The doubly periodic Scherk-Costa surfaces

We present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before.


Introduction
In the Euclidean three-space, among all complete embedded minimal surfaces known to date, on the one hand the doubly periodic class still remains less numerous in examples. On the other hand, the richness in the triply periodic class counts to a great extent on the Conjugate Plateau Construction, a powerful tool but not always applicable (see [RB2]) or extendable to infinite frames (see [JS]). Recently, the non-periodic class became also very rich due to important works like [Kp] and [T3]. Through [T1-2] and [Web2] the same happened to the singly periodic, which was already quite numerous with tree possible kinds of ends: planar, Scherk or helicoidal.
In the 20 th century, the known examples were obtained thanks to their high order symmetry groups, a resource already used up nowadays. Therefore, potentially new examples normally lack in symmetries, which makes it so hard to prove their existence. One might opt for keeping a high order symmetry group with an increase in the genus, but this leads to the same hurdle, namely too many period problems.
Some modern constructions do close many periods at once, like in [Kp], [T1-3] and [Web2]. However, such methods are not portable outside the class and types of ends they direct to. For at most three period problems, however, it is still feasible to handle the Weierstraß Data together with adequate methods (see [BRB], [L], [LRB], [MRB] and [Web1]). By the way, in our paper we apply the limit-method described both in [L] and [LRB].
However, what is the purpose of constructing a new minimal surface? The following reason motivates this present work. Minimal surfaces model many structures, like crystals and co-polymers, but several symmetry groups are not yet represented by any them (see [H] and [LM]). As explained above, the doubly periodic class still lacks in examples, even after very rich works like [HKW], [K1], [W] and [PRT]. The purpose of this paper is then to present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before. For the converse, there are symmetry groups that admit more than one representative, even restricted to a certain conformal type (see [RB3]). Although not embedded, these examples easily hint at embedded ones.
Our surfaces are inspired in the examples called L b in [RB4,p 482]. By taking the picture of L b in that page, if one replaces the catenoidal ends by curves of reflectional symmetry parallel to Ox 1 x 2 , the resultant surface will then come out as in Figure 1. Its symmetry group is D 2d or42m in Schönflies' or simplified Hermann-Maugin notation, respectively. The same procedure for C b from [RB4,p 483] could also result in a new doubly periodic example. However, it would then have the same symmetry group as M π/4,π/2,0 from [PRT]. Of course, they would differ in genus, but one still might go round it by adding handles to the latter, a widely applicable technique. That is why our paper is totally devoted to the example in Figure 1.
We formally state our result in the following theorem: Theorem 1.1. There exists a one-parameter family of complete embedded doubly periodic minimal surfaces in R 3 such that, for each member of this family the following holds: (a) The quotient by its translation group has genus three. (b) The surface is generated by a fundamental piece, which is a surface with boundary in R 3 . The fundamental piece has two Scherk-ends (modulo translation), and a symmetry group generated by 180 • -rotations around a straight line and 180 • -rotations around a straight segment. The segment crosses the line orthogonally and both determine a plane Π. (c) The boundary of the fundamental piece consists of two parallel lines in Π and two planar closed curves of reflectional symmetry. The curves are parallel to but not contained in Π, and one is the image of the other under the symmetries of the fundamental piece. By successive 180 • -rotations around the lines of the boundary and reflections in the closed curves, one generates the doubly periodic surface. This work refers to part of our doctoral theses [L] and [RB0], supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and DAAD (Deutscher Akademischer Austausch Dienst), respectively. Professor Hermann Karcher, from the University of Bonn in Germany, was adviser of the second author, who thanks him for his dedication, which greatly helped in the realisation of this work. The second author was advisor of the first.

