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1 Math. Ann. 31, (1998 Mathematische Annalen c Springer-Verlag 1998 Riemann bilinear relations on minimal surfaces Joaquín Pérez Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, E-1871 Granada, Spain ( jperez@goliat.ugr.es Received 13 July 1995 / Revised version 24 May 1996 Mathematics Subject Classification (1991: 53A1, 53C42 Introduction From a physical point of view, minimal surfaces in R 3 are objects submitted to a balanced force system, consisting in the forces associated to non zero onedimensional homology classes in the surface. Several geometric properties can be studied in terms of those forces, as embeddedness, symmetries and deformations, leading up to uniqueness and non existence results, see [11 13]. More precisely, each closed curve γ in a minimal surface S R 3 carries a force that expresses the stress produced by an unit conormal vector field η along γ on the whole surface. The action of this conormal field provides a tendency of translation, or linear momentum F, and another one of rotation around an axis, or angular momentum M. The first one is given by the force vector η and the second can be expressed γ by the torque vector p η, where p denotes the position vector field on the γ surface. These objects have been deeply studied by Kusner in a more general context [5, 6], obtaining the balancing conditions for the above force system, that is, the global force and torque are equal to zero. Also Korevaar and Kusner [4], see also the references there, study certain spaces of constant mean curvature surfaces in R 3 in terms of these forces and torques. Kusner s balancing equations for a complete immersed minimal surface M R 3 with finite total curvature and embedded ends are consequences of the fact that the sum of the residues of certain meromorphic differentials defined on the conformal compactification M of M [1] is zero. Indeed, the classical Riemann bilinear relations applied on the Weierstrass one-forms of the surface [1] not only imply that the global torque must balance but also let us write down relations that reveal how the Research partially supported by a DGICYT Grant No. PB

2 38 J. Pérez distribution of the ends constrains the possible values of the period and the force vectors corresponding to the homology of M. For instance, the forces F(γ i, F(Γ i, 1 i k associated to a homology basis of M, k being the genus of M, satisfy the equation ( F(γ i F(Γ i, e 3 = π aj 2 aj 2, C provided that all the ends are horizontal, where e 3 =(,, 1, a j is the logarithmic growth of the end p j and C resp. C is the set of ends with limit normal vector e 3 resp. e 3, see Theorem 1. The last equation gives us information about the rank of the R 3 -valued cohomology class Force for a complete embedded minimal surface S of finite total curvature: The rank zero and one cases are only attached respectively by the plane and the Catenoid [11]. When rank(force= 2 and S has an odd number of ends, the last equation imposes severe restrictions on the allowed distribution of the logarithmic growths at the ends, see Corollary 1: A properly embedded minimal surface with finite total curvature, rank(force= 2and an odd number of ends has an odd number of planar ends and its catenoid ends can be associated by pairs with the same logarithmic growth. As a particular case, A properly embedded minimal surface in R 3 with finite total curvature and three ends has rank(force= 3. The method in this paper also persuades to think about a similar establishment for minimal surfaces in any non compact flat three-manifold with a Weierstrass representation, that is, manifolds isometric to R 3, R 3 /T or T R, where T is a non trivial translation and T R is the quotient of R 3 by a lattice generated by two independent translations. Similar constraint equations are obtained in any case, and by imposing suitable conditions we deduce uniqueness results, as The only properly embedded minimal torus in R 3 /T with two planar ends is a Riemann example, provided that it is symmetric by the reflection in a plane. C 1 An integral formula Let M be a compact genus k Riemann surface and let Φ =(φ 1,φ 2,φ 3 bea C 3 -valued meromorphic differential on M. Take 2k closed curves γ 1,...,γ k, Γ 1,...,Γ k on M such that the following properties are satisfied: 1. For each i {1,...,k}, γ i cuts once to Γ i, this intersection is transversal, and γ i does not cut to any other of the 2k 2 remaining curves;

3 Riemann bilinear relations on minimal surfaces For each i {1,...,k 1}, we can take a curve α i on M joining the intersection points γ i Γ i and γ i+1 Γ i+1, and such that α i does not cut to γ 1,...,γ k,γ 1,...,Γ k,α 1,...,α i 1, α i+1,...,α k 1 except at the points γ i Γ i, γ i+1 Γ i+1 ; 3. If we cut M along γ 1,...,γ k,γ 1,...,Γ k,α 1,...,α k 1, we obtain a simply connected surface M with boundary M. M consists in a pair of copies of each one of the above curves, that will be denoted by γ 1,...,γ k, Γ 1,...,Γ k, α 1,...,α k 1 γ 1,..., γ k, Γ1,..., Γ k, α 1,..., α k 1, where γ i is the corresponding copy of γ i, and the same holds for the remaining curves. Consider over those curves the orientation they inherit as boundary of M. Identifying the copies γ 1,...,γ k,γ 1,...,Γ k on M with the original curves on M, we have an orientation on this last set of curves. Now consider the set {p 1,...,p r }= {p M / Res p Φ/=}. We can suppose that p i lies in the interior of M, 1 i r. Let δ i be a small closed curve around p i, contained in the interior of M, such that δ i δ j =, i /= j. Each one of these curves is the boundary of a disk D i that contains p i which can be assumed to be a conformal disk on M. Choose a point x M, and take r curves β 1,...,β r M such that 4. For each i {1,...,r}, β i joins δ i with x, β i M = {x }, β i cuts once to δ i but it does not cut to any other of the remaining δ-curves; 5. Two of these curves β i, β j only can intersect at x. If we cut M along the curves δ 1,...,δ r,β 1,...,β r and remove the disks D 1,...,D r, we will obtain another simply ( connected surface M M, which r boundary consists in M = M δ j β j β j, where β j denotes a copy of β j obtained by cutting M along β j. Now, Φ is a meromorphic differential form on M without residues at its poles, so there exists a meromorphic C 3 -valued map H on M such that dh = Φ on M. Let {p 1,...,p s}= {p M / p is a pole of Φ and Res p Φ =}. We can suppose that {p 1,...,p s} in contained in the interior of M. For each i {1,...,s}, let δ i be a small closed curve around p i, contained in the interior of M, such that δ i δ j = whenever i /= j. Each curve δ i is the boundary of a conformal disk D i containing p i. By removing the disks D 1,...,D s from M, we obtain a new ( compact surface s M M which boundary consists in M = M δ j. The orientation on M induces in a natural way another one on M, that coincides with the classical orientation as the boundary of M if we consider on this surface the induced orientation as a subset of M, so we have oriented the curves γ 1,...,γ k,γ 1,...,Γ k,δ 1,...,δ r,δ 1,...,δ s, which can be viewed on M.Onthe other hand, the divergence theorem on M can be applied to the holomorphic differential form H Φ, giving

