Martin Traizet
email: martin.traizet@lmpt.univ-tours.fr

HOLLOW VORTICES AND MINIMAL SURFACES

Keywords:

We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.

Introduction

We consider the following overdetermined problem in the plane, known as the hollow vortex problem:

(1)

   ∆u = 0
in Ω u constant on each component of ∂Ω ||∇u|| = 1 on ∂Ω

Here Ω ⊂ R 2 is an unbounded domain with smooth, non-empty boundary, and u : Ω → R is a smooth function. Observe that we require u to be constant on each boundary component, but we do not ask that the constant is the same for all boundary components. The fact that ||∇u|| = 1 on the boundary is equivalent to the Neumann condition ∂u ∂ν = ±1, where ν is the (interior) unit normal to the boundary. Again, the sign of ∂u ∂ν may depend on the boundary component.

Problem [START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF] is overdetermined because both Neumann and Dirichlet boundary conditions are prescribed, which is not possible for general domains Ω. A domain Ω admitting a function u solving Problem (1) will be called a domain with hollow vortices. The hollow vortices refer to the components of R 2 \ Ω. The name comes from the following physical interpretation of Problem (1): the stationary flow of an inviscid, incompressible fluid in a domain Ω is described by Euler equations:

div v = 0, ( v • ∇) v = - 1 ρ ∇p
Partially supported by ANR-11-ISO1-0002 grant.

1 Figure 1. A Von Karman vortex street, from Van Dyke Album of Fluid Motion [START_REF] Van Dyke | An Album of Fluid Motion[END_REF] where v denotes the velocity vector, p the pressure and ρ the mass density of the fluid. In the 2-dimensional case, we can write v = ( ∂u ∂y , -∂u ∂x ) for some function u called the stream function. If we assume moreover that the flow is irrotational, then ∆u = 0. The condition that u is constant on a boundary component γ of Ω means that γ is a stream line. The condition that ||∇u|| is constant on γ means that the norm of the velocity is constant, which is equivalent to constant pressure by Bernoulli law. It is crucial for the work in this paper that the constant is the same for all boundary components. That constant is chosen to be equal to 1 by scaling.

We can think of γ as bounding a spinning bubble of air with constant pressure inside, or "hollow vortex". Hollow vortices have been proposed as a model for some periodic configurations of vortices observed in the turbulent flow past an obstacle known as Von Karman vortex streets. The authors of [START_REF] Crowdy | Analytical solutions for von Karman streets of hollow vortices[END_REF] have found hollow vortex solutions corresponding precisely to what is beeing observed in Figure 1.

In the particular case where u = 0 on ∂Ω and u > 0 in Ω, solutions to the hollow vortex problem have been studied in [START_REF] Hélein | A note on some overdetermined elliptic problem[END_REF], [START_REF] Khavinson | An overdetermined problem in potential theory[END_REF] and completely classified by the author in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF], by establishing a correspondence with a certain type of minimal surfaces and using classification results in minimal surface theory. It turns out that the correspondence extends to the general case of Problem [START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF], only to a wider class of minimal surfaces. Our goal in this paper is to describe this correspondence and give examples.

The corresponding overdetermined problem for minimal surfaces is the following:

(2)

     (1 + v 2 y )v xx + (1 + v 2 x )v yy -2v x v y v xy = 0 in Ω v constant on each component of ∂ Ω ||∇v|| → ∞ on ∂ Ω
Here Ω ⊂ R 2 is an unbounded domain with non-empty boundary and v : Ω → R is a smooth function. The subscripts denote partial derivatives. The first equation is the minimal surface equation: it says that the graph of v, denoted M , is a minimal surface. The limit in the last condition means the following: for any z 0 ∈ ∂ Ω, lim z→z 0 ∂v ∂ν = ±∞, the limit being uniform on compact sets of ∂ Ω. Geometrically speaking, this means that the Gauss map (i.e. the unit normal vector) of M is horizontal on the boundary. Such minimal surfaces can be smoothly extended by reflection in the horizontal plane containing each boundary component. For this reason, we call them minimal graphs bounded by horizontal symmetry curves.

For simplicity, we assume that all domains Ω and Ω considered in this paper satisfy the following finiteness hypothesis: Hypothesis 1. Either:

• Ω has a finite number of boundary components, • or Ω is invariant by a translation T and the quotient Ω/T has a finite number of boundary components (simply-periodic case), • or Ω is invariant by two independent translations (doubly-periodic case).

Under this hypothesis, the main result of this paper is

Theorem 1. There is a 1:1 correspondence between:

• solutions (Ω, u) of Problem (1) such that ||∇u|| < 1 in Ω, • solutions ( Ω, v) of Problem (2).
We describe how the correspondence works in Section 2. An interesting feature of the correspondence is that the domain with hollow vortices Ω and the corresponding minimal graph M are conformally related. Also, each component of ∂ Ω is a translation of the corresponding component of ∂Ω.

If Ω is a domain with hollow vortices, then in general, extending the corresponding minimal surface M by reflection will not give an embedded minimal surface. However, there are two particular cases where M can be extended to a complete embedded minimal surface:

• Case I: u > 0 in Ω and u = 0 on ∂Ω (this is the case already considered in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF]).

In this case, the corresponding minimal surface M lies in the half-space x 3 > 0 and has boundary in the horizontal plane x 3 = 0. It can be extended by reflection to a complete, embedded minimal surface. • Case II: 0 < u < c in Ω and u = 0 or u = c on each boundary component of ∂Ω.

In this case, M lies in the slab 0 < x 3 < c and has boundary in the horizontal planes at height 0 and c. It can be extended by iterated reflections in horizontal planes into a complete, embedded, periodic minimal surface with vertical period 2c.

We know a lot of such minimal surfaces, including some triply-periodic minimal surfaces discovered in the 19th century by Schwarz. We will review some of the classical examples in Section 3.