Preliminaries
In this section we state some well known theorems on minimal surfaces. For details, we refer the reader to [K2], [LMa], [N] and [O]. In this paper all surfaces are supposed to be regular.
Theorem 2.1. (Weierstraß representation). Let R be a Riemann surface, g and dh meromorphic function and 1-differential form on R, respectively, such that the zeros of dh coincide with the poles and zeros of g. Consider the (possibly multi-valued) function X : R → R 3 given by Then X is a conformal minimal immersion. Conversely, every conformal minimal immersion X : R → R 3 can be expressed as above for some meromorphic function g and 1-form dh.
Definition 2.1. The pair (g, dh) is the Weierstraß data and φ 1 , φ 2 , φ 3 are the Weierstraß forms on R of the minimal immersion X : R → X(R) = S ⊂ R 3 .
Definition 2.2. A complete, orientable minimal surface S is algebraic if it admits a Weierstraß representation such that R = R \ {p 1 , . . . , p r }, were R is compact, and both g and dh extend meromorphically to R. From now on we consider only algebraic surfaces. The function g is the stereographic projection of the Gauß map N : R → S 2 of the minimal immersion X. This minimal immersion is well defined in R 3 /G, but allowed to be a multivalued function in R 3 . The function g is a covering map ofĈ and the total curvature ofS is −4πdeg(g).

The compact Riemann surfaces M and the functions z
Denote by M the surface represented in Figure 1, and let M be the quotient of M by its translation group. A compactification of the Scherk ends of M will lead to a compact Riemann surface that we call M . The fundamental piece of M is represented in Figure 2(a), together with some special points on it. The Scherk ends are E 1 and E 2 . We have that M is invariant under reflections in the closed bold curve, indicated in Figure 2(a). The images of S, F, E 1 and E 2 under this reflection will be called S ′ , F ′ , E 1 and E 2 , respectively.
Let us denote by ρ the 180 • -rotation around the axis x 3 , indicated in Figure 2(a). It is easy to see that M has genus 3 and ρ has 4 fixed points, namely S, T, S ′ , F ′ . Therefore, the Euler-Poincaré formula gives Because of that, ρ(M ) is a torus T . A horizontal reflectional symmetry of M is induced by ρ on T , and since this symmetry has two components, we conclude that T is a rectangular torus, represented in Figure 2(b). Now we can choose an elliptic function Z on T , define z := Z • ρ and then try to deduce Weierstraß data of M on M in terms of z. Consider Figure 3(a) and the points marked with a black square ( ) on it, which are the branch points of a certain meromorphic function Z : T → C with deg(Z)=2. Let us now take an angle α ∈ (0, π/4). As indicated in Figure 3(a), we choose Z such that it takes the values ±i tan α and ±i cot α at its branch points. Up to a biholomorphism, such a function is unique, and α determines the rectangular torus. The torus is square if and only if α = π/8. The most important values of Z are indicated in Figure 3(a). We have Z(ρ(F )) = −Z(ρ(S)) = x for some x ∈ (0, ∞), and Z(ρ(E 1,2 )) = y, where y −1 ∈ (−x −1 , x −1 ). This means, we include the possibility of y to be ∞.
In the next section, we shall write the function g on M in terms of z := Z •ρ. However, this task will be simpler if we introduce another function W : T → C, of which the important values are presented in Figure 3(b). In fact, W is a "shift" of iZ. We can write W in terms of Z and Z ′ by using an addition theorem for elliptic functions, or apply some easier arguments which will be explained in the next paragraph. Nevertheless, they will give us an explicit formula for W 2 instead of W .
Consider the following picture where ξ is a pure imaginary value to be determined later: Let us define From Figures 3(a) and 4 it is easy to see that, for a certain complex constant c, the equality cf 2 = W 2 − tan 2 α W 2 − cot 2 α holds. Based on Figure 3(a), we can easily write down an algebraic equation of T as follows: Therefore, Z ′ Z 2 = −(Z 2 + Z −2 + tan 2 α + cot 2 α) and consequently we fix ξ = i(y 2 + y −2 + tan 2 α + cot 2 α) 1 2 . From Figure 3, it is not difficult to The explicit relation between W 2 and Z, Z ′ can be given as follows ( Based on Figures 3 and 5(a), one easily concludes that Of course, a priori both sides of 3 are just proportional. However, since the unitary normal vector is expected to be horizontal on the closed bold curve in Figure 2(a), this must imply that g is unitary there. Both z and w are pure imaginary on this curve. Therefore, the proportional constant at (3) must be unitary. Moreover, based on Figures 2(a) and 3, the first picture suggests that g is real for z ∈ (−x, x) and pure imaginary for z ∈ R \ (−x, x). Since w ∈ [− cot α, cot α] whenever z is real, we conclude that the unitary proportional constant is 1. Thus, (3) itself is already consistent with our analysis. From now on, we define M as a member of the family of compact Riemann surfaces given by (3).
By applying the Riemann-Hurwitz formula to (3) Now we can summarize our study of the symmetries of M and the be-haviour of g in the following table: 5. The height differential dh on M Now we are going to write down an explicit formula for dh, which will take into account the regular points and types of ends we want the surface M to have. Based on Figures 2(a) and 3(a), one sees that S and F correspond to regular points of M, at which the normal vector is vertical. The same is valid for S ′ and F ′ . Therefore, dh({S, F, S ′ , F ′ }) = {0}. Since M has only Scherk-type ends, all in the x 2 -direction, then dh has no poles and is holomorphic on M . Moreover, since deg(dh)=4, we conclude that all zeros of dh are simple. They are represented in Figure 5(b).
For convenience of the reader, we reproduce here the algebraic equation of T already established in (1): From Figure 5(b) one immediately verifies that A priori, both sides of (6) are just proportional, but since z is real on the straight lines of M , from (5) we get pure imaginary values for Z ′ • ρ on these lines. Therefore, the proportional constant at (6) must be real, and we choose it to be 1. This will imply that (6) is also consistent with Re{dh} = 0 on the closed bold curve in Figure 2(a). There we have z = it, tan α < |t| < cot α, which leads to real values for Z ′ • ρ.
Exactly at this point, we need to prove that M really has the planar geodesics and straight lines of our initial assumption. This task is summarized in the following table: From this table, it is immediate to verify that dh · dg/g is real on the expected planar geodesics of M, and pure imaginary on the expected straight lines of M.