4 31 J. Pérez = s H Φ + + δ j ( r ( + + β j β j δ j γ i +, k 1 ( + γ i + Γ i + Γ i + α i α i where each integral is computed with the corresponding orientation as a subset of M all the above integrals have the same argument. The integrals along the -curves can be written in terms of the remaining integrals: For a fixed i {1,...,k}, the curves γ i and γ i come from the same curve on M. Thus, we can consider a reversing-orientation diffeomorphism f : γ i γ i that maps each point on γ i on the same point viewed on γ i, and H f = H + Φ, f Φ = Φ, (1 Γ i hence H Φ = f (H Φ = H Φ Φ Φ. (2 γ i γ i γ i Γ i γ i A similar argument can be applied on each pair of curves, obtaining H Φ = H Φ + Φ Φ, H Φ = H Φ Γ i Γ i γ i Γ i α i α i H Φ = H Φ + Φ Φ. β j β j δ j β j So we have s ( r = H Φ +2 Φ Φ + H Φ + Φ Φ. γ i Γ i δ j δ j β j δ j Note that if we change the orientation, we will find the opposite expression on each term in the above equation. Moreover, if we exchange the γ-curves for the Γ -curves, the second term in the last expression becomes the opposite, but the others not. But the equation (1 that we used in order to compute (2 holds when γ i cuts through Γ i from the left to the right, if we see both curves on M. In other words, we are requiring to {γ 1,...,γ k,γ 1,...,Γ k } to be a canonical homology basis for M, see [1]. Now we can state the following general integral formula: ( r s 2 Φ Φ = Φ Φ + H Φ H Φ. (3 γ i Γ i δ j β j δ j We can obtain a scalar version of (3 that allows us to remove the first term on the right-hand side under suitable conditions. For, consider two meromorphic differential forms φ a,φ b on M, and define {p 1,...,p r } ={p M / Res p φ a /=}. δ j

5 Riemann bilinear relations on minimal surfaces 311 Reasoning as above, we obtain a simply connected surface M that carries a meromorphic function h a such that dh a = φ a on M. Consider the points {p 1,...,p s} = {poles of h a φ b }. Now cutting M and applying the divergence theorem to h a φ b, we have ( γ i φ a φ b Γ i Γ i φ a γ i φ b = ( r s δ j δ j φ a φ b + β j h a φ b δ j h a φ b. (4 Let ψ : M {p 1,...,p n } N be a proper conformal minimal n-ended immersion of a finitely punctured compact genus k Riemann surface M in N = R 3, R 3 /T or T R, where T is a non trivial translation in R 3 and T R is the quotient of R 3 over the isometry group generated by two independent translations. Suppose that ψ has finite total curvature when N = R 3 /T or N = T R this condition is verified if ψ is one-to-one, see [9].LetΦ =(φ 1,φ 2,φ 3 bethe Weierstrass one-form of ψ, which is well-defined in all the cases, see [8 1]. In this situation, (3 and (4 hold. On the other hand, Φ γ i Φ = Γ i ( φ 3 Γ i φ 2 γ i Γ i φ 2 γ i φ 3, φ 3 φ 1 φ 3 φ 1, φ 1 φ 2 φ 1 φ 2, (5 γ i Γ i Γ i γ i γ i Γ i Γ i γ i and we can use (4 on each component of the above expression. Observe that the p j -points and the p j -points could be different in each component. If the geometry of the surface determines a special direction, we can suppose that only one of the differentials φ 1,φ 2,φ 3 has residues at its poles for instance, when N = R 3 and ψ has parallel embedded ends, up to rotation in R 3 all the ends will have vertical limit normal, thus only φ 3 will have residues at the catenoid type ends, obtaining the whole formula (5 by integration of the differentials without residues, and the first term of (4 does not appear in any component. 2 Minimal surfaces with finite total curvature We will apply the general integral formula to a properly immersed minimal surface in R 3 with finite total curvature. Before this, we are going to recall some facts related with the torque vector associated to a closed curve in a minimal surface, that can be found in [5].

6 312 J. Pérez 2.1 The torque vector Given ρ ], 1], denote by D (ρ ={z C / < z ρ}, and D = D (1. We will also use the identification R 3 C R for the remainder of the paper. Let ψ : D R 3 be a conformal minimal immersion, and let η be the inner unit conormal to ψ D (ρ along the curve C ρ = { z = ρ}. The torque vector associated to ψ is defined as T = ψ(s η(s ds, C ρ where s is the arc parameter of ψ(c ρ. As the R 3 -valued differential ψ dψ has divergence zero, the above integral does not depend on ρ. Note that if we translate ψ by a vector v R 3, the new torque vector T v = C ρ (ψ + v η ds verifies T v = T + v F, where F is the force vector of ψ possibly vanishing. So, T does not depend on a translation in the direction of F. The conormal η can be viewed as a force acting at any point of ψ(c ρ that represents the stress produced by ψ on a minimal surface containing ψ(d as a piece. The momentum of η(s respect to the axis <v>with v S 2 = {x R 3 / x =1}, is given by M (s,v= ψ(s η(s,v v, so the total momentum is a map from S 2 to R 3 that applies each v S 2 on M (v = C ρ M (s,vds = T,v v. The maximum of M (v is attached at v = ± T T, hence the torque vector points to the direction with the largest tendency of rotation of ψ(d around it. It is well-known that a properly embedded finite total curvature minimal annulus with compact boundary must be a punctured disk, its Weierstrass representation Φ =(φ 1,φ 2,φ 3 extends in a meromorphic way through the punctures and it is asymptotic to the end of a Catenoid or a plane. Those cases will be called, as usual, a catenoid type end or a planar end. For more details, see [1, 13]. Kusner [5] found expressions for the torque vector T associated to such an end ψ. In terms of its Weierstrass form Φ =(φ 1,φ 2,φ 3, it is not hard to prove that in the planar end case, T =Re [Res (h 2 φ 3, h 1 φ 3, h 1 φ 2 ], (6 where H =(h 1, h 2, h 3 =ψ + iψ is a primitive of Φ here ψ is a well-defined conjugate minimal immersion of ψ. Up to a rotation in R 3, we can suppose that the extended Gauss map of ψ points to (,, 1 at the puncture. Calling g to the stereographic projection from the North Pole of this Gauss map, take a conformal coordinate (D(ε, z centered at the end such that g(z =z k. Then, (6 becomes { T = if k > 2, π ( Im(a 2, Re(a2, π(ia 2, if k =2, where a is the non zero complex number given by φ 3 = ( a + O (z 2 dz, and O (z l denotes a holomorphic function with a zero of order l at z =. (7