For some reason, specialists in the field of minimal surfaces are mostly interested in embedded surfaces, so the known examples only correspond to domains with hollow vortices of type I or II. However, some interesting methods have been developed to construct minimal surfaces, for example: the conjugate Plateau construction, see H. Karcher [START_REF] Karcher | The Triply Periodic Minimal Surfaces of Alan Schoen and their Constant Mean Curvature Companions[END_REF], or the flat structure method of M. Weber and M. Wolf [START_REF] Weber | Teichmüller theory and handle addition for minimal surfaces[END_REF]. Relaxing the constraint that we want the minimal surface to be embedded, it should be possible to adapt these methods to construct more general domains with hollow vortices.

In Section 4, we discuss another method, which was developed by the author and collaborators to construct minimal surfaces with small catenoidal necks. We will see how it can be adapted to construct domains with small hollow vortices.

1.1. Related works. A family of periodic solutions to the hollow vortex problem was constructed by Baker, Saffman and Sheffield in [START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF]. It corresponds to the family of horizontal Scherk surfaces (see Figure 2, top left) -the authors were of course not aware of that relationship. The same solution was derived again by Crowdy and Green in [START_REF] Crowdy | Analytical solutions for von Karman streets of hollow vortices[END_REF], together with another family of solutions called "staggered vortex streets". The corresponding minimal graphs are periodic and take on two different values on the boundary as in Case II. However, they are asymptotic to half-planes of non-zero slope at infinity, so extending these surfaces by reflection yields complete minimal surfaces which are not embedded.

Under the additional assumption that u > 0 in Ω and u = 0 on ∂Ω, solutions to the hollow vortex problem are called exceptional domains. Hauswirth, Hélein and Pacard studied the problem in [START_REF] Hélein | A note on some overdetermined elliptic problem[END_REF] and discovered an exceptional domain which, as it turns out, corresponds to the horizontal catenoid. Partial classification results were obtained by Khavinson, Lundberg and Teodorescu in [START_REF] Khavinson | An overdetermined problem in potential theory[END_REF]. A complete classification is given in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF].

In a recent paper, Eremenko and Lundberg [START_REF] Eremenko | Quasi-exceptional domains[END_REF] have investigated the hollow vortex problem under the assumption that u > 0 in Ω and ∂u ∂ν > 0 on ∂Ω. Solutions are called quasi-exceptional domains. Two examples are constructed. The first one corresponds to the minimal surface one gets if one tries to add a vertical handle to the horizontal catenoid. The second one corresponds to the minimal surface one gets if one tries to deform the horizontal Scherk surface so that the "holes" have different sizes. Both constructions are known to fail because one cannot solve the vertical period problem. On the hollow vortex side, this means that the function u takes on different values on the boundary components, which is perfectly fine.

In [START_REF] Crowdy | Hollow vortices, capillary water waves and double quadrature domains[END_REF], the authors compute solutions to the hollow vortex problem which are a mixture of smooth hollow vortices and point-vortices. These point-vortices can probably be regularized into small hollow vortices in the spirit of what we do in Section 4, but the resulting solution will not solve Problem (1) because ||∇u|| will take on different constant values on the different boundary components (it will be very large on the boundary of the small hollow vortices). This more general hollow vortex problem, where the constant value of ||∇u|| depends on the boundary component, has been studied by many authors.

Solutions do not correspond to minimal surfaces, at least not in the way described in this paper.

2. The correspondence 2.1. Preliminary observations. We first discuss the hypothesis ||∇u|| < 1 in the statement of Theorem 1. Let (Ω, u) be a solution to the hollow vortex problem [START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF]. The function

u z = 1 2 (u x -iu y ) is holomorphic in Ω and satisfies |2u z | = 1 on ∂Ω.
Let us rule out the trivial case where u z is constant, in which case Ω is a half-plane or a band bounded by two parallel lines. It is known that ||∇u|| < 1 in Ω in the following cases:

(1) if Ω and u z are doubly periodic, by the maximum principle for holomorphic functions in the quotient, (2) if ||∇u|| is bounded, by a Phragmen Lindelöf-type result of Fuchs [START_REF] Fuchs | A Phragmen Lindelöf Theorem conjectured by D[END_REF],

(3) if u is bounded from below (or above) in Ω, by the proof of Lemma 2 in [START_REF] Eremenko | Quasi-exceptional domains[END_REF]. (In this paper, the authors assume that ∂u ∂ν = +1 on the boundary, but only the condition ||∇u|| = 1 is used in the proof of Lemma 2.) Let us also mention that provided ||∇u|| < 1, the domain Ω must be strictly concave: see the proof of Proposition 4 in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF].

Weierstrass representation.

For the reader not familiar with minimal surfaces and to fix notations, we recall the Weierstrass representation formula:

(3)

X(z) = (x 1 (z), x 2 (z), x 3 (z)) = X 0 + Re z z 0 1 2 (g -1 -g)ω, i 2 (g -1 + g)ω, ω
In its local form, g is a meromorphic function on a simply connected domain Σ ⊂ C and ω = f (z)dz where f is a holomorphic function on Σ having a zero at each zero or pole of g, with the same multiplicity. z 0 ∈ Σ is an arbitrary base point and X 0 is some constant vector. Then X : Σ → R 3 is a conformal parametrization of a minimal surface M . Moreover, the Gauss map of M is given by

N = 2 Re(g) |g| 2 + 1 , 2 Im(g) |g| 2 + 1 , |g| 2 -1 |g| 2 + 1 .
In other words, g is the stereographic projection of the Gauss map.

In its global form, Σ is a Riemann surface, g is a meromorphic function and ω is a holomorphic 1-form on Σ.

2.3.

The correspondence "vortex → minimal". Let (Ω, u) be a solution to the hollow vortex problem [START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF]. Consider the minimal surface M given by the Weierstrass representation formula (3) with

g = -1 2u z , ω = 2u z dz, X 0 = (0, 0, u(z 0 )). Let ψ(z) = x 1 (z) + ix 2 (z) : Ω → C.
Proposition 1. In the above setup:

(1) x 3 (z) = u(z).

