongnclosed curves.

Both surfaces are invariant under the action of the elements of S_n and, as n tends to infinity, the sequenceΣn converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of B³ while the sequence Σ˜n converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of B³− {0}.

Even though we do not have a proof of this fact, it is very likely that (up to the action of an isometry ofR³), the surfaces Σncoincide with the surfaces already constructed by R. Schoen and A. Fraser. In contrast, the existence of ˜Σnis new and does not follow from the results in [3]. The parameterization of the free boundary minimal surfaces we construct is not explicit, nevertheless our construction being based on small perturbations of explicitly designed surfaces, it has the advantage to give a rather precise description of the surfaces Σn and ˜Σn. Naturally, the main drawback is that the existence of the free boundary minimal surfaces is only guaranteed whenn, the number of boundary curves, is large enough.

2. Plan of the paper.

In section 3, we study the mean curvature of surfaces embedded inB³which are graphs over the horizontal diskD²×{0}. In section 4 we analyse harmonic functions which are defined on the unit punctured disk in the Euclidean 2-plane and have log type singularities at the punctures. In section 5 for everyn∈Nlarge enough we construct a family of genus 0 surfaces Sn and a family of genus 1 surfaces ˜Sn

embedded inB³which are approximate solutions to the minimal surface equation,

meet the unit sphereS²orthogonally and havenboundary components. In section 6 we consider all embedded surfaces inB³ which are close toSn and ˜Sn and meet the sphereS² orthogonally. In section 7 we analyse the linearised mean curvature operator about Sn and ˜Sn. Finally, in the last section we explain the Fixed-Point Theorem argument that allows us fornlarge enough to deformSn and ˜Sn into free boundary minimal surfaces Σn and ˜Σn satisfying the theorem (1.1).

3. The mean curvature operator for graphs in the unit3-ball We are interested in surfaces embedded inB³which are graphs over the horizon- tal diskD²×{0}. To define these precisely, we introduce the following parametriza- tion of the unit ball

X(ψ, φ, x3) := 1

coshx3+ cosψ sinψ e^iφ,sinhx3

,

whereψ∈(0, π/2), φ∈S¹ andx3∈R. The horizontal diskD²× {0}corresponds tox3= 0 in this parametrization and the unit sphereS²corresponds to ψ=π/2.

Also, the leafx3=x⁰₃is a constant mean curvature surface (in fact it is a spherical cap) with mean curvature given by

H= 2 sinhx⁰₃,

(we agree that the mean curvature is thesum of the principal curvatures, not the average) moreover, this leaf meetsS²orthogonally.

In these coordinates, the expression of the Euclidean metric is given by X^∗geucl= 1

(coshx3+ cosψ)² dψ²+ (sinψ)²dφ²+dx²₃ . We consider the coordinate

z= sinψ 1 + cosψe^iφ,

which belongs to the unit diskD²⊂C. We then defineX by the identity X(z, x3) =X(ψ, φ, x3),

wherez and (ψ, φ) are related as above. Then

X(z, x3) =A(z, x3)(z, B(z) sinhx3), where the functionsB and Aand explicitly given by

B(z) =1

2 1 +|z|²

, A(z, x3) := 1

1 +B(z)(coshx3−1).

In the coordinates z ∈ D² and x3 ∈ R the expression of the Euclidean metric is given by

X^∗geucl=A²(z, x3) dz²+B²(z)dx²₃ .

In the next result, we compute the expression of the mean curvature of the graph of a functionz7→u(z) inB³, and by such a graph we mean a surface parametrized by

z∈D² 7→ X(z, u(z))∈B³. We have the:

Lemma 3.1. The mean curvature with respect to the metric X^∗geucl of the graph of the functionu, namely the surface parametrized by(z, u(z)), is given by

H(u) = 1

A³(u)B div A²(u)B²∇u p1 +B²|∇u|²

! + 2p

1 +B²|∇u|²sinhu,

where by definition A(u) =A(·, u). In this expression, the metric used to compute the gradient of u, the divergence and the norm of ∇u is the Euclidean metric on D².

Proof. The area form of the surface parametrized by z =x1+i x2 7→ (z, u(z)) is given by

da:=A²(u)p

1 +B²|∇u|²dx1dx2, and hence the area functional is given by

Area(u) :=

Z Z

D²

A²(u)p

1 +B²|∇u|²dx1dx2. The differential of the area functional atuis given by

DArea|_u(v) = Z Z

D²

−A²(u)B²∇u· ∇v

p1 +B²|∇u|² + 2A(u)∂x3A(u)p

1 +B²|∇u|²v

!

dx1dx2. But

∂x3A=−A²B sinhx3, and hence we conclude that

DArea|_u(v) =

− Z Z

D²

div A²(u)B²∇u p1 +B²|∇u|²

!

+ 2A³(u)Bp

1 +B²|∇u|²sinhu

!

v dx1dx2. To conclude, observe that the unit normal vector to the surface parametrized by z7→ X(z, u(z)) is given by

N:= 1 A(u)

1 p1 +B²|∇u|²

−B∇u+ 1 B ∂x3

, and hence

geucl(N, ∂x3) = A(u)B p1 +B²|∇u|², so that

geucl(N, ∂x3)da=A³(u)B,

and the result follows from the first variation of the area formula DArea|u(v) =−

Z Z

D²

H(u)geucl(N, ∂x3)v da.