7 Riemann bilinear relations on minimal surfaces 313 The torque vector T associated to a planar end ψ : D R 3 whose Gauss map has ramification order 2 at the end also describes the intersection of ψ with the limit affine tangent plane Π: this intersection is asymptotic to a straight line r Π in the direction of T, r = {p + λt /λ R}, and the points of ψ(d outside a tubular neighbourhood of r are situated in the opposite quadrants Q 1 Q 2 = {p R 3 / p p, N ( det(p p, T, N ( < }, where N ( is the limit normal vector at the puncture. This assertion can be proved easily from the Weierstrass data g(z =z 2, φ 3 = ( a + O (z 2 dz. At a catenoid type end ψ : D R 3 with Gauss map g, we can assume up to a rotation that g( =, hence our surface is given by the meromorphic data ( g(z =z, φ 3 = a z + cz + O (z 2 dz, (8 where a R {} is the logarithmic growth of the end and c C, see [1, 13]. Now the torque vector T can be computed as T = ( { z =ρ} ψ dγ ds N ds, where N is the S 2 -Gauss map of ψ and γ = γ(s is a parametrization by the arclength of ψ({ z = ρ}. When ρ goes to zero the above integral converges to T =a ( Re(c 2, Re(c ( 1,, where c 1, c 2 are complex numbers obtained by integration of φ 1 = 1 2 g 1 g ( φ 3, φ 2 = i 2 g 1 + g φ 3, respectively. Calling E = (Re(c 1, Re(c 2, and F = Force(ψ =(,, a, the symmetry axis of the half-catenoid asymptotic to ψ is given by E+ < F >. We will refer to this straight line as the axis of the end, so the torque vector at a catenoid type end is given by T = E F. Kusner [5] also proved a balancing formula involving forces and torques for properly immersed constant mean curvature surfaces M R 3. This condition insures that for a nulhomologous sum of cycles Γ Γ n on M, the corresponding forces F 1,...,F n and torques T 1,...,T n must balance, that is, F F n ==T T n. (9 When the cycles Γ 1,...,Γ n correspond to the curves around the ends of a proper minimal immersion ψ : M {p 1,...p n } R 3 with finite total curvature and embedded ends, the above formula can be obtained by taking imaginary parts in the general integral formula ( A new balancing condition and applications Let ψ : M {p 1,...p n } R 3 be a proper minimal immersion with finite total curvature and parallel embedded ends, M being a compact genus k surface. Using the notation in Sect. 1, the points p M where Res p Φ/= are the catenoid type ends, and the poles of Φ without residue are the planar ends. Denote by

8 314 J. Pérez F(γ i =Im γ i Φ the force vector associated to the curve γ i, and in a similar way for the Γ i -curve. Rotate the surface in such a way that the ends have vertical limit normal. Taking real parts in (5, and using (4, F(γ i F(Γ i = n Im [ Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 ]. If p j is a planar end, it is an elementary exercise to conclude from the Weierstrass representation that Im [ Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 ] = 1 T j N (p j, (1 where T j is the torque vector associated to the end p j and N (p j is the value of the Gauss map at this puncture. Now if p j is a catenoid type end and g(p j =, we can use (8 to conclude that Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 =( ac 2, ac 1, i 2 a2, where a is the logarithmic growth of the end and c 1, c 2 are complex ( numbers appearing in the local expressions of the primitives h 1, h 2 of φ 1 = 1 2 g 1 g ( φ 3, φ 2 = i 2 g 1 + g φ 3, respectively: (h 1, h 2 (z =(c 1, c ( a 2 z +(c + az, ia +(c aiz + O (z 2. z A similar argument at a catenoid type end with g(p j = gives Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 =( ac 2, ac 1, i 2 a2. Thus, Im [ Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 ] = 1 E (p j F j 1 2 a2 e 3, where F j is the force vector at the end, e 3 =(,, 1, (resp. + holds when g(p j = (resp. g(p j =, and E (p j = (Im(c 1,Im(c 2,. Denote by ψ a conjugate minimal immersion to ψ. ψ is not well-defined on M {p 1,...,p n }, so we can consider it defined on a certain covering of this surface. Each catenoid type end of ψ produces a helicoidal type end of ψ, which has a vertical axis defined as the axis of the Helicoid asymptotic to the end. This vertical axis is not well-defined in R 3, but the relative position between all the axes defined in this way does not depend on the lifting of ψ.asψ = Im Φ, the axis of a helicoidal type end p j of ψ is given by E (p j + < F j >. Note that E (p j F j catenoidal ends is not multivalued, because F j =. Now we can state the following

9 Riemann bilinear relations on minimal surfaces 315 Lemma 1. Let ψ : M {p 1,...,p n } R 3 be a proper minimal immersion with finite total curvature and embedded ends with limit normal vectors in the direction of the x 3 -axis. Then F(γ i F(Γ i = N (p j T j planar ends catenoidal ends F j E (p j +π ( C aj 2 aj 2 e 3, (11 C where C (resp. C denotes the set of catenoid type ends where the extended Gauss map points to the South Pole (resp. the North Pole. Remark 1. As the residue of the Weierstrass form Φ at a planar end is zero, the above Lemma can be directly extended to the case in which only the catenoid type ends are parallel. The second term of the right-hand side of (11 is horizontal, and the same holds for the first one provided that all the ends are parallel. Taking in this case the vertical component of (11 we will have a scalar equation in terms of ψ: Theorem 1. Let ψ : M {p 1,...,p n } R 3 be a proper minimal immersion with finite total curvature and embedded ends with limit normal vectors in the direction of the x 3 -axis. Then, ( F(γ i F(Γ i, e 3 = π aj 2 aj 2. (12 C C Equation (12 gives information about the Force map associated to a complete minimal embedding ψ : M R 3 with finite total curvature, defined as the R 3 - valued cohomology class on M determinated by the closed one form (dψ J, where J is the complex structure of M. The maximum principle implies that the plane is the only example with rank(force =, and the Catenoid can be characterized as the only example with rank one [11]. In the rank two case, all the forces are contained in the same vertical plane in R 3 up to a rotation, we can assume that the Gauss map extends vertically to the ends,so the left-hand side of (12 vanishes. This gives an obstruction on the allowed distribution of ends: Corollary 1. Let ψ : M {p 1,...,p n } R 3 be a properly embedded minimal surface with finite total curvature and horizontal ends. Then,