(2) ψ(z) is well defined in Ω, namely does not depend on the integration path from z 0 to z.

(3) dψ = dz along ∂Ω. Assume moreover that ||∇u|| < 1 in Ω. Then (4) For any z = z in Ω, 0 < |ψ(z ) -ψ(z)| < |z -z|. (5) ψ is a diffeomorphism from Ω to Ω = ψ(Ω). (6) The boundary of Ω is ψ(∂Ω).
The proof of this proposition is essentially the same as the proof of Theorem 9 in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF]. For completeness, we give the details in Appendix A.

From Points (1) and ( 5), we see that M is the graph of v = u • ψ -1 over the domain Ω. Since |g| = 1 on ∂Ω, the Gauss map is horizontal on the boundary, so ||∇v|| → ∞ on ∂ Ω. From Point (2), we see that each component of ∂ Ω is a translation of the corresponding component of ∂Ω. Moreover, from Point (4), we see that ψ moves the boundary components toward each other.

Remark 1. In [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF] we took g = 2u z , so |g| < 1 in Ω and the Gauss map of M was pointing down. For a graph it is more natural to choose the upward pointing normal so that the horizontal projection preserves orientation. The correspondence has better properties with the antipodal choice g = -1 2uz .

2.4. The correspondence "minimal → vortex". Let ( Ω, v) be a solution to Problem (2) and M be the minimal surface given as the graph of v. We orient M by its upward pointing normal. Then M is parametrized on some Riemann surface (with boundary) Σ by the Weierstrass representation formula (3). We have |g| > 1 on Σ and |g| = 1 on ∂Σ.

We observe that even though Σ is diffeomorphic to the planar domain Ω, in practice, it will not be given explicitely as a domain in the plane (see examples in Section 3), so it is better to leave it as an abstract Riemann surface. Let

ψ(z) = x 1 (z) + ix 2 (z). Since M is a graph, ψ is a diffeomorphism from Σ to Ω. Define F : Ω → C by F (ψ(z)) = - z z 0 gω.
Proposition 2. In the above setup:

(1) F is well defined in Ω.

(2) dF = dz on ∂ Ω.

(

) For any z = z in Ω, |F (z) -F (z )| > |z -z |. (4) F is a diffeomorphism from Ω to Ω = F ( Ω). (5) The function u(z) = v(F -1 (z)) solves Problem (1) and satisfies ||∇u|| < 1 in Ω. 3
The proof of this proposition is essentially the same as the proof of Theorem 10 in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF]. For completeness, we give the details in Appendix B.

The maps (Ω, u) → ( Ω, v) and ( Ω, v) → (Ω, u) defined by Propositions 1 and 2 are inverse of each other, provided we identify two domains which differ by a translation. See Theorem 11 in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF].

Remark 2. A computation shows that dF is given in term of v by

dF = dx + idy + (1 + v 2 x )dx + v x v y dy W + i v x v y dx + (1 + v 2 y )dy W where W = 1 + v 2 x + v 2 y .
Alternately, one could take this as a definition of F . (The minimal surface equation implies that dF is closed.) With this definition, the proof of Proposition 2 is more computational but avoids Weierstrass representation.

Classical examples

We focus on examples which are bounded by closed curves, as they are probably more interesting from the hydrodynamics point of view. All these examples admit deformations, and a lot more examples are known, see [START_REF] Karcher | The Triply Periodic Minimal Surfaces of Alan Schoen and their Constant Mean Curvature Companions[END_REF]. The Weierstrass data for all these examples is explicit and one can compute numerically the corresponding domain with hollow vortices: see Figure 2.

(1) The horizontal Scherk surface, a periodic minimal surface with horizontal period:

g = 1 z , ω = z dz z 4 + 6z 2 + 1 .
(2) Karcher toroidal halfplane layers, a family of doubly periodic minimal surfaces with one horizontal and one vertical periods:

g = 1 z , ω = dz (z 2 + a 2 )(z 2 + a -2 )
, 0 < a < 1.

(3) Schwarz P-surface, a triply periodic minimal surface:

g = 1 z , ω = z dz √ z 8 -14z 4 + 1 .
(4) Schwarz H-surfaces, a family of triply periodic minimal surfaces:

g = 1 z , ω = z dz z(z 3 + a 3 )(z 3 + a -3 ) , 0 < a < 1.
In all these examples, the Riemann surface Σ is a branched cover of the unit disk D(0, 1) punctured at z = ±i( √ 2 -1) in Case (1) and z = 0 in Case [START_REF] Connor | The construction of doubly periodic minimal surfaces via balance equations[END_REF]. The residues at the punctures and the multivaluation of the square roots are responsible for the periods of the minimal surfaces and the corresponding domains Ω.

Domains with small hollow vortices

In this section, we give a general method to construct domains with small hollow vortices, first in the finite connectivity case, then in the periodic case. In what follows, we identify points and vectors in the plane R 2 with complex numbers.

F i = j =i c i c j p i -p j .
We say a configuration is balanced if

F i = 0 for i = 1, • • • , n.
We say a balanced configuration is non-degenerate if the n × n jacobian matrix ∂F i ∂p j has complex rank n -2. Observe that we always have ( 4)

n i=1 F i = 0 (5) n i=1 p i F i = i<j c i c j .
Hence n -2 is the maximum rank that the jacobian matrix may have. Also, [START_REF] Eremenko | Quasi-exceptional domains[END_REF] gives a restriction on the weights for a balanced configuration to exist.

Theorem 2. Given a balanced, non-degenerate configuration, there exists a 1-parameter family of solutions (Ω t , u t ) of the hollow vortex problem (1), depending on a small parameter t > 0, such that:

(1) Ω t has n boundary components, denoted γ 1,t • • • , γ n,t , all of them closed curves.