This completes the proof of the result.

Using the above Lemma, we obtain the expression of the linearised mean curva- ture operator aboutu= 0. It reads

(3.1) Lgrv= ∆(Bv) = ∆

1 +|z|²

2 v

, where ∆ is the (flat) Laplacian onD².

Lemma 3.2. Take a change of variables in D² :z=r e^iφ, r∈(0,1), φ∈S¹ and a functionu∈ C¹(D²), such that ^∂u_∂r

r=1= 0. Then the graph ofuinB³meets the sphereS² orthogonally at the boundary.

Proof. The surface parametrized by (r, φ) 7→ X(r e^iφ, u(r, φ)) is embedded inB³ and meets ∂B³ atr= 1. The result follows from the fact that the tangent vector

Tr(1) =∂rX(r e^iφ, u(r, φ))

r=1= 1

cosh²u(1, φ) e^iφ,sinhu(1, φ) , is collinear to the normal vector

NS= 1

coshu(1, φ) e^iφ,sinhu(1, φ) ,

to the sphereS²at the pointX(e^iφ, u(1, φ)).

4. Harmonic functions with singularities defined on the unit disk Take some number n ∈ N. Our goal is to construct a graph in B³ which has bounded mean curvature, is invariant under the transformation z 7→ z¯ and the rotations by ^2π_n and is close to a half-catenoid in small neighbourhoods of then-th roots of unity

zm=e^2πimn ∈∂D², m= 1, . . . n,

(and, in the case of the second construction, to a catenoid in a small neighbourhood ofz= 0).

The parametrisation of a standard catenoidC in R³ is X^cat(s, θ) = coshs e^iφ, s

, (s, φ)∈R×S¹. It may be divided into two piecesC^±, which can be parametrized by

z∈C\D² 7→ z,±log|z| ∓log 2 +O(|z|⁻²)

, as |z| → ∞.

We would like to find a function Γn, which satisfies (4.2)

( LgrΓn = 0 in D² (D²\ {0})

∂rΓn= 0 on ∂D²\ {z1, . . . , zn} ,

and which has logarithmic singularities at z = zm (and z = 0). Notice that the operator Lgr in the unit disk with Neumann boundary data has a kernel which consists of the coordinate functionsx1, x2. This corresponds to tilting the unit disk D²× {0}in B³. The kernel can be eliminated by asking Γn to be invariant under the action of a group of rotations around the vertical axis.

Notice also that the constant functions are not in the kernel ofLgr: by moving the disk in the vertical direction in the cylinder D²×Rwe do not get a minimal but a constant mean curvature surface inB³.

Take a functionGn, such thatGn(zⁿ) =B(z) Γn(z). Then the problem (4.2) is equivalent to

(4.3)

( ∆Gn= 0 in D² (D²\ {0})

∂rGn−¹_nGn= 0 on∂D²\ {1}.

We constructGn explicitly. For all integern≥2, we put

(4.4) Gn(z) :=−n

2 + Re





∞

X

j=1

n z^j nj−1



. Writing

1 nj−1 =

∞

X

k=0

1 (nj)^k+1, we see that also have the expression

(4.5) Gn(z) :=−n

2 + Re

∞

X

k=0

Hk(z) n^k

! , where, for allk∈N, the functionHk is given by

(4.6) Hk(z) :=

∞

X

j=1

z^j j^k+1. Observe, and this will be useful, that

(4.7) H0(z) =−ln(1−z).

Obviously, Gn is harmonic in the open unit disk. Making use of (4.6), we see that, for allk≥1,

∂r(ReHk) = ReHk−1, on∂D², while it follows from (4.7) that

∂r(ReH0) =1 2,

again on∂D²− {1}. Therefore, we conclude from (4.5) that n ∂rGn−Gn= 0,

on∂D².

For all integern≥1, we define inD²− {0}, the function ˜Gn by

(4.8) G˜n(z) :=−n−log|z|.

Again ˜Gn is harmonic in D²− {0} and we also have n ∂rG˜n−G˜n= 0, on∂D².

To complete this paragraph, we define Γn(z) := 1

B(z)Gn(zⁿ) and Γ˜n(z) := 1

B(z)G˜n(zⁿ).

By construction,LgrΓn= 0 inD²and∂rΓn= 0 on∂D², away from then-th roots of unity ; whileLgrΓ˜n= 0 inD²− {0}and∂rΓ˜n= 0 on∂D².

4.1. Matching Green’s function. Take two parameters 0< ε <1 and 0<ε <˜ 1 and consider a catenoidCε˜inR³, parametrized by

X_ε˜^cat: (s, φ)∈R×S¹ 7→ ε˜coshs e^iφ,εs˜ , andnhalf-catenoidsCε,m,m= 1, . . . , n