10 316 J. Pérez 1. If rank(force =2, then C a 2 j = C a 2 j. 2. If rank(force =2and n is odd, then the catenoid type ends can be associated by pairs with the same logarithmic growth, so under the natural x 3 -increasing order we find b pairs of catenoid type ends with negative logarithmic growth, p planar ends with p odd and t pairs of catenoid type ends with positive logarithmic growth, with n =2b + p +2t. 3. If ψ has three ends, then rank(force =3. Proof. The comment before the statement of this Corollary proves 1. By writing the ends under the x 3 -increasing order we find logarithmic growths a 1... a r < b n r... b 1, where a 1 is the logarithmic growth of the lowest end and b 1 is defined as well for the highest end. Note that the possible planar ends are included in the a-values. As n is odd, the ends corresponding to a 1, b 1 have the same limit normal, so 1 can be written as (a 2 1 a 2 2 +(a 2 3 a (b 2 1 b 2 2+(b 2 3 b =. This equation and the above ordering of the a, b-coefficients yield the second assertion. 3 is a direct consequence of 2. This completes the proof. Take a proper minimal immersion ψ : M {p 1,...,p n } R 3 with finite total curvature. Suppose that all the ends are planar with minimum branching order of the extended Gauss map. From (9 and (11 we have n T j =, (13 F(γ i F(Γ i = n N (p j T j. (14 Suppose that the left-hand side of (14 is zero. So, N (p n = 1 T n 2 T n ( n 1 N (p j T j.ifn (p 1,...,N (p n 1 are parallel, then N (p n = 1 T n 2 T n (N (p 1 T = 1 [ Tn T n 2, T N (p 1 T n, N (p 1 T ], where T is a linear combination of T 1,...,T n 1. As (13 gives that the last term in the right-hand side vanishes, all the ends are parallel and the maximum principle insures that ψ is a plane. So, we have

11 Riemann bilinear relations on minimal surfaces 317 Proposition 1. Let ψ : M {p 1,...,p n } R 3 be a proper minimal immersion with finite total curvature, such that all its ends are planar with minimum branching order of the Gauss map. If F(γ i F(Γ i =, (15 then ψ can not have n 1 parallel ends. Hence the minimum number of planar ends with minimum branching order is three, and two of them can not be parallel provided that (15 holds. Moreover, a complete minimal immersion in R 3 with finite total curvature, exactly three planar ends of minimum branching order and verifying (15 will provide a solution of the following system we use the same notation we are working with : { T1 + T 2 + T 3 =, N (p 1 T 1 + N (p 2 T 2 + N (p 3 T 3 =. We claim that every solution of the above system has two of the vectors N (p i colinear. A sketch of the proof of this claim is as follows. If T 1, T 2 are colinear we directly deduce from the equations that N (p 1 and N (p 2 are also colinear. If T 1, T 2 span a plane U, the system insures that we can find an isometry f of R 3 mapping each T i in N (p i T i, i =1, 2, 3. It follows that f (x, x = for all x U. Now it is easy to prove that f leaves U invariant, so N (p 1, N (p 2 are in the orthogonal space U, hence they are colinear. Now we have Proposition 2. There are no complete minimal immersions in R 3 with finite total curvature and exactly three planar ends of minimum branching order verifying (15. Remark 2. As a consequence of (14 we can prove that a properly immersed minimal torus in R 3 with finite total curvature and all its ends of planar type with branching order greater than two has all its forces in only one direction, so admits a deformation that preserves the conformal structure and the ends type up to a rotation so that the forces are vertical, the deformation ψ λ (λg, 1 λ ω, λ>, see [11], can be applied to ψ. 3 Minimal surfaces in R 3 /T with planar ends Let M = M {p 1,...,p n } R 3 /T be a properly embedded minimal surface with finite total curvature and horizontal planar ends, where M is a compact genus k Riemann surface and T is a translation of vector P R 3 {}. Using the notation of Sect. 1, from (4 and (5 we have

12 318 J. Pérez Φ γ i Φ = i Γ i n Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2. (16 Separating real and imaginary parts in the above equation, and taking into account (6, (1 we can state F(γ i F(Γ i = n T j N (p j, [P(γ i F(Γ i +F(γ i P(Γ i ] = n T j, where P(γ i =Re γ i Φ, F(γ i =Im γ i Φ are the period and the force vectors associated to the curve γ i respectively, and we use a similar notation for Γ i. Also, T j denotes the torque vector associated to the end p j of M, which lifts to a planar end in R 3. Note that T j does not depend on the chosen lift because the force vector at a planar end vanishes. The last formula can be simplified by choosing a suitable homology basis {γ 1,...,γ k,γ 1,...,Γ k } of M : M produces a singly periodic minimal surface S R 3. Take a fundamental domain on S bounded by two horizontal planes Π 1 and Π 2 = Π 1 + P. We can choose those planes in such a way that S Π 1 consists in a finite number of Jordan curves γ 1,...,γ s. The curves γ i and γ i + P determine the same non nulhomotopic curve in M, which we also call γ i. On the other hand, the fundamental domain bounded by Π 1,Π 2 on S is a non compact surface with boundary and genus k s. Take a point q i on each γ i, and a curve Γ i contained in joining q i with q i + P, 1 i s. Now we can choose a homology basis of, γ s+1,...,γ k,γ s+1,...,γ k, such that γ i,γ j, i,,...,k satisfy the properties of Sect. 1. Moreover, So we have P(γ 1 =...= P(γ s = P(γ s+1 =...= P(γ k =, P(Γ s+1 =...= P(Γ k =, P(Γ 1 =...= P(Γ s = P. Theorem 2. Let M = M {p 1,...,p n } R 3 /T be a genus k properly embedded minimal surface with finite total curvature and n horizontal planar ends, T being a translation of vector P R 3 {}. Then, F(γ i F(Γ i = n N (p j T j, P F(C = n T j, (17 where F(C is the force of M along a cycle C obtained by cutting the surface with a horizontal transversal plane.