(2) The circulation C(γ i,t ) is equal to 2πc i t.

(3) As t → 0, γ i,t shrinks to the point p i . Moreover, its asymptotic shape is circular.

Here is a simple example of balanced configuration, with dihedral symmetry of order n -1 (n ≥ 3):

c j = 1, p j = e 2πij/(n-1) for 1 ≤ j ≤ n -1, c n = 1 - n 2 , p n = 0.
The configuration is balanced by symmetry and Equation [START_REF] Eremenko | Quasi-exceptional domains[END_REF]. One can easily break the symmetries by perturbing the weights.

The proof of Theorem 2 follows [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF] in the minimal case and is omitted. It is also very similar to the proof of Theorem 3 below, which we give in Appendix C. 4.2. Periodic domains. In this section, we construct periodic domains with hollow vortices and period T in the x-direction, and such that the velocity vector has a limit as y → ±∞. The limit velocities cannot be arbitrary, as the following proposition shows: Proposition 3. Let Ω be a domain with hollow vortices. Assume that (1) Ω and the velocity vector v are periodic with period T in the x-direction.

(2) The quotient Ω/T has a finite number of boundary components, denoted γ 1 , • • • , γ n , all of them closed curves.

(3) The velocity vector v has a limit as y → +∞ and y → -∞, denoted respectively v + and v -. Then either:

(a) v + = v -and n i=1 C(γ i ) = 0, or (b) v + = -v -= 1 2T n i=1 C(γ i ).
Here we see the vectors v + and v -as complex numbers. In Case (b), the limit velocity must be real numbers, while there is no such restriction in Case (a).

Proof. Since we identify vectors with complex numbers, we can write v = -2iu z . Hence

lim y→±∞ 2u z = -iv ± .
If γ is a stream line, the circulation of the velocity is related to u z by ( 6)

γ 2u z dz = γ u x dx + u y dy -i γ u y dx -u x dy = -i C(γ).
For large R, consider the domain Ω R = Ω ∩ {-R < y < R}. By Cauchy Theorem,

0 = ∂(Ω R /T ) 2u z dz = n i=1 γ i 2u z dz + T -iR z=-iR 2u z dz + iR z=T +iR 2u z dz.
We let R → ∞ and obtain [START_REF] Hoffman | The strong halfspace theorem for minimal surfaces[END_REF]

n i=1 C(γ i ) = T (v + -v -).
Since u is constant on γ i , we have du = u z dz + u z dz = 0 along γ i . Hence using

|2u z | = 1 on γ i , γ i (2u z ) 2 dz = - γ i 4u z u z dz = - γ i dz = 0.
Using Cauchy theorem again with the function (2u z ) 2 and letting R → ∞ gives ( 8)

T ((v + ) 2 -(v -) 2 ) = 0.
Proposition 3 follows from ( 7) and (8). 2

Definition 2. A periodic vortex configuration is a finite set of non-zero complex numbers p 1 , • • • , p n with weight c 1 , • • • , c n which are non-zero real numbers, together with a non-zero complex number c 0 . We assume that either: Case (a)

c 1 + • • • + c n = 0, or Case (b) c 1 + • • • + c n + 2c 0 = 0.
We define forces by

F i = j =i c i c j p i + p j p i -p j + 2c i c 0 in Case (a) 0 in Case (b)
We say a configuration is balanced if F i = 0 for 1 ≤ i ≤ n. We say a balanced configuration is non-degenerate if the jacobian matrix ∂F i ∂p j has complex rank n -1. Observe that in either case, F 1 + • • • + F n = 0, so n -1 is the maximum rank that the jacobian matrix may have.

Theorem 3. Given a balanced, non-degenerate periodic configuration, there exists a 1parameter family of solutions (Ω t , u t ) of the hollow vortex problem (1), depending on a small parameter t > 0, such that:

(1) Ω t is a periodic domain with period T = 2π.

(2) The quotient Ω t /T has n boundary components, denoted γ 1,t • • • , γ n,t , all of them closed curves. (3) The circulation C(γ i,t ) is equal to 2πc i t. (4) As t → 0, γ j,t shrinks to the point q j = i log p j . Moreover, its asymptotic shape is circular. [START_REF] Eremenko | Quasi-exceptional domains[END_REF] In Case (a), the limit of the velocity as y → ±∞ is tc 0 . In Case (b), the limit of the velocity as y → ±∞ is ∓tc 0 .

We prove this theorem in Appendix C. The proof follows [START_REF] Connor | The construction of doubly periodic minimal surfaces via balance equations[END_REF] in the minimal case. Please take care that the limit position of the vortices is q j = i log p j and not p j as in Theorem 2. The 2πi multivaluation of the complex logarithm is responsible for the period T = 2π of the domain. In term of the points q j , the forces are given by

F i = -i j =i c i c j cot q i -q j 2 + 2c i c 0 in Case (a) 0 in Case (b) 4.3.
Examples of periodic configurations of type (a). We take n = 2, c 1 = 1 and c 2 = -1. Solving F 1 = 0 gives

q 2 = q 1 + i log 2c 0 -1 2c 0 + 1 , c 0 = ±0.5.
The points q i for various values of c 0 are represented on Figures 3,4,5.

If we take n = 5, c 1 = c 2 = c 3 = 1 and c 4 = c 5 = -1.5, we obtain an uneven vortex street: see Figure 6.

Examples of periodic configurations of type (b).

First assume that all c i are equal to 1. The configuration p j = e 2πij/n is balanced by symmetry: all forces F i are equal and their sum is zero. This gives q j = -2πj/n so the vortices are regularly spaced. This configuration gives the family of domains corresponding to the family of horizontal Scherk surfaces when it is close to its catenoidal limit. To get a more interesting example, take n = 3, c 1 = c 2 = 1 and leave c 3 as a parameter. We may normalize p 3 = 1. Computations show that p 1 and p 2 are the roots of the polynomial P (z) = z 2 + 2c 3 c 3 + 1 z + 1. We obtain a two lanes vortex street, see Figure 7.