X_ε,m^cat : (s, θ)∈R× π

2,3π 2

7→ εcoshs e^iθ+zm, εs ,

centered at then-th roots of unityzm. In a neighbourhood of z= 0 (z=zm), we can take a change of variables

z= ˜εcoshs e^iφ, (z=zm+εcoshs e^iθ), φ∈S¹, (θ∈[−θε+ 2πm/n, θε+ 2πm/n]) s∈[−sε˜,0] and s∈[0, sε˜], (s∈[−sε,0] and s∈[0, sε]), for certain parameterssε˜,sε∈(0,+∞) andθε∈(0, π/2). We can parametrize the lower and the upper parts ofC˜εandCε,mas graphs

z 7→ z,±G^cat_ε˜

, z 7→ z,±G^cat_ε,m , where in some small neighbourhoods ofz= 0 (z=zm),

G^cat_˜ε (z) = ˜εlogε˜

2 −˜εlog|z|+O ε˜³/|z|² , G^cat_ε,m(z) =εlogε

2−εlog|z−zm|+O ε³/|z−zm|² Our goal is to find positive parametersτ and ˜τ and a connection betweenεand ˜ε, such that the function

z 7→ τ Gn(zⁿ) + ˜τG˜n(zⁿ),

would be close toG^cat_˜ε in a neighbourhood ofz= 0 and toG^cat_ε,min a neighbourhood ofz=zm. We denote

fn(z) :=

∞

X

k=0

Hk(zⁿ) n^k =

∞

X

k=0

1 n^k

∞

X

j=1

z^nj j^k+1,

and remind that Gn(zⁿ) = −ⁿ₂ + Refn(z). It is easy to verify the function fn(z) satisfies

∂fn

∂z (z) =−d

dzlog(1−zⁿ) +1 zfn(z), which yields

d dz

fn

z

= nzⁿ⁻² zⁿ−1.

We can write nzⁿ⁻²

zⁿ−1− 1

zm(z−zm) =

nzⁿ⁻²zm−

n−1

P

k=0

z^n−1−kz_m^k zm(zⁿ−1) =

n−1

P

k=0

−z^n−1−kz_m^k +zmzⁿ⁻² zm(zⁿ−1) =

zⁿ⁻²(zm−z) +zm n−1

P

k=2

z^n−1−k z^k−1−z_m^k−1 zm(z−zm)

n−1

P

k=0

z^n−1−kz^k_m

=

−zⁿ⁻²+zm n−1

P

k=2

z^{n−1−kk−2}P

l=0

z^k−2−lz^l_m zm

n−1

P

k=0

z^n−1−kz_m^k

:=hn(z).

The functionhn(z) is continuous in a small neighbourhood ofz=zmand

|hn(zm)| ≤c n,

for a constantcwhich does not depend onn. So, we have d

dz fn

z

+ 1

zm(z−zm)=hn(z), which yields that in a neighbourhood ofz=zm

fn(z) z + 1

zm

log(z−zm) = 1 zm

z→zlimm

(fn(z) + log(z−zm)) + Z zm

z

hn(z)dz, where the integral is taken along the segment of the straight line passing from z tozm and by log we mean the principal value of complex logarithm defined in the unit disc deprived of a segment of a straight line which doesn’t pass thought any of then-th roots of unity. We have

∞

X

k=1

1

n^kHk(zⁿ_m) =

∞

X

k=1

∞

X

j=1

1

n^kj^k+1 ≤ π² 6n. Moreover,

Re lim

z→zm

(−log(1−zⁿ) + log(z−zm)) =−log|n z_mⁿ⁻¹|=−logn.

So, in the neighbourhood ofz=zm, we have Gn(zⁿ) =−n

2 +c(n) + log|z−zm|+O(|z−zm|log|z−zm|) +O(n|z−zm|), where|c(n)| ≤clognfor a constantc which does not depend onnand

τ Gn(zⁿ) + ˜τG˜n(zⁿ) =











−n(˜τ+τ /2)−τ n˜ log|z|+O(τ|z|ⁿ), as |z| →0

−n(˜τ+τ /2) +τ c(n)−τlog|z−zm|+

O(τ|z−zm|log|z−zm|) +O(τ n|z−zm|), as |z−zm| →0 .

We should takeτ=εandn˜τ= ˜ε. Moreover, we should have

−˜ε−εn

2 = ˜εlogε˜

2 and −ε˜−εn

2 +ε c(n) =εlogε 2.

This gives us the relation logε

˜ ε+ε˜

ε−n 2

ε

˜ ε =−n

2 +c(n) + 1, and

ε

˜

ε =g⁻¹_n (−n

2 +c(n) + 1) =:d(n), where

gn(t) : t∈(0,+∞) 7→ logt−n 2 t+1

t ∈(−∞,∞), is an everywhere decreasing function. Finally, we find

˜

ε= 2e^{−1−n2d(n)} and ε=d(n) ˜ε.

(In the case, where we do not have a singularity at z = 0 we just need to take ε=e^−n2+c(n)). We obtain for all β∈(0,1)

(4.9) τ Gn(zⁿ) =εlog ε

2|z−zm| +O(ε^1−β|z−zm|), as |z−zm| →0

τ Gn(zⁿ) + ˜τG˜n(z) =







˜ εlog ε˜

2|z|+O(˜ε^1−β|z|ⁿ), as |z| →0 εlog ε

2|z−zm|+O(ε^1−β|z−zm|), as |z−zm| →0 Finally, we put Gn(z) =τ Gn(zⁿ)/B and ˜Gn(z) =

˜

τG˜n(zⁿ) +τ Gn(zⁿ)

/B and

G˜n(z) =







2 ˜εlog ε˜

2|z|+O(˜ε^1−β|z|²), as|z| →0 εlog ε

2|z−zm|+O(ε^1−β|z−zm|), as|z−zm| →0

Remark 4.1. We can now explain why our construction works only for large n.