Solution of the period problems
Let us consider Figure 6. It reproduces Figure 2(a) and its image under z with some special paths indicated there. Figure 6: The fundamental piece of M and its image under z.
Around the punctures of M, namely w −1 ({± cot α}), we consider small curves given by w(t) = ± cot α + δe it , where δ is a positive real and t varies in the interval [0, 4π] (we recall that w takes the values ± cot α with multiplicity 2). All these curves are homotopically equivalent for sufficiently small values of δ. Therefore, by letting δ → 0 an immediate calculation leads to Re (φ 1 , φ 3 ) = 0, and up to a minus sign, Re φ 2 = Res(φ 2 )| w=cot α , where Based on this analysis and (7), it is clear that Re (1) φ 3 = 0. Moreover, (7) also gives us Re (2) φ 3 = 0. The curve (2) from Figure 6 is homotopically equivalent to the sum of (1) with its image under the maps (g, z) → (g, z) and (g, z) → (−g, z), composed in this order (see Table 4). Actually, this composition corresponds to the rotation ρ, explained at the beginning of Section 3. Since ρ•(1) (φ 1 , φ 2 ) = − (1) (φ 1 , φ 2 ), then Re (2) (φ 1 , φ 2 ) = 0. It remains to prove that In Figure 6, the curve (3) is symmetric with respect to the geodesic (2). Because of that, the only non-zero component of the period vector Re (3) φ 1,2,3 must be the third one. Moreover, Re (3) φ 3 = 0 because (7) implies that dh is real and never vanishes on the dashed lines of Figure 3 We have just reduced the period problems to the proof of (9). For this purpose, we shall apply the limit-method cited at the introduction. Let us show that the Weierstraß data (3) and (6) converge to the Weierstraß data from Callahan-Hoffman-Meeks's surface of genus 3 (see Figure 7).
σ applied to (1). This fact can be verified in the complex plane. Hence

Re
(1) We split the vertical loop as β := β + ∪β − , where β + is the ascending curve from −x to x and β − the path from x to −x. The rotation ρ, introduced in Section 3, corresponds to (g, z) → (−g, z), whence g → −g and dh → dh under its action. Therefore, REMARK 6.2: Figure 10 represents the image under g of a fundamental domain D, namely a smallest subset of X(M) that fully generates it by isometries of R 3 . The left image corresponds to g| D referring to the "front piece", which contains β − . The right image concerns the "back piece", which contains β + .
Case II x = 1 > ∼ y. Now β + is still given by z(t) = −e it with 0 ≤ t ≤ π and 1+z 1−z = −i sin t 1+cos t . However, since x > ∼ y we have that cot α+w cot α−w and consequently g 2 (t) vary according to Figure 13. Again from Remarks 6.1 and 6.2, the branch of square root for g 2 (t) is now indicated in Figure 12(b). Thus Re (1) gdh > 0. In Figure 14 we indicated the behaviour of a doubly periodic Scherk-Costa surface for the above cases and x = y = 1.
Therefore, Re (1) gdh = 0 for some values of (x, y) in a neighbourhood of (1, 1). At this limit-point, the function Re (1) φ 2 depends only on the parameter α ∈ (0, π/4). This is the one-dimensional period problem for [CHM]. According to [MR], it has only one zero that we call α * .
We can illustrate this fact by taking a vertical axis ν and plotting a graph on the plane Oαν, which crosses the horizontal axis at α * . Back to our surfaces, if the extra parameters (x, y) were restricted to a curve (x(κ), y(κ)) with an extreme at (1, 1), then we could visualize κ as a third axis to Oαν. In this way, both Re (1) φ 1,2 turn out to be dependent on two variables, namely (α, κ), and their graphs are surfaces like Figure 15 suggests. Notice that we cannot provide numeric pictures of this fact, since our analyses include limit-values. They typically make unreliable any computational image. This is the second step to solve (9).

Embeddedness of the fundamental piece
This chapter is strongly based in the ideas of [MRB] and [RB4]. We begin with by identifying a fundamental domain D of X(M) in Figure 17. In the previous section we proved the existence of a curve (x(t), y(t)), 0 ≤ t < 1, along which (9) holds. Moreover, lim t→0 α(t) = α * . Let us fix t ∈ (0, 1) and consider the minimal immersion X t : D \ {E 1,2 } → R 3 , defined by (3) and (6). Each branch of square root of g takes any q ∈ D \ {E 1,2 } to a pair of points in R 3 , say X t (q) + and X t (q) − . If X t (0) is the origin, then each point is the image of the other by a 180 • rotation about Ox 3 .
We consider a fundamental piece P of M. Let P − be the image of D \ {E 1,2 } in R 3 under X t , and P + the image of P − in R 3 under a 180 • rotation around X t ([−x, x]). Thus P = P + ∪ P − .
Let K be a subset of D such that D \ K = V E , where V E is a connected neighbourhood of E 1,2 . From (7) and (10) we see that (g, dh) converge uniformly to the Weierstraß data of the embedded CHM-surface. Let us denote this minimal embedding by X 0 . When t → 0, X t | V E approaches a planar end for sufficiently small V E . For t close to zero, the projection of X t | ∂V E onto x 3 = 0 consists of two curves C ± which determine two simply connected open regions R + and R − . Since g(V E ) is contained in a half-sphere, then (x 1 , x 2 )| V E is an immersion onto R ± because x 2 is bounded for any fixed t ∈ (0, 1). Since ∂R ± are the monotone curves C ± , then X T | ∂V E is a graph of x 3 as a function of (x 1 , x 2 ). C + R R + C − − Figure 18: Regions R ± and curves C ± .
We observe that X 0 | K is a compact embedded minimal surface in R 3 . Since its boundary does not have self-intersections, then X t | K is still embedded for sufficiently small t. Moreover, X t | K does not intercept X t | V E , otherwise there would be a ball in R 3 containing the whole boundary of X t | K but not all the rest of it. This is impossible according to the maximum principle. Hence, the pieces X t | K and X t | V E make together a minimal embedding X t : D \ {E 1,2 } → R 3 , for t sufficiently close to zero.
Again by the maximum principle, we may extend this conclusion for all t ∈ (0, 1). Therefore, P + is embedded in R 3 , and since P − is its image under a 180 • rotation about the segment of P + , the whole piece P will not have self-intersections. Since the immersion is proper, then P is embedded in R 3 . Now P ⊂ R 3 /G, where G is the group of R 3 generated by (x 1 , x 2 , x 3 ) → (x 1 , x 2 , −x 3 + 2Re β + dh) and (x 1 , x 2 , x 3 ) → (x 1 , x 2 + Re φ 2 , x 3 ). In the horizontal faces of ∂(R 3 /G) we have the reflection curves of P . In the vertical faces we have the straight lines of P . By applying G to P one generates M, which is then complete, doubly periodic and embedded in R 3 .