• • • • • • • • • • • • • • • -→ -→ Figure 6.
Appendix A. Proof of Proposition 1

Proof of Point (1):

x 3 (z) = u(z 0 ) + Re z z 0 2u z dz = u(z 0 ) + z z 0 (u z dz + u z dz) = u(z 0 ) + z z 0 du = u(z).
Proof of Point (2): consider the differential

dψ = dx 1 + idx 2 = Re 1 2 (g -1 -g)ω + i Re i 2 (g -1 + g)ω = 1 2 (g -1 ω -gω).
With our choice of g and ω,

(9) dψ = 1 2 dz -4(u z ) 2 dz .
We have to prove that dψ is an exact differential (namely, the differential of a globally defined function ψ). If t → γ(t) is a parametrization of a boundary component of Ω, then since u is constant on γ:

du(γ ) = 0 = (u z dz + u z dz) (γ ). (10) dψ(γ ) = 1 2 (dz(γ ) + 4u z u z dz(γ )) = 1 2 (1 + ||∇u|| 2 )dz(γ ) = dz(γ ).
Hence if γ is a closed component of ∂Ω, γ dψ = 0. Since Ω is a planar domain, this implies that dψ is an exact differential. Also [START_REF] Khavinson | An overdetermined problem in potential theory[END_REF] proves Point (3).

Proof of Point (4): We prove that

(11) |2(ψ(z ) -ψ(z)) -(z -z)| < |z -z|
which implies Point (4) by triangular inequality. We may decompose the segment

[z, z ] into n segments [z i , z i+1 ] for 1 ≤ i ≤ n, such that z 1 = z, z n+1 = z , z i ∈ ∂Ω for 2 ≤ i ≤ n
and for each i, the open segment (z i , z i+1 ) is either included in Ω or its complement. In the first case, we have by Equation ( 9) and using ||∇u|| < 1

(12) |2(ψ(z i+1 ) -ψ(z i )) -(z i+1 -z i )| = z i+1 z i 4(u z ) 2 dz ≤ z i+1 z i 4|u z | 2 |dz| < |z i+1 -z i |.
In the second case, since Ω is a concave domain, z i and z i+1 must be on the same component of ∂Ω. By Point (3), ψ(z i+1 ) -ψ(z i ) = z i+1 -z i so ( 12) becomes an equality. Summing for 1 ≤ i ≤ n gives [START_REF] Masur | The extension of the Weil-Petersson metric to the boundary of Teichmuller space[END_REF].

Proof of Point (5): Equation ( 9) and ||∇u|| < 1 implies that dψ is an isomorphism so ψ is a local diffeomorphism. Point (4) implies that ψ is injective, so is a global diffeomorphism onto its image.

Proof of Point (6): since ψ is a homeomorphism from Ω to Ω and extends continuously to Ω, we have ψ(∂Ω) ⊂ ∂ Ω by elementary topology. (Here Ω denotes the closure of Ω.) Assume by contradiction that ψ(∂Ω) = ∂ Ω and let a 0 ∈ ∂ Ω \ ψ(∂Ω).

intersects M for t > 0 small enough. Let t 0 < 1 be the largest time so that C t intersects M . Then C t 0 intersects M at a boundary point. Since C t lies in the half-space x 2 < a < ε, that point must be on γ 1 or γ 2 . Since x 3 < 0 on γ i and ν = (0, 0, 1), M will still intersect C t for t slightly larger than t 0 , a contradiction.

2 Appendix B. Proof of Proposition 2
Proof of Point (1): consider the differential dϕ = -gω on Σ. We have to prove that dϕ is an exact differential. Since ω has a zero at each pole of g, dϕ is holomorphic in Σ. Let γ be a component of ∂Ω. Since |g| = 1 on γ and ω(γ ) is imaginary, ( 13)

dψ(γ ) = 1 2 g -1 ω(γ ) -gω(γ ) = -gω(γ ) = dϕ(γ ).
Since dψ is an exact differential and Σ is diffeomorphic to a planar domain, dϕ is the differential of a globally defined function ϕ.

Then F = ϕ • ψ -1 is well defined. Point (2)
is a consequence of [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF].

Proof of Point (3): let τ be the unit vector in the direction of z -z. We prove that ( 14)

F (z ) -F (z), τ > z -z, τ
which implies Point (3). We may decompose the segment

[z, z ] into n segments [z i , z i+1 ] for 1 ≤ i ≤ n, such that z 1 = z, z n+1 = z , z i ∈ ∂ Ω for 2 ≤ i ≤ n and for each i, the open segment (z i , z i+1
) is either included in Ω or its complement. In the first case, let α(t) : (0, ) → Σ be such that ψ(α(t)) is the parametrization of the segment (z i , z i+1 ) at constant speed τ , in other words dψ(α ) = τ . Then

dϕ(α ), τ = dϕ(α ), dψ(α ) = dϕ(α ) -dψ(α ), dψ(α ) + ||dψ(α )|| 2 = 1 4 |gω(α )| 2 -|g -1 ω(α )| 2 + 1 > 1 since |g| > 1.
Integrating from t = 0 to , we obtain ( 15)

F (z i+1 ) -F (z i ), τ > = z i+1 -z i , τ .
If the segment (z i , z i+1 ) is included in the complementary of Ω, then since Ω is a concave domain, z i and z i+1 must be on the same component of ∂ Ω, so F (z i ) = F (z i+1 ) by Point (2). Hence (15) become an equality. Summing for 1 ≤ i ≤ n gives [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF].

Proof of Point (4): since |g| > 1 in Σ, dϕ = 0 in Σ so ϕ and F are local diffeomorphisms. By Point (3), F is injective, so is a global diffeomorphism onto its image Ω. By Point (3), F is proper, so ∂Ω = F (∂ Ω).