On one hand, in order to match the graph of the Green’s functionG˜nwith catenoids we need to truncate the cantenoids far enough and scale them by a small enough factor. On the other hand, in the neighbourhood of singularities the constant term ofG˜n depends on the number of singularitiesnand, as constant functions are not in the kernel of the linearised mean curvature operator, this gives the correspondence between the scaling factors of the catenoids and n.

5. Catenoidal bridges and necks

In this section we explain the construction of the surface Sn, invariant under the action of the group Sn, which has bounded mean curvature, meets the unit sphereS² orthogonally at the boundary and is close to two horizontal disks ”glued together” with the help of ”catenoidal bridges” in the neighbourhood then-th roots of unity. We will also denote ˜Sn the genus 1 surface, obtained fromSnby attaching a ”catenoidal neck” atz = 0. We will explain now what we mean by ”catenoidal bridges” and ”catenoidal neck”.

5.1. Catenoidal bridges. One of the possible constructions would be to ”glue”

the graph X(z,Gn(z)) together with minimal stripes obtained by intersecting eu- clidean catenoids centered at z =zm with the unit sphere. The difficulty of this approach is that those stripes would not meet the sphere orthogonally. We prefer to find a way to put half-catenoids in the unit sphere in the orthogonal way loosing the minimality condition.

Remark 5.1. We describe below the construction of the surfaceS˜n. The construc- tion of the surface Sn can be obtained by replacing the function G˜n by Gn which has the same expansion in the terms of ε atz =zm, taking into account that the relation between the parameters nandεchanges.

We use the notationC₋ for the half-plane{z∈C|Re(z)<0}. Form= 1, . . . , n consider the conformal mappings

λm:C₋−→D², λm(ζ) =e^2iπmn 1 +ζ 1−ζ.

These mappings transform a half-disk in the C₋ centered atζ = 0 and of radius ρ <1 to a domain obtained by the intersection of the unit diskD² with a disk of radius _1−ρ^2ρ2 and a center at _1−ρ^1+ρ22e^2iπmn . Let (ζ=ξ1+iξ2, ξ3) be the coordinates in C₋×R, then we define the mapping

Λm:C₋×R−→D²×R, Λm(ζ, ξ3) = (λm(ζ),2ξ3). Consider the half-catenoidCε/2in C₋×R, parametrized by

X_ε/2^cat : (σ, θ)∈R× π

2,3π 2

7→ ε

2coshσ e^iθ,ε 2σ In the regions, where σ >0 orσ <0 we can take the change of variables

ζ= ρ

2e^iθ= ε

2coshσ e^iθ, θ∈ π

2,3π 2

and, having in mind that the function ˜Gn defined in ¯D²\ {z1, . . . , zn} is invariant under rotations by the angle ^2π_n, consider a vertical graph overC₋:

(ρ, θ) 7→

ρ 2e^iθ,1

2G¯n(ρ, θ)

, where ¯Gn(ρ, θ) = ˜Gn(λm(ρ/2e^iθ))

In the neighbourhood of ρ = 0, we have |λm(ρ/2e^iθ)−zm| =ρ+O(ρ²). So, using the expansion (4.9) for the function ˜Gn in the neighbourhood of z=zm, we obtain a similar expansion for ¯Gn in the neighbourhood of 0:

G¯n(ρ, θ) =εlog ε

2ρ+O(ε^1−βρ), ∀β∈(0,1).

At the same time the lower and the upper part of theCε/2 can be seen as graphs of the functions

±G^cat_ε/2(ρ) =±ε 2log ε

2ρ+O(ε³/ρ²).

Now take a function ¯Υ which is defined in the neighbourhood of |ζ|=εby Υ(ρ, θ) = (1¯ −η¯ε(ρ))1

2G¯n(ρ, θ) + ¯ηε(ρ)G^cat_ε/2(ρ),

where ¯ηεis a cut-off function, such that

¯

ηε≡1, for ε < ρ <1/2ε^2/3, η¯ε≡0, for ρ > ε^2/3. Using this, we can parametrize the surfaceSn in the region

Ω^m_cat:=λm

nε/2 coshσ e^iθ : εcoshσ <1/2ε^2/3, θ∈[π/2,3π/2]o , by (σ, θ)7→ X ◦Λm ε/2 coshσ e^iθ,^ε₂σ

and as a bi-graph X {(z,2 Υ^m(z))∪(z,−2 Υ^m(z))},

forz∈Ω^m_glu:=λm

nρ/2e^iθ : 1/2ε^2/3< ρ <2ε^2/3, θ∈[π/2,3π/2]o , where Υ^m(z) is a function, such that Υ^m(λm(ρ/2e^iθ)) = ¯Υ(ρ, θ).

Remark 5.2. Orthogonality at the boundary

In a neighbourhood of itsm-th component of the boundary the surface ˜Sn (Sn) can be seen as the image by the mapping X ◦Λm of a surface (which we denote S¯n) contained in the half-space C₋×R. Consider a foliation of the half-space by horizontal half-planes. It is clear that every leaf of this foliation is orthogonal to

∂C₋×R. Thus, the normal to ∂C₋×R at a point is tangent to the horizontal leaf passing through this point. So, if there existed a tangent vector field along ¯Sn, horizontal at∂S¯n, then it would have to be collinear to the normal to∂C₋×R.