13 Riemann bilinear relations on minimal surfaces 319 The Riemann examples {R λ } λ> see [2, 3] satisfy the hypotheses of the above Theorem. These are tori with two planar ends. Moreover, each R λ is symmetric by the reflection respect to a vertical plane and respect to a straight line that cuts this plane orthogonally permuting the ends, so the torque vectors associated to the ends are the same. Hence, from (17 we deduce that the forces F(γ and F(Γ are colinear. In fact, it can be proved that one of these vectors vanishes. This planar symmetry characterizes the Riemann examples: Theorem 3. Let M = M {p 1, p 2 } R 3 /T be a properly embedded genus one minimal surface with two planar ends. If M is invariant by a reflection in a plane of R 3, then it is a Riemann example. Proof. We can suppose that the Gauss map g of M verifies g(p 1 =,g(p 2 =. Moreover, from [9] we deduce that g has degree two. So g has minimum branching order at both ends, and (17 says that Up to conformal transformations, we can put F(γ F(Γ =e 3 (T 2 T 1. (18 M = {(z,w C 2 /w 2 = z(z a(bz +1}, g(z,w=z, (z,w M, (19 where a, b C {}. Hence, p 1 =(,, p 2 =(,. As φ 3 is holomorphic on M, we deduce that φ 3 = A dz w, with A C {}. On the other hand, M yields a properly embedded singly periodic minimal surface S R 3, which is symmetric respect to the reflection in a plane Π. As the ends are horizontal, Π only can be horizontal or vertical. The first case can not occur, because in this situation the force of S could be computed as the force along Π S and this last vector is vertical, which contradicts [11]. So Π has to be vertical, and up to a rotation around the x 3 -axis we can suppose that Π = {x 2 =}. The reflection in Π produces an antiholomorphic involution B on M such that B φ 1 = φ 1, B φ 2 = φ 2, B φ 3 = φ 3. ( Now is easy to conclude that B(z,w= (z, AA w. Hence, a(bz +1, (z,w M, and using (19 we obtain A A 2 w 2 = z(z ( 2 A (z a(bz +1=(z a(bz +1, z C. (2 A Evaluation at z = a and z = 1 b gives (a a(ab +1=, (b b(ab +1=. (21 Comparing coefficients in (2 we have A2 a, A2 b R. On the other hand, (6 gives expressions for the torque vectors at the ends:

14 32 J. Pérez T 1 = 4π (, A2 a,, T 2 = 4π (, A2 b,. So (18 yields ( A 2 F(γ F(Γ =4π b A2 a,,. (22 S cuts to the plane {x 3 = t} in an embedded curve γ t which is closed for t distinct of the end heights. By the symmetry assumption, each compact curve γ t cuts the symmetry plane Π twice. For t equal to an end height, γ t only cuts once to Π one of the points of γ t Π diverges as t approaches to an end height. We can define in this way two curves Γ 1 = Γ 1 (t,γ 2 = Γ 2 (t contained in Π, which can be viewed as closed curves in M passing through the ends. Now take γ = γ t, one of the above compact curves, so Γ 1 cuts once to γ, {γ,γ 1 } is a canonical homology basis for M and B [γ] = [γ], B [Γ 1 ]=[Γ 1 ], Re Φ =. Choose a curve Γ in the same homology class of Γ 1, by modifying slightly Γ 1 in a neighbourhood of each end in such a way that the integral formula can be applied to {γ,γ}. Moreover, Φ = B ( Φ = φ1, φ 2, φ 3, γ γ γ hence F(γ =(α,,β for α, β R. Analogously we can obtain F(Γ =(,δ,, with δ R. As a consequence, F(γ F(Γ =( βδ,,αδ. But α/=, because F(γ can not be vertical see [11],soδ = and from (22 we have a = b. If ab +1 /=, from (21 we have a = b R, soa 2 R and M is a Riemann example. If ab + 1 =, then A2 A2 a = b 2 b, so (22 contradicts that δ =. This proves the theorem. γ Remark 3. López, Ritoré and Wei [7] have improved recently this result by removing the symmetry condition. 4 Minimal surfaces in R 3 /T with helicoidal type ends Let M = M {p 1,...,p n } R 3 /T be a genus k properly embedded minimal surface with helicoidal type ends, where T is a translation of vector P R 3 {}. We can suppose that the Gauss map extends vertically through the ends. Hence, the period vector at each end p j is vertical and non zero, the force vanishes, φ 1,φ 2 have no residues at the ends, and we can take a conformal coordinate z around p j such that φ 3 = iβ dz, (23 z

15 Riemann bilinear relations on minimal surfaces 321 where β is a real number that measures the slope of the Helicoid asymptotic to the end, see [9]. As a consequence, (4 and (5 can be applied by integrating φ 1, φ 2, and (16 holds. As the Gauss map g has a zero or a pole at each end p j,it can be locally expressed as g(z =z k t(z, or g(z =z k t(z, (24 where k N is the winding number of the end and t is a holomorphic function satisfying t( /=. From (23 and (24 it is straightforward to obtain expressions for the residues that appear in (16: ( iβc 2, iβc 1, iβ2 2k if g(p j = k, Res pj (h 2 φ 3, h 1 φ 3, h 1 φ 2 = (25 ( iβc 2, iβc 1, iβ2 if g(p j = k, where c 1, c 2 are complex numbers obtained by integration of φ 1,φ 2, respectively. As M is embedded, all the end slopes are the same up to sign, and all the winding numbers coincide. As Res pj φ 3 =, we have the same number of ends with positive slope β>and of ends with negative slope β. We claim that the ends with g(p j = have the same slope, say β, and the ends with g(p j = have the opposite slope. At an end where g(p j =, (24 holds, and calling F(z = 1 z 1 2 g φ 3 = c + 2k iβ 2t(kz k + O G(z = 1 z gφ 3 = c + O (z k, 2 the projection of M over the (x 1, x 2 -plane is given by (x 1 + ix 2 (z =(F G(z =c iβ 2kt(z k + O At an end with g(p j =, a similar argument yields (x 1 + ix 2 (z =c iβt( 2kz k + O ( 1 z k 1 ( 1 z k 1, ( 1 z k 1. and in both cases the x 3 -coordinate is x 3 (z =k +βarg(z. If we had two helicoidal type ends with the same slope but opposite limit normal vectors, then the traces of those ends on a vertical cylinder C R of radius R large enough would consist in two curves very close to helices of slope β on C R this is clear if both ends have the same axis; if not, take R much bigger than the distance between the two axes so we can repeat the same argument, and these two helices would have x 3 -coordinate increasing, but their projections on the (x 1, x 2 -plane would rotate in opposite directions, so they would intersect, a contradiction. Thus we can suppose that the ends with g(p j = have positive slope β and the opposite slope is taken by those ends with g(p j =. Denote by k the common winding number at the ends. Then, from (16 and (25 we obtain,

16 322 J. Pérez [ ] Φ Φ = (βc 2,j, βc 1,j, (βc 2,j, βc 1,j,, (26 γ i Γ i E E where c 1,j, c 2,j C are obtained by expressing locally h 1, h 2 around the end p j and E (resp. E denotes the set of ends where g(p j = (resp. g(p j =. Put E(p j = ( Re(c 1,j, Re(c 2,j,, the axis of a helicoidal type end is E(p j + < P j >, where P j is the period vector at the end. In our situation, P j =(,, ±β, where + holds if and only if g(p j =, and taking real parts in (26 we deduce that F(γ i F(Γ i = E(p j P j + E(p j P j. E E Writing all the end periods in terms of one fixed period vector P, it follows that Theorem 4. Let M = M {p 1,...,p n } R 3 /T be a properly embedded genus k minimal surface with n helicoidal type ends, T being a translation of vector P R 3 {}. Then, F(γ i F(Γ i = E(p j E(p j P, EP E P where E P (resp. E P denote the set of ends with period vector P (resp. P. 5 Minimal surfaces with Scherk type ends Let S R 3 be a properly embedded doubly periodic minimal surface. We can suppose that the lattice generated by the translations leaving invariant to S consists of horizontal vectors. Take a fundamental domain D in S. By glueing related edges of D we will obtain a properly embedded minimal surface M in the quotient T R of R 3 over two independent translations that generate the lattice. For the remainder, we will suppose that this last surface is orientable and has finite topology. Then we can write M = M {p 1,...,p n }, where M is a compact genus k Riemann surface and p 1,...,p n are the ends of M. Take a homology basis {γ 1,..., γ k,γ 1,...,Γ k } of M such that the conditions in Sect. 1 hold, and cut M along those curves in the above basis so that the remaining closed curves do not produce periods of M, obtaining a compact surface with boundary possibly empty M 1, which contains in its interior to the ends. Consider n curves β 1,...,β n starting from the same point x M 1 if M 1 is empty, choose x M 1 {p 1,...p n }, joining x with each end, such that the β-curves do not cut to the curves in the original basis unless possibly at x. Now cut M 1 along this curves, obtaining a new compact surface with boundary M 2, such that we can lift M 2 to a well-defined minimal embedding ψ into R 3. Moreover, this process can be done in such a way that ψ(m 2 =D.

17 Riemann bilinear relations on minimal surfaces 323 In the above situation, M has finite total curvature, separates T R in two components, n is even, and the Gauss map g of M extends in a holomorphic and non zero way to the ends. Each end can be conformally parametrized by D = {z C / < z 1}, and in this parametrization, the third coordinate function can be written x 3 (z =k ln z + k, z D, where k, k R, k /=. When k < (resp. k > the end is called a top end (resp. a bottom end. From the last equation, φ 3 = k z dz, where Φ =(φ 1,φ 2,φ 3 is the Weierstrass one-form of M which is well-defined on M. The intersection of any top end with T {t} converges, as t, to a geodesic on T with length P, where P denotes the period vector associated to the end. P does not depend on the top end, and the embedding property of M also insures that all the P-vectors at the top ends are the same up to sign. The same holds for the bottom ends. Moreover, the number of top ends (resp. bottom ends is even, which will be denoted by 2n(T (resp. 2n(B. For more details, see the work of Meeks and Rosenberg [8]. Denote by W the component of (T R M where the S 2 -Gauss map points to. M ( T {t}, t, is as close as we want to a set of 2n(T parallel geodesics in T {t}. If we assign to each geodesic the value of the Gauss map at the corresponding end we can associate the top ends in n(t pairs, each one consisting in two ends with opposite limit normal, which traces in T {t} are the boundary of a component of W ( T {t}.ast, this component converges to a horizontal strip with boundary consisting in the two parallel geodesics obtained as the limit of the level sets of the corresponding ends. In this way we have obtained n(t parallel top strips. Calling S T to one of these strips, we can define its width W (S T as the distance measured in T {t} between the geodesics in S T. The bottom strips and their widths are defined as well. For instance, the top strips and the bottom strips in the classical doubly periodic Scherk surfaces or in any covering of these examples have the same width, but this is not the case in Wei s examples [14]. Now we will obtain an analytic expression for the width of a top or a bottom strip. Take an end p parametrized by g(z =a + z m t(z, φ 3 = kz 1 dz, z D, where a C {}, k R {}, and t is a holomorphic function in D such that t( /=. Hence the residue R of Φ =(φ 1,φ 2,φ 3 at the end is R = ( ( a a ( k, i 1 2 a + a k, k,so the period and the force vectors are respectively given by P πk ( ( 1 i a + a, k 1 N 2 3 (N 2, N 1,, (27 ( 1 F πk a a, 2 k ( N1 1 N3 2 N 3, N 2 N 3, N3 2 1, (28 where N (p = (N 1, N 2, N 3 is the value of the Gauss map at the end. As a consequence, ( 1 P = π k a + a k =( 1. (29 1 N 2 2 3

18 324 J. Pérez As two ends of the same type i.e. both top ends or both bottom ends have the same P and the same N3 2, from (29 their k-coefficients will coincide. Denote by k T < (resp. k B > the k-coefficient of any top end (resp. bottom end. From (27 and (28 it follows that two ends of the same type have equal forces, and their period vectors coincide if and only if the Gauss map takes equal values at the ends. Otherwise, the period vectors are opposite. Let F T (resp. F B bethe force at any top end (resp. bottom end. As the total force at the ends vanishes, n(t F T + n(bf B =. (3 Moreover, it is well-known [8] that n(t =n(b provided that the top ends are parallel to the bottom ends. In this situation, F T = F B and from (28, k T = k B. On the other hand, if the ends are not parallel then they are vertical and (28 gives F =ke 3, thus from (3, n(t k T + n(bk B =, which generalizes the equation obtained in the parallel case. Also note that if all the ends are parallel, then the period vectors associated to ends of distinct types coincide if and only if their limit normal vectors are opposite. We can write M locally at an end as z ψ(z = Re Φ =Re ( C + log(zr + O (z = Re(C ln z F + arg(zp + Re(O (z, (31 where C =(c 1, c 2, c 3 C 3 and O (z isac 3 -valued holomorphic map that vanishes at z =. Suppose that we are working at a top end. For t, the trace of the end on T {t} has the form ψ(ρe iθ =α(θ+re ( O (ρe iθ, θ [, ], where ρ ], 1[ depends on t and α(θ = Re(C + 1 θ ln ρf + P, θ [, ] is a parametrization of a horizontal segment at height t, in the direction of P and 1 with length P. Henceforth, α(θdθ is the position vector of the center of that segment, that depends on t. Calling N (p to the limit normal vector, the real number d = 1 α(θdθ, N (p = 1 [ ψ(ρe iθ Re ( O (ρe iθ ] dθ, N (p measures the signed distance from the origin in R 3 to the limit affine tangent plane at the end, viewed in the fundamental domain D of the doubly periodic surface S as in the beginning of this Section. Clearly this distance does not depend on t, hence we can take limits as ρ goes to zero: d = lim 1 ψ(ρe iθ dθ, N (p. (32 ρ The above distance depends on the choice of the coordinate origin, but if we take two top ends p 1, p 2 which determine a top strip S T and call d 1, d 2 to the corresponding distances defined as above, the width of the strip can be computed as

19 Riemann bilinear relations on minimal surfaces 325 W (S T =d 1 + d 2, (33 which does not depend on the choice of the origin. Now suppose that all the ends are parallel, so n(t =n(b. We can group the ends in 2n(T pairs, in such a way that each pair consists in two ends of distinct types with the same limit normal vector: Consider a fundamental domain D in the doubly periodic surface S. Let Π be a plane parallel to the limit tangent plane at the ends, such that Π S is transversal. We shall denote by {Π d / d R} the foliation of R 3 by parallel planes to Π with distance(π d,π =d, d R. We can suppose that D d> Π d by choosing a suitable Π. So, there exist positive real numbers d1 T < d 2 T <... < d 2n(T T, d 1 B < d 2 B <... < d 2n(T B, such that Π d1 T,...,Π d2n(t T (resp. Π d B 1,...,Π d B 2n(T coincide with the limit affine tangent planes at the top ends (resp. bottom ends that lie in D. Also we can choose D in such a way that the ends corresponding to Π d T 1, Π d B 1 have the same limit normal vector. Hence the same holds for the ends corresponding to Π d T j, Π d B j,1 j 2n(T. These are the pairs we talked about above. Note that each pair defines a slab S j R 3 which boundary is Π d T j Π d B j, and the width of this slab can be computed as A(S j =dj T dj B, understanding that the width could be negative or zero, with the obvious meaning. A(S j can be obtained from (32: Calling pj T,pj B to the top and the bottom ends that bound the slab S j and d T,d B to the corresponding distances defined at (32, then A(S j =d T d B. ( General posing In a properly embedded minimal surface in T R or R 3 /T with Scherk type ends, we can always assume that φ 1, φ 2, φ 3 have simple poles at each end. As we want to study (4 in both cases, we will work in a general situation that includes these two particular cases. Let M be a compact genus k Riemann surface, and let φ a,φ b be two meromorphic differentials on M satisfying 1. Poles(φ a = Poles(φ b ={p 1,...,p n }. 2. At each p j, φ a and φ b have simple poles. Using the same notation as in Sect. 1, (4 gives ( ( n φ a φ b φ a φ b = φ a φ b + h a φ b. γ i Γ i Γ i γ i δ j β j δ j (35 Let us study the right-hand side of (35. Let (D (1 + ε, z be a conformal coordinate centered at p j. Then φ a = ( R j (az 1 + f a,j (z dz, z D (1 + ε, where R j (a = Res pj φ a and f a,j is a holomorphic function in D(1 + ε. A primitive h a of φ a can be written as h a (z =c a,j + R j (a log(z+o a (z, (36

20 326 J. Pérez where c a,j C, O a (z is a holomorphic function that vanishes at z = and log(z is the branch of the logarithm defined on D(1 + ε {z R / z < 1+ε} satisfying (see Fig. 1 log(origin of β j =i, log(end of β j =. Similar expressions can be written for φ b. Hence, δ j φ a Fig. 1. φ b + h a φ b =ir j (a [ h b (end of β j h b (origin of β j ] + β j δ j +ic a,j R j (b 2 R j (ar j (b+r j (a log(zf b,j (zdz. { z =1} Calling F b to a primitive of f b,j in D(1+ε, it follows that { z =1} log(zf b,j (zdz = i [F b (1 F b (]. (36 gives that we can take F b (z =c b,j + O b (z, so taking into account how we have chosen the branch of the logarithm, log(zf b,j (zdz = i [h b (end of β ] j c b,j z =1 = i [ ir j (b+h b (origin of β j c b,j ]. On the other hand, h b (end of β j = h b (x +i h j R h(b. The last three ( equations let us write (35 as φ a φ b φ a φ b = γ i Γ i Γ i γ i = i n ( ca,j R j (b c b,j R j (a + 2 ( Rj (ar h (b R h (ar j (b. (37 h<j Now take three meromorphic differentials φ 1,φ 2,φ 3 on M, such that 1. Poles(φ 1 = Poles(φ 2 = Poles(φ 3 ={p 1,...p n }, 2. At each p j, φ i has a simple pole, 1 i 3, 1 j n.

21 Riemann bilinear relations on minimal surfaces 327 For the remainder of this Section we will use the following notation: Φ =(φ 1,φ 2,φ 3, H =(h 1, h 2, h 3 = Φ, R j =(R j (1, R j (2, R j (3 = Res pj Φ P j =Re δ j Φ = Im(R j F j =Im 1 j n δ j Φ =Re(R j P(γ i =Re } γ i Φ F(γ i =Im and the same for Γ γ i Φ i, 1 i k (38 Locally at a pole p j, H can be written as H (z =C j + log(zr j + O (z, (39 where C j =(c 1,j, c 2,j, c 3,j C 3, log(z is the branch of the logarithm we are working with and O (z isac 3 -valued holomorphic map that vanishes at z =. Now we can write (37 as n Φ Φ = i C j R j + 2 R j R h. (4 γ i Γ i h<j Note that C j depends on the conformal coordinate z, but this does not occur with C j R j. 5.2 Doubly periodic case Take M,S and D as in the beginning of this section. Using the above notation, (39 yields C j R j = 1 C j R j dz = 1 H (ρe iθ dθ R j. i z { z =ρ} Observe that the above expression does not depend on ρ. By separating in real and imaginary parts, ( Re(C j R j = 1 4π 2 ψ(ρe iθ dθ F j + ψ (ρe iθ dθ P j, (41 ( Im(C j R j = 1 4π 2 ψ(ρe iθ dθ P j + ψ (ρe iθ dθ F j, (42 where ψ denotes the conjugate immersion. Moreover, R j R h = 1 4π 2 P j P h i ( Fj 4π 2 P h + P j F h, where we have used (3 in order to cancel F j F h. If now we separate (4 in real and imaginary parts,

22 328 J. Pérez [P(γ i P(Γ i F(γ i F(Γ i ] = ( n ψ(ρe iθ dθ P j = h<j P j P h [P(γ i F(Γ i +F(γ i P(Γ i ] = ( = 1 n ψ(ρe iθ dθ F j + 1 ( Fj P h + P j F h. 2 h<j ψ (ρe iθ dθ F j ψ (ρe iθ dθ P j We can also suppose that p 1,...,p 2n(T are the top ends, and p 2n(T +1,...,p n the bottom ends. Hence the last term in the right-hand side of (43 vanishes. On the other hand, we can remove in (43, (44 those terms with ψ by considering the components of the formulae in suitable directions: Let F = n(t F T = n(bf B. Then, P(γ i P(Γ i F(γ i F(Γ i, F = n(t 1 top ends +n(b 1 bottom ends On the other side, (27,(28,(29 give ψ(ρe iθ dθ, P j F T ψ(ρe iθ dθ, P j F B. P j F T = P j 2 N (p j if p j is a top end, P j F B = P j 2 N (p j if p j is a bottom end. So the right-hand side of the last expression is n(t P 2 T top ends n(b P 2 B 1 1 bottom ends ψ(ρe iθ dθ, N (p j } ψ(ρe iθ dθ, N (p j, (43 (44 (45 where P T is the length of a period vector at a top end, and P B is defined as well at the bottom ends. Taking limits as ρ goes to zero, (32 and (33 give

23 Riemann bilinear relations on minimal surfaces 329 P(γ i P(Γ i F(γ i F(Γ i, F = n(t P 2 T d j n(b P 2 B d j = = n(t P 2 T n(t top ends n(b W (Sj T n(b P 2 B W (S B j, bottom ends where W (S T j,w (S B j denote the widths of the top strips and of the bottom strips, respectively. Now if we want to remove the ψ -terms in (44 we need all the period vectors at the ends to have the same direction. So let us suppose that all the ends are parallel. Hence n(t =n(b and F T = F B. Fix an end, say a top one p 1, and let P 1 be its period vector. It follows from (44 that P(γ i F(Γ i +F(γ i P(Γ i, P 1 = n 1 ψ(ρe iθ dθ, F j P 1. And from (45 that P j 2 N (p j if p j is a top end with N (p j =N(p 1, or p F j P 1 = j is a bottom end with N (p j = N(p 1, P j 2 N (p j if p j is a top end with N (p j = N(p 1, or p j is a bottom end with N (p j =N(p 1. Hence, P(γ i F(Γ i +F(γ i P(Γ i, P 1 = = P 1 2 T+ 1 ψ(ρe iθ dθ, N (p j +, +,,, T B + B where T + (resp. T denotes the set of top ends with N (p j = N (p 1 (resp. N (p j = N (p 1, and B +, B are defined as well in the set of bottom ends, and all the inner products have the same argument. Taking limits as ρ goes to zero and using (34, the right-hand side becomes P 1 2 N (p j =N (p 1 A(S j + N (p j = N (p 1 A(S j, where A(S j is the width of the slab S j defined in the beginning of this section and the above sum is running in the set of ends. Now we can state the following Theorem 5. Let M = M {p 1,...,p n } T R be a properly embedded genus k minimal surface. Then,

24 33 J. Pérez i P(γ i P(Γ i F(γ i F(Γ i, F = = n(t P 2 T n(t n(b W (Sj T n(b P 2 B W (S B j, where 2n(T is the number of top ends, 2n(B is the number of bottom ends, F = n(t F T = n(bf B and W (Sj T, W (Sj B are the widths of the corresponding top strips or bottom strips, respectively. ii Moreover, if all the ends are parallel, P(γ i F(Γ i +F(γ i P(Γ i, P 1 = P 1 2 ends with N (p j =N (p 1 A(S j ends with N (p j = N (p 1 A(S j, where A(S j denotes the width of each slab determined by two parallel ends with the same normal vector. 5.3 Singly periodic case Let S R 3 be a non flat properly embedded minimal surface invariant by a translation T of vector P R 3 {}, which we will suppose vertical. S induces in a natural way an orientable properly embedded minimal surface in the quotient R 3 /T. Assume that this surface has finite topology, so from [9] it is conformally a finitely punctured compact Riemann surface, say of genus k, M = M {p 1,...,p n } R 3 /T, where p 1,...,p n are the ends of M. Suppose that these ends are of Scherk type, i.e. they are asymptotic to flat vertical annuli. Up to a suitable rotation around the x 3 -axis in R 3 we can insure that the differentials φ 1,φ 2,φ 3 of the Weierstrass form of M which are well-defined in this case have simple poles at the ends, see [9]. As the period vector at each end p j is vertical, Res pj (φ 1,φ 2 R 2 and we can take a conformal coordinate (D, z centered at p j such that φ 3 = ( R j (3z 1 + f 3 (z dz, where f 3 is a holomorphic function in D. The limit normal vector has to be horizontal, say g(p j =e iθ, θ R. Hence g can be written locally as g(z =e iθ +O (z p for certain p N and we have Res pj (φ 1,φ 2 =ir j (3( sin θ, cos θ, thus R j (3 ir. Put R j (3 = ai, a R {}. Using the same notation as in (38, } R j = a(sin θ, cos θ, i, (46 P j =(,, a, F j =a(sin θ, cos θ,. As a consequence, all the a-coefficients are the same up to sign. We can talk about consecutive ends by cutting the singly periodic surface S with a vertical

25 Riemann bilinear relations on minimal surfaces 331 cylinder C R of radius R large enough so the end traces on C R are very close to vertical straight lines. As the force vector at each end points to the simply connected component of R 3 C R and the Gauss map points to a fixed component of R 3 S, consecutive ends have opposite a-coefficients and opposite period vectors. So, the number of ends is even and we have as much ends with positive a-coefficient as with negative one. Also note that from F j = and (46 we have n a j N (p j =, (47 where a j is the corresponding a-coefficient at the end p j. Denote by a > the positive a-coefficient. (31 remains true, and the number d j defined in (32 measures the signed distance from the origin in R 3 to the limit affine tangent plane of the lifted surface at the end p j. Such a distance depends on the choice of the origin, but we know from (47 that this does not happen with d j d j. (48 P j,e 3 > P j,e 3 < The above expression also has a physical meaning in terms of the resultant of a system force. For, consider a vertical cylinder C R of radius R large enough so S only cuts to C R in a neighbourhood of each end. Hence we have a non compact minimal surface S R S lying in the convex component of R 3 C R, whose boundary consists of n curves on C R similar to vertical straight lines, and each force expresses the stress produced by the corresponding end on S R. Take an end p j with period and force vectors P j, F j respectively. The momentum of F j respect to the x 3 -axis is given by M j = d j N (p j F j = d j P j, where we have used (46, and the resultant momentum is n M j = d j P, P j,e 3 > P j,e 3 < provided that P, e 3 >. Thus a balancing condition on S will be stated by proving n F j =, n M j =. The first condition is clearly satisfied, while the second one is a consequence of the following d j Theorem 6. Let M = M {p 1,...,p n } R 3 /T be a genus k properly embedded minimal surface, T being a translation of vector P R 3 {}, with Scherk type ends. Then, d j = d j, P j =P P j = P where P j is the period vector at each end p j and d j is the signed distance from the origin to the limit affine tangent plane at p j defined in (32.