Proof of Point (5): we have

u = v • F -1 = v • ψ • ϕ -1 = x 3 • ϕ -1 .
Since x 3 is harmonic and ϕ is biholomorphic, u is a harmonic function. Differentiating u(ϕ(z)) = x 3 (z), we obtain

2u z (ϕ(z)) × (-g(z)ω) = 2 ∂x 3 ∂z dz = ω.
Hence

2u z (ϕ(z)) = -1 g(z) which implies that ||∇u|| < 1 in Ω and ||∇u|| = 1 on ∂Ω. 2 Appendix C. Proof of Theorem 3
We need to construct a meromorphic function g and a holomorphic differential ω on a domain Σ ⊂ C such that |g| > 1 in Σ, |g| = 1 on ∂Σ and ω is imaginary along ∂Σ. Then we proceed as in Section 2.4 for the correspondence "minimal → vortex". Everything depends on the small parameter t > 0.

C.1. The domain Σ t and the function g t . Consider the function

f (z) = c 0 + n i=1 a i z z -p i .
Here a 1 , • • • , a n are non-zero complex parameters such that ( 16)

a 1 + • • • + a n = 0 in Case (a) -2c 0 in Case (b)
For t > 0, let Σ t be the domain |tf (z)| < 1. For t small enough, Σ t has n boundary components which we label γ 1 , • • • , γ n . We define the meromorphic function g t on Σ t by

g t (z) = -i tf (z)
.

We have

|g t | > 1 in Σ t and |g t | = 1 on ∂Σ t .
C.2. Opening nodes. To define the holomorphic differential ω t we need to construct the "double" of the domain Σ t . We do this by "opening nodes". Consider two copies of the complex plane, denoted C 1 and C 2 . As f has a simple pole at p i , there exists a neighborhood V i ⊂ C 1 of p i , a neighborhood W i ⊂ C 2 of p i and ε > 0 such that the holomorphic functions

v i (z) = 1 f (z) : V i → D(0, ε) and w i (z) = 1 f (z) : W i → D(0, ε) are biholomorphic. Consider the disjoint union C 1 ∪ C 2 . For i = 1, • • • , n, remove the disks |v i | < t 2 ε and |w i | < t 2 ε . Identify the point z ∈ V i with the point z ∈ W i such that (17) v i (z)w i (z ) = t 2 .
This defines a Riemann surface of genus n -1 which we denote Σ t . We also define Σ 0 as the (singular) Riemann surface with n nodes (or double points) obtained by identifying

p i ∈ C 1 with p i ∈ C 2 for 1 ≤ i ≤ n.
The anti-holomorphic involution σ which exchanges z ∈ C 1 with z ∈ C 2 is well defined on Σ t by the following computation:

z ∼ z ⇒ v i (z)w i (z ) = t 2 ⇒ v i (z)w i (z ) = t 2 ⇒ w i (z)v i (z ) = t 2 ⇒ σ(z) ∼ σ(z ).
We see Σ t as a domain in C 1 . The involution σ exchanges the disjoint domains Σ t ⊂ C 1 and Σ t ⊂ C 2 and its fixed set is ∂Σ t , indeed:

z = σ(z) ⇔ z ∼ z ⇔ v i (z)w i (z) = t 2 ⇔ |tf (z)| 2 = 1.
Hence we can see Σ t as the disjoint union of Σ t and its mirror image Σ t glued along their boundaries by the map z → z. In other words, Σ t is the double of Σ t . The point of constructing Σ t by opening nodes is that it will allow us to understand the limit t → 0. Finally, we can extend the definition of the function g t to Σ t by

g t (z) =      -i tf (z) = -i v i (z) t in C 1 -i tf (z) = -i t w i (z) in C 2
The identification [START_REF] Weber | Teichmüller theory and handle addition for minimal surfaces[END_REF] implies that g t is well defined on Σ t . Moreover, g t has the symmetry g t • σ = 1/g. C.3. The holomorphic differential ω t . We compactify Σ t by adding the points at infinity in C 1 and C 2 , denoted ∞ 1 and ∞ 2 . We also denote 0 1 and 0 2 the points z = 0 in C 1 and C 2 .

Proposition 4. For t = 0, there exists a unique meromorphic differential ω t on Σ t with 4 simple poles at 0 1 , 0 2 , ∞ 1 and ∞ 2 , such that

γ j ω t = -2πic j for 1 ≤ j ≤ n Res 0 1 ω t = c 0 Res 0 2 ω t = -c 0 .
Here c 0 , c 1 , • • • , c n are given from the configuration. Moreover:

(1) ω t has the following symmetry: σ * ω t = -ω t .

(2) ω t extends analytically at t = 0 with

(18) ω 0 = c 0 dz z + n i=1 c i dz z -p i in C 1 (
3) The residues of g t ω t at 0 1 and ∞ 1 are given by

(19) Res 0 1 g t ω t = -i t (20)
Res ∞ 1 g t ω t = i t Proof: the existence of ω t follows from the standard theory of compact Riemann surface: the curves γ 1 , • • • , γ n-1 are the A-cycles of a canonical homology basis. In general, one can define a meromorphic differential with simple poles by prescribing its periods on these cycles and the residues at the poles, with the only restriction that the sum of the residues is zero. In our case, prescribing the residue at ∞ 1 is the same as prescribing the period along the last cycle γ n .

Proof of Point (1):

γ j σ * ω t = σ(γ j ) ω t = - γ j ω t = 2πic j , Res 0 2 σ * ω t = Res 0 1 ω t = c 0 ,
Res 0 1 σ * ω t = -c 0 . Hence the meromorphic differentials σ * ω t and -ω t have the same poles, periods and residues, so they are equal.