On the other hand, the image of the the foliation by horizontal half-planes by the mappingX ◦Λm gives a foliation of the unit ball by spherical caps which are orthogonal toS²at the boundary. The horizontal vector field tangent to∂C₋×R is sent by this mapping to a vector field tangent to the sphere and to a spherical cap leaf. The result follows from the fact that the restriction ofX ◦Λm to horizontal half-planes is conformal.

Finally in our case, the existence of the horizontal tangent vector field follows from the fact that ∂θX_ε/2^cat is horizontal and that ∂θG¯n = ∂θG^cat_ε/2 = ∂θη¯ε = 0 at θ∈ {π/2,3π/2}.

LetHdenote the mean curvature of the surface ˜Sn.

Proposition 5.1. There exists a constantc which does not depend onεsuch that in the region

Ω^m_cat=λm

nε/2 coshσ e^iθ : εcoshσ <1/2ε^2/3, θ∈[π/2,3π/2]o , we have

H(λm(ε/2 coshσ e^iθ) ≤ c

coshσ

Proof. The proof consists of calculating the mean curvature of Cε/2 with respect to the ambient metric

(X ◦Λm)^∗geucl(ζ, ξ3) =A²(Λm(ζ, ξ3)) dζ²+B²(Λm(ζ, ξ3))dξ²₃

= 4

[|1−ζ|²+ (1 +|ζ|²)(cosh(2ξ3)−1)]² dζ²+ (1 +|ζ|²)²dξ²₃

=a²(ζ, ξ3) dζ²+b²(ζ)dξ₃²

=:gm(ζ, ξ3)

wherea(ζ, ξ3) = 2

|1−ζ|²+ (1 +|ζ|²)(cosh(2ξ3)−1) andb(ζ) = 1 +|ζ|².

Let∇^εdenote the Levi-Civita connection corresponding to this metric. We have the following estimates for the Christoffel symbols in a neighborhood of (ζ, ξ3) = (0,0):

Γ¹₁₁=−Γ¹₂₂= Γ²₁₂= ¹_a∂ξ^∂a

1 =O(1), Γ²₁₁=−Γ²₂₂=−Γ¹₁₂= ¹_a∂ξ^∂a

2 =O(1) Γ¹₁₃= Γ²₂₃= Γ³₃₃=_a¹_∂ξ^∂a

3 =O(ξ3), Γ³₁₁= Γ³₂₂=−_ab¹2 ∂a

∂ξ3 =O(ξ3), Γ¹₃₃=−(^b_a²_∂ξ^∂a

1 +b_∂ξ^∂b

1) =O(1), Γ²₃₃=−(^b_a²_∂ξ^∂a

2 +b_∂ξ^∂b

2) =O(1), Γ³₁₃= ¹_a∂ξ^∂a

1 +¹_b∂ξ^∂b

1 =O(1), Γ³₂₃= ¹_a∂ξ^∂a

2 +¹_b∂ξ^∂b

2 =O(1) Γ¹₂₃= Γ²₁₃= Γ³₁₂= 0

Using|ζ|=ε/2 coshσ, ξ3=ε/2σ,and

∇^ε_∂p∂q =∂p∂qX_ε/2^cat+h

∂pX_ε/2^catiih

∂qX_ε/2^catij

Γ^k_ij∂k, where∂p and∂q stand for∂σ or∂θ and∂k =∂ξk, k= 1,2,3. We get

h∇^ε_∂p∂q−∂p∂qX_ε/2^catii

(σ, θ)

≤c ε²cosh²σ, i= 1,2

h∇^ε_∂p∂q−∂p∂qX_ε/2^cati3

(σ, θ)

≤c ε²coshσ.

The unit normal toCε/2 with respect to the metricgm is

N(σ, θ) = 1

aq

b²

cosh²σ+ tanh²σ

− b

coshσe^iθ,1 btanhσ

.

Using the the expansions for aandb in the neighbourhood of 0 and fact that the third coordinate of the vector∂p∂qX_ε/2^cat is zero for allpandqwe get the following expression for the second fundamental form :

h_ε(σ, θ) =ε(dσ²−dφ²) +hε(σ, θ), where

(hε)_pq(σ, θ)

≤c ε²coshσ. On the other hand, we can write the expansion of the metric induced onCε/2 fromgm

gε(σ, θ) =ε²cosh²σ(dσ²+dφ²) +gε(σ, θ), where

(gε)_pq(σ, θ)

≤cε³cosh³σ.

Finally,

H(ε/2 coshσ e^iθ) =

tr g⁻¹_ε hε (σ, θ)

≤ c coshσ.

5.2. Catenoidal neck atz= 0. In a small neighborhood ofz= ˜εtake the change of variablesz=r e^iφ, φ∈S¹. Then, in the neighbourhood ofz= 0, we have

B(r e^iφ) =B(r) = 1

2+O(r²).

We remind that thatτ Gn(zⁿ) + ˜τG˜n(zⁿ) = ˜GnB is a function whose graph is close to the lower part of the euclidean catenoid scaled a factor ˜ε. Then, the graph of ˜Gn

is close to the lower part of the surface ˜Cε˜, parametrized by X˜_˜ε^cat : (s, φ)∈R×S¹ 7→ (˜εcoshs e^iφ,2 ˜εs).

Let us define a cut-off functionr 7→ η⁰_˜ε(r), such that η_ε⁰_˜(r)≡1 for r∈

0,1/2 ˜ε^1/2

, η⁰_ε˜(r)≡0 for r >2 ˜ε^1/2.

Taking the change of variables: ˜εcoshs e^iφ =z =r e^iφ, fors > 0 ors <0 we can parametrize the lower and the upper part of ˜Cε˜as vertical graphs

z 7→ (z,±2G^cat_˜ε ), where

G^cat_˜ε (r) =ε˜ 2log

ε˜ 2r

+O

ε˜³ r²

. We define the function

Υ : ¯˜ D² \ {0, z1, . . . , zm} −→R,

Υ(r, φ) = (1˜ −η⁰_˜ε(r)) ˜Gn(r e^iφ) + 2η_ε⁰_˜G^cat_ε (r), and parametrized ˜Sn in the region

Ω⁰_cat:=n

˜

εcoshs e^iφ : ˜εcoshs <1/2 ˜ε^1/2, φ∈S¹o , by (s, φ) 7→ X ◦X˜_ε˜^cat(s, φ) and as a bi-graph :

(r, φ) 7→ Xn

(r e^iφ,Υ(r, φ))˜ ∪(r e^iφ,−Υ(r, φ))˜ o , for z∈Ω⁰_glu:=n

r e^iφ : 1/2 ˜ε^1/2< r <2 ˜ε^1/2, φ∈S¹o . We use the notations :

B(s) =B(˜εcoshs) =1

2(1 + ˜ε²cosh²s) and

A(s) =A(˜εcoshs,2 ˜εs) = 1

1 +B(s) (cosh(2 ˜εs)−1). Proposition 5.2. There exists a constantc which does not depend onεsuch that in the region

Ω⁰_cat=n

˜

εcoshs e^iφ : ˜εcoshs <1/2 ˜ε^1/2, φ∈S¹o , the mean curvature of the surfaceS˜n satisfies

H(˜ε coshs e^iφ)

≤cε˜^1−β, ∀β∈(0,1).

Proof. As in the proposition 5.1 we would like to calculate the mean curvature of C˜εwith respect to the ambient metric

X^∗geucl(z, x3) =A²(z, x3) dz²+B²(z)dx²₃

=: ˜g(z, x3).

We denote by ˜∇^˜εthe Levi-Civita connection corresponding to this metric. Then, in the neighborhood of (z, x3) = (0,0) the Cristoffel symbols satisfy :

Γ˜¹₁₁=−Γ˜¹₂₂= ˜Γ²₁₂= _A¹_∂x^∂A₁ =O(|z|x3), Γ˜²₁₁=−Γ˜²₂₂=−Γ˜¹₁₂= _A¹_∂x^∂A₂ =O(|z|x3) Γ˜¹₁₃= ˜Γ²₂₃= ˜Γ³₃₃= _A¹_∂x^∂A

3 =O(x3), Γ˜³₁₁= ˜Γ³₂₂=−_AB¹2

∂A

∂x3 =O(x3), Γ˜¹₃₃=−(^B_A²_∂x^∂A

1 +B_∂x^∂B₁) =O(|z|), Γ˜²₃₃=−(^B_A²_∂x^∂A

2 +B_∂x^∂B₂) =O(|z|), Γ˜³₁₃=_A¹_∂x^∂A

1+_B¹_∂x^∂B

1 =O(|z|), Γ˜³₂₃= _A¹_∂x^∂A

2 +_B¹_∂x^∂B

2 =O(|z|) Γ˜¹₂₃= ˜Γ²₁₃= ˜Γ³₁₂= 0

Using|z|= ˜εcoshs, x3= 2 ˜εs,and

∇˜^ε_∂^˜_p∂q =∂p∂qX˜_ε^cat_˜ + [∂pX˜_ε^cat_˜ ]ⁱ[∂qX˜_˜ε^cat]^jΓ˜^k_ij∂k, where∂p and∂q stand for∂sor ∂t, we get

[ ˜∇^ε_∂^˜_p∂q−∂p∂qX˜_ε^cat_˜ ]ⁱ(s, θ)

≤cε˜³cosh³s, i= 1,2

[ ˜∇^ε_∂^˜_p∂q−∂p∂qX˜_ε˜^cat]³(s, θ)

≤cε˜^3−βcosh²s, ∀β∈(0,1).

The normal vector field to ˜Cε˜with respect to the metricX^∗geuclis N˜(s, φ) = 1

Aq

4B²

cosh²s+ tanh²s

− 2B

coshse^iφ, 1 Btanhs

.

As the third coordinate of the vector∂p∂qX˜_ε^cat_˜ is zero for all pand q, we get the following expression for the second fundamental form :

˜h_˜ε(s, φ) = ˜ε(ds²−dφ²) + ˜hε˜(s, φ), where

˜hε˜

pq(s, φ)

≤cε˜^3−βcosh²s.On the other hand the metric induced onC˜ε

fromX^∗geucl can be written as

˜

g˜ε(s, φ) = ˜ε²cosh²s(ds²+dφ²) + ˜gε˜(s, φ), where

(˜gε˜)_pq(s, φ)

≤cε˜^4−β cosh²s. Finally, H(˜εcoshs e^iφ)

= tr

˜ g⁻¹_ε˜ ˜h_ε˜

(s, φ)

≤cε˜^1−β, ∀β ∈(0,1).

5.3. The graph region. Away from the catenoidal bridges and the catenoidal neck, that is in the region

Ωgr:=n

z∈D² : 2 ˜ε^1/2<|z|<1o

\ ∪ⁿ

m=1λm

nζ∈C₋ : |ζ|<2ε^2/3o , we parametrize the surface ˜Sn as a bi-graph

Xn

(z,G˜n(z))∪(z,−G˜n(z))o .

Proposition 5.3. There exists a constantcwhich does not depend on ε, such that in the region

Ωgr∪Ω⁰_glu ∪ⁿ

m=1Ω^m_glu

=

z∈D² : 1/2 ˜ε <|z|<1 \ ∪ⁿ

m=1λm

nζ∈C₋ : |ζ|<1/2ε^2/3o , the mean curvatureHof S˜n satisfies

(5.10) |H(z)| ≤c ε^3−β 1

|z|⁴ +

n

X

m=1

1

|z−zm|⁴

!

, ∀β ∈(0,1).

Proof. According to the lemma (3.1), the mean curvature of the graphX(z, u(z))), u∈ C²(D²) satisfies :

H(u) = 1

A³(u)B div A²(u)B²∇u p1 +B²|∇u|²

! + 2p

1 +B²|∇u|²sinhu

= 2

A²(u)

B∇u∇A(u) p1 +B²|∇u|² + 2

A(u)

∇B∇u

p1 +B²|∇u|² + 1 A(u)

B∆u p1 +B²|∇u|²

− 1 A(u)

B²∇B∇u|∇u|² (1 +B²|∇u|²)^3/2 − 1

2A(u)

B³Hessu(∇u,∇u) (1 +B²|∇u|²)^3/2 + 2p

1 +B²|∇u|²sinhu

= ∆ (Bu) +P3(u,∇u,∇²u)

whereP3is a bounded nonlinear function which can be decomposed in entire series inuand the components of∇uand∇²uforkukC¹ ≤1 with terms of lowest order 3 and where the components of ∇uappear with an even power and the components of∇²uonly with the power 1.

In the region Ωgr the surface ˜Sn is parametrized as a graph of one of the functions

±G˜n, where

B(z) ˜Gn(z) =−εn

2 +εRe(fn(z)) + ˜εlog|z|+ ˜ε,

Studying the behaviour of the functionfn(z) one can easily verify that

G˜n(z)

≤cε |logε|+|log|z||+

n

X

m=1

|log|z−zm||

! ,

∇G˜n(z)

≤cε 1

|z|+

n

X

m=1

1

|z−zm|

! ,

∇²G˜n(z)

≤cε 1

|z|²+

n

X

m=1

1

|z−zm|²

!

As ∆ BG˜n

= 0, analysing carefully the terms in P3( ˜Gn,∇G˜n,∇²G˜n) one can see that (5.10) is true in Ωgr.

In the regions Ω^m_glu the surface ˜Sn is parametrized as a graph of one of the functions ±2 Υ^m, where

Υ^m(λm(ρ/2e^iθ)) = ¯Υ(ρ, θ) = (1−η¯ε(ρ))1

2G¯n(ρ, θ) + ¯ηε(ρ)G^cat_ε/2(ρ).

In the neighbourhood ofζ= 0 we have:

∇z= 1

2∇ζ(1 +O(|ζ|)), ∇²_z= 1

4∇²_ζ(1 +O(|ζ|)).

Using that

λm(ρ/2e^iθ)−zm

=ρ+O(ρ²) , we obtain G¯n ∼G^cat_ε/2=O(εlogε), G¯n−G^cat_ε/2=O(ε^1−βρ),

∇G¯n

∼

∇G^cat_ε/2

=O

ε ρ

∇( ¯Gn−G^cat_ε/2)

=O(ε^1−β), ∇²G¯n

∼

∇²G^cat_ε/2 =O

ε ρ²

,

∇²( ¯Gn+G^cat_ε/2)

=O(ε^1/3−β) We introduce the function

B(ρ, θ) =¯ B λm

ρ 2e^iθ

= 1 +

1 + ^ρ₂e^iθ

/ 1−^ρ₂e^iθ

2 = 1 +O(ρ).

We have

∂ρB¯ ∼∂_ρ²B∼∂ρ∂θB¯ =O(1), ∂θB¯∼∂_θ²B¯ =O(ρ).

On the other hand, the cut-off function ¯ηεsatisfies

¯

ηε=O(1), d¯ηε

dρ =O 1

ρ

, d²η¯ε

dρ² =O 1

ρ²

. Using that

∇Υ =¯ 1

2(1−η¯ε)∇G¯n+ ¯ηε∇G^cat_ε/2+∇¯ηε

G^cat_ε/2−1 2G¯n

,

∇²Υ =¯ 1

2(1−η¯ε)∇²G¯n+ ¯ηε∇²G^cat_ε/2 + 2∇η¯ε

∇

G^cat_ε/2−1 2G¯n

t

+∇²η¯e

G^cat_ε/2−1 2G¯n

, we get the estimates

∇Υ¯ =O

ε^1−β ρ

,

∇²Υ¯ =O

ε^1−β ρ²

, and

P3(Υ) =O ε^3−β

ρ⁴

=O(ε^1/3−β), for allβ∈(0,1). We also have

∆z=1

4|1−ζ²|∆ζ,

where ∆z and ∆ζ are Laplacian operators in coordinatesz andζ. So,

∆

B G¯ ^cat_ε/2

=Gε/2∆ ¯B+ 2∇B¯∇G^cat_ε/2+ ¯B∆G^cat_ε/2=O ε³

ρ⁴

.

Putting this calculations together, we check that in Ω^m_gluthe mean curvature of ˜Sn

satisfies

H=O

ε^1/3−β

, ∀β∈(0,1).

An identical proof shows that in Ω⁰_glu we haveH=O(ε^1−β).

6. Perturbations of S˜n

Recall that the surface ˜Sncan be seen as an image by the mappingXof a surface S˜n (constructed in the previous paragraph) which is contained in the unit cylinder D²×R. We would like to calculate the mean curvature of small perturbations of S˜n and to this end we calculate the mean curvature of small perturbations of the surface ˜Sn with respect to the metric ˜g=X∗geucl.

Let, as before, ˜N denote the unit normal vector field to ˜Cε˜with respect to the metric

˜

g(z, x3) =A²(z, x3) dz²+B²(z)dx²₃ .

and take a functionw∈ C²( ˜Sn) small enough, invariant under rotations by the angle

2π

n and the transformationz7→z. We denote ˜¯ Sn(w) the surface parametrized by (s, φ)∈R×S¹ 7→ X˜_ε^cat_˜ (s, φ) +w(s, φ) ˜N(s, φ),

in region Ω^cat₀ . Furthermore, in the region Ωgr∪Ω⁰_glu=n

z∈D² : 1/2 ˜ε^1/2<|z|<1o

\ ∪ⁿ

m=1λm

nζ∈C₋ : |ζ|<2ε^2/3o , we parametrizeSn(w) by

z 7→

z,±Υ(z)˜

±w(z) ˜Ξ˜ε(z), where Ξ˜ε˜= (1−η_ε⁰_˜)∂x3+η_ε⁰_˜1 2N˜, where η⁰_˜ε(|z|) is the cut-off function defined in the paragraph 6.2. Notice that in Ω⁰_glu

kΞ˜˜εk˜g∼ k∂x3k˜g=1

2 +O(˜ε).

In the neighbourhood of the n-th root of unityzm the surface ˜Sn can be seen as an image by the mapping Λm of a surface ¯Sn contained in C₋×R. We put

¯

w=w◦λm∈ C²( ¯Sn) and parametrize ˜Sn(w) in the region Ω^m_cat by (σ, θ)∈R×[π/2,3π/2] 7→ Λm

X_ε/2^cat(σ, θ) +a

2w¯N(σ, θ) ,

where N is the unit normal vector field to the half-catenoidCε/2 with respect to the metric

gm(ζ, ξ3) = (Λm◦ X)∗geucl=a²(ζ, ξ3) dζ²+b²(ζ)dξ₃² . In the regions Ω^m_glu, we parametrize ˜Sn(w) by

ζ 7→ Λm ζ,±Υ(ζ)¯

± w(z) ¯¯ Ξε(ζ)

, where Ξ¯ε˜=1

2((1−η¯ε)∂ξ3+ ¯ηεaN), and ¯ηε(|ζ|) be the cut-off function, introduced in the paragraph 6.1. Notice that in Ω^m_glu, we have

kΞ˜εkgm ∼ 1

2k∂ξ3kgm= 1 +O(ε^4/3).

Remark 6.1. We multiply the vector fieldN by the factora in order to make the vector field

∂θ(aN) (ζ)|_θ∈{^π2,^3π₂},

horizontal. In this case, the condition sufficient for S˜n(w)to be orthogonal to the unit sphere is

∂θ( ¯w)|_θ∈{^π2,^3π₂} = 0.

Notation 6.1. LetΩdenote a coordinate domain we work in. From now on, when we don’t need a more detailed information, we use the following notations :

• Lfor any bounded second order linear differential operator defined inΩ(in other words L w is a linear combination of w and the components of ∇w and∇²w with coefficients which are bounded functions inΩ, where ∇and

∇² are the gradient and the Hessian in the chosen coordinates).

• Q^k(w,∇w,∇²w),k∈N, for any nonlinear function, which can be decom- posed in entire series with terms of lowest order k, and where the com- ponents of ∇²w appear only with power 1. We will also use the notation Q^k(w)for brevity.

• Let γ ∈ C^∞(Ω) be a positive real function. We denote L^γw and Q^k,γ(w) functions which share the same properties as L w andQ^k(w)with the only difference that the components of the gradient and the Hessian of w are calculated with respect to the metricγ⁻²geucl.

For example, if we work in the coordinates (r, φ) and take γ =r, then L^γ will be a linear differential operator inr²∂_r²,∂_φ²,r∂²_rφ,r∂r and∂φ. 6.1. Mean curvature of the perturbed graph. In the region

Ωgr={z∈D² : 2 ˜ε^1/2<|z|<1} \ ∪ⁿ

m=1λm{ζ∈C₋ : |ζ|<2ε^2/3}, we suppose thatkwkC²(Ωgr)<1. Then, the mean curvature of ˜Sn(w) satisfies

H(w) =H(0) + ∆(B w) +P3

G˜n+w .