Proof of Point (2): we know from the theory of opening nodes that ω t extends analytically at t = 0, and ω 0 is a meromorphic differential on Σ 0 with at most simples poles at the nodes (see [START_REF] Masur | The extension of the Weil-Petersson metric to the boundary of Teichmuller space[END_REF], [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF] or [START_REF] Traizet | Opening infinitely many nodes[END_REF]). The residues at the poles are determined by the prescribed periods. (Here γ i is oriented as a boundary of Σ t so has clockwise orientation.) Proof of Point (3): we have

f (0) = c 0 , g t (0 1 ) = -i tc 0 , Res 0 ω t = c 0 .
This gives (19). In Case (a), we have, using

a 1 + • • • + a n = c 1 + • • • + c n = 0, f (∞) = c 0 , g t (∞ 1 ) = -i tc 0 , Res ∞ 1 ω t = -c 0 .
In Case (b), we have, using

a 1 + • • • + a n = c 1 + • • • + c n = -2c 0 , f (∞) = -c 0 , g t (∞ 1 ) = i tc 0 , Res ∞ 1 ω t = c 0 .
In either cases, this gives (20).

C.5. The period problem. From now on, we assume that a i (t) has the value given by Proposition 5.

Proposition 6. For t small enough, one can adjust p 1 , • • • , p n so that for

1 ≤ i ≤ n, (22)
γ i g t ω t = 0.
Moreover, p i (t) is a smooth function of t and p i (0) is given by the configuration.

Proof: by Point (3) of Proposition 4 and the residue theorem , we have

(23) n i=1 γ i g t ω t = 2πi ( Res 0 1 g t ω t + Res ∞ 1 g t ω t ) = 0.
So it suffices to solve (22) for 1 ≤ i ≤ n -1. By symmetry and definition of g t ,

γ i g t ω t = γ i 1 g t (-ω t ) = it γ i f ω t .
So we want to solve (24)

γ i f ω t = 0 for 1 ≤ i ≤ n -1.
We solve (24) using the implicit function theorem at t = 0. We compute C.6. Solution of the hollow vortex problem. We proceed as in Appendix B, with ω replaced by t ω t . We define ϕ t on Σ t ⊂ C 1 for t = 0 by ϕ t (z) = i log z 0 -z z 0 g t t ω t .

Proposition 7.

(1) ϕ t : Σ t → C/2πZ is a well defined holomorphic map. (2) ϕ t extends analytically at t = 0 (away from the points p 1 , • • • , p n ) with ϕ 0 (z) = i log z. (3) ϕ t is a diffeomorphism onto its image Ω t = ϕ t (Σ t ) ⊂ C/2πZ. (The domain Ω t lifts to a periodic domain in the plane with period 2π.) (4) The function u t defined on ϕ t (Σ t ) by u t (ϕ t (z)) = Re Proof: by Propositions 5, g t ω t is holomorphic and non-zero in Σ t . By Proposition 6, the only periods of g t ω t come from the residues at 0 and ∞. By Point (3) of Proposition 4, ϕ t is well defined modulo 2π, which proves Point (1). To prove Point (2), we write ϕ t (z) = i log z 0 + i z z 0 ω t f and we use Equation (21). Regarding Point (3), we already know that ϕ t is a local diffeomorphism because its derivative does not vanish. Consider a component γ i of ∂Σ t . The unit normal to ϕ t (γ i ) is g t . Since the function g t is a diffeomorphism from γ i to the unit circle, ϕ t (γ i ) is a small convex curve. Consider then the well-defined holomorphic function ψ = exp(iϕ t ) : Σ t → C. Observe that ψ has a simple pole at 0 and a simple zero at ∞. Since each ψ(γ i ) bounds a disk in the Riemann sphere, we can extend ψ into a local homeomorphism ψ from the Riemann sphere to itself. Since the Riemann sphere is compact and simply connected, ψ must be a diffeomorphism. Hence ϕ t is injective, which proves Point (3).

Proof of Point (4): As in the proof of Point (5) of Proposition 2, we have

2 ∂u t ∂z (ϕ t (z)) = -1 g t (z) = -i tf (z).
Formula (25) for the velocity follows. From the definition of u t we obtain ϕ * t (2u z dz) = tω t . By [START_REF] Hélein | A note on some overdetermined elliptic problem[END_REF], the residue theorem and the definition of ω t , we have

C(ϕ t (γ i )) = i ϕt(γ i ) 2u z dz = i γ i ϕ * t (2u z dz) = i γ i tω t = 2π t c i .
2

Figure 2 .

 2 Figure 2. Top: the periodic domains corresponding to the horizontal Scherk surface (left) and a toroidal half-plane layer with a = 0.5 (right). Bottom: the doubly-periodic domains corresponding to Schwarz P-surface (left) and H-surface with a = 0.5 (right). The curves are the stream lines. Computed with Maple.

4. 1 .

 1 Domains with a finite number of hollow vortices. First some definitions. Let (Ω, u) be a solution to Problem[START_REF] Baker | Structure of a linear array of hollow vortices of finite cross-section[END_REF]. Let γ be a closed curve in the boundary of Ω. Let C(γ) = γ v • d be the circulation of v on the curve γ. Since || v|| = 1 on γ, the absolute value of C(γ) is equal to the length of γ, but it can have either sign, depending on whether the vortex is "spinning" left or right. Definition 1. A vortex configuration is a finite set of n ≥ 2 distinct points p 1 , • • • , p n in the complex plane with weights c 1 , • • • , c n which are non-zero real numbers. Forces are defined by

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. A periodic configuration of type (a) with n = 2, c 0 = 0.25. Circles represent vortices with right spin (c i > 0), bullets represent vortices with left spin (c i < 0). The arrows indicate the direction of the velocity at infinity. Three fundamental domains are represented. Compare with Figure 1. • •

Figure 7 .

 7 Figure 7. A periodic configuration of type (b) with n = 3, c 1 = c 2 = 1 and c 3 = -1.5.

= 2πi c 2 i + 2c 0 c i + 2 j =i c i c j p i p i -p j = 2πi c 2 i + 2c 0 c i + j =i c i c j p i + p j p i -p j + 1 = 2πi n j=1 c 2 Remark 5 .

 221j=125 p i c 2 i z (z -p i ) 2 + 2c i z -p i c 0 + j =i c j z z -p j i c j + 2c 0 c i + j =i c i c j p i + p j p i -p j = 2πiF iwhere F i is as in Definition 2. Since the configuration is balanced and non-degenerate, we can solve (24) using the implicit function theorem. Since F 1 + • • • + F n = 0, it was crucial to have the relation (23) amongst the periods for all t.

  solves Problem (1) on Ω t . Moreover, the velocity v t and its circulation are given by(25) v t (ϕ t (z)) = tf (z) C(ϕ t (γ i )) = 2π t c i .

The finiteness hypothesis (Hypothesis 1) ensures that ψ(∂Ω) is closed. Indeed, for each component γ of ∂Ω, ψ(γ) is a translate of γ so is closed, and the finiteness hypothesis prevents them from accumulating.

Let ε = d(a 0 , ψ(∂Ω). Choose a point a 1 ∈ Ω such that |a 0 -a 1 | ≤ ε 4 . Let a 2 be a point on ∂ Ω whose distance to a 1 is minimum. Then d(a 2 , ψ(∂Ω)) ≥ ε 2 and the semi-open segment [a 1 , a 2 ) is entirely included in Ω. Let α(t) : [0, ) → Ω be a path such that ψ(α(t)) is the parametrization at unit speed of the segment [a 1 , a 2 ). We must have ||α(t)|| → ∞ as t → ∞, else a 2 would be in ψ(Ω). This implies that the path α has infinite length. Now the conformal metric induced by the minimal immersion X is given by

Hence, the curve X(α(t)) on M has infinite length. This curve is the graph of the function v on the segment [a 1 , a 2 ). By standard results in minimal surface theory (see the proof of Lemma 2 in [START_REF] Traizet | Classification of the solutions to an overdetermined elliptic problem in the plane[END_REF] for the details), this implies that lim z→a 2 v(z) = ±∞. Moreover, there exists a divergence line L, containing a 2 and contained in ∂ Ω, such that v → ±∞ on L. By connectedness, Ω must be on one side of L. To derive a contradiction, we distinguish two cases:

• If d(L, ψ(∂Ω)) > 0, then the function v satisfies the minimal surface equation in a band with boundary value ±∞ on one side. This is impossible by Proposition 1 in [START_REF] Mazet | Quelques résultats de non-existence pour l'équation des surfaces minimales[END_REF].

Remark 3. If all components of ∂Ω are closed curves, then the finiteness hypothesis implies that d(L, ψ(∂Ω)) > 0.

• If d(L, ψ(∂Ω) = 0, then there is an unbounded component of ∂Ω, say γ 1 , such that ψ(γ 1 ) is asymptotic to L. Also, there can be at most two such components. Label γ 2 the other component asymptotic to L, if any. Let us write γ i = ψ(γ i ). There exists ε > 0 so that all other components of ψ(∂Ω) are at distance greater than ε of L. We obtain a contradiction using the catenoid as a barrier as in the proof of the strong half-space theorem of Hoffman Meeks [START_REF] Hoffman | The strong halfspace theorem for minimal surfaces[END_REF]. The only difference is that M is not complete so we have to mind its boundary.

Without loss of generality, we may assume that L is the line x 2 = 0 in the horizontal plane, M lies in the half-space x 2 > 0, and also v < 0 on γ 1 and γ 2 and v → +∞ on L. Let ν be the interior conormal to the boundary of M . Since γ 1 and γ 2 are horizontal symmetry curves, ν is vertical. Evaluating the vertical flux in the subdomain of Ω defined by R < |x 1 | < 2R and 0 < x 2 < ε for large values of R, we obtain that ν = (0, 0, 1) on γ 1 and γ 2 .

Let C be the horizontal half-catenoid x 2 1 + x 2 3 = cosh 2 x 2 , x 2 < 0. Let C t = (0, a, 0) + t C 1 , where 0 < a < ε and 0 < t ≤ 1. If a is small enough then C 1 does not intersect M . Also, as t → 0, C t converges to the vertical plane x 2 = a, so C t C.4. Zeros of ω t . Proposition 5. For t small enough, one can adjust the parameters a 1 , • • • , a n so that ω t has a zero at each zero and pole of g t . Moreover, a i (t) is a smooth function of t and a i (0) = c i .

Proof: the meromorphic differential ω t has 4 poles on a genus n -1 compact Riemann surface so has 2n zeros. By symmetry, ω t has n zeros in C 1 , which we call

We have to solve f (ζ i (t)) = 0 for 1 ≤ i ≤ n. We use the implicit function theorem at t = 0. When t = 0, an obvious solution is to take a i = c i which gives by (18) ( 21)

Then we compute ∂f

The determinant of this n × n matrix is a Cauchy determinant so it is invertible. The problem to apply the implicit function theorem is that the parameters a 1 , • • • , a n are constrained by Equation ( 16), so we have in fact only n -

restricted to the space a 1 + • • • + a n = 0, is an isomorphism.

Proof: let (a 1 , • • • , a n ) be in the kernel of the partial differential of h. Then

Consider the meromorphic differential on

Then µ has n -

By Lemma 1 and the implicit function theorem, we can solve f

The meromorphic differential g t ω t has 4 simple poles at 0 1 , ∞ 1 , 0 2 , ∞ 2 , and at most a simple pole at ζ . By Point (3) of Proposition 4 and by symmetry, the sum of the residues of g t ω t at 0 1 , ∞ 1 , 0 2 and ∞ 2 is zero. By the residue theorem, the residue at ζ is zero, so g t ω t is actually holomorphic at ζ , which means that ζ = ζ n . This proves Proposition 5.

2

Remark 4. We tacitly assumed that the zeros ζ 1 , • • • , ζ n are distinct, which is the generic case. In case there are multiple zeros, the proof must be fixed using the Weierstrass preparation theorem. See details in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF].