#### Abstract

The Schwarz reflection principle states that a minimal surface in is invariant under reflections in the plane of its principal geodesics and also invariant under 180Β°-rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.

#### 1. Introduction

During the *Clay Mathematics Institute 2001 Summer School on the Global Theory of Minimal Surfaces*, M. Weber introduced the following terminology in his first lecture entitled *Embedded minimal surfaces of finite topology*:

βA

horizontal symmetryis a reflection at a vertical plane or a rotation about a horizontal line. Avertical symmetryis a reflection at a horizontal plane or a rotation about a vertical line.β

With this terminology, he proved that such symmetries induce symmetries in the *cone metrics* determined by , , and from a *WeierstraΓ pair * of a minimal surface (see [1, 2] for details).

By classifying the symmetries this way, we sort out the space groups that might admit one, both, or none of them. Since minimal surfaces may model some natural structures, like crystals and copolymers, an example within a given symmetry group might fit an already existing compound, or even hint at nonexisting ones. However, several symmetry groups are not yet represented by any minimal surface (see [3, 4] for details and comments).

Restricted to symmetries given by reflections in the plane of principal geodesics and by 180Β°-rotations about straight lines contained in the surface, outside the triply periodic class it is easy to find complete embedded minimal surfaces in of which these symmetries are either only horizontal or only vertical. For instance, the Costa surface (see [5β7]) has only horizontal symmetries. The doubly periodic examples found by Meeks and Rosenberg in [8] have only vertical symmetries (see also [9] for nice pictures).

In the class of triply periodic minimal surfaces almost all known examples have either both or none of such symmetries, after suitable motion in . In fact, this must be true because most of the surfaces in this class have a cubic symmetry group. Examples with only horizontal symmetries do not seem to be well known. Besides the surfaces shown herein, perhaps there are only the βTT-surfacesβ as Karcher named them in [10, pp. 297, 328-9] (see also [11]), and a surface from Fischer-Koch [12], which is, however, presented without rigorous proof.

The βTT-surfacesβ are generated by an annulus, of which the boundary consists of two twisted equilateral triangles. For edge length and height 1, they coincide with the Schwarz P-surface, and hence have further symmetries besides the horizontal ones. Moreover, when a TT-surface has only horizontal symmetries, its translation group cannot be given by an orthogonal lattice.

In the present work, we give existence proofs for examples that are probably the first triply periodic minimal surfaces with only horizontal symmetries, of which the translation group is given by an orthogonal lattice. They are constructed by Karcherβs method [6, 7, 10], although the purpose alone of few symmetries could be accomplished by modern methods introduced, for instance, by Traizet [13] and Fujimori and Weber [14]. However, Traizetβs method is not explicit (in the sense explained in [15]) whereas Fujimori-Weberβs method may turn it hard to analyse the so-called *period problems*. These are equations involving elliptic integrals with interdependent parameters.

Regarding examples with only vertical symmetries, we believe they have not been found yet.

The examples presented herein are inspired in the surfaces and from [16, 17]. Any of those is generated by a fundamental piece, which is a surface with boundary in with two catenoidal ends. The fundamental piece resembles the Costa surface with its planar end replaced by either symmetry curves or line segments. By suppressing the catenoidal ends, if we pile up several copies of the fundamental piece, we get the pictures in Figures 1 and 2(b). They are also named and .

**(a)**

**(b)**

**(a)**

**(b)**

The reader will notice that the surfaces , also described in [16, 17], were not mentioned beforehand. This is because, for them, the βpiling upβ procedure naturally forces extra symmetries to exist, and one goes back to another famous surface from H. Schwarz (see Figure 2(a)). Notice, for instance, the vertical straight line that comes out in the surface.

We are going to prove the following results:

Theorem 1.1. *There exists a one-parameter family of triply periodic minimal surfaces in , of which the members are called , and for any of them the following holds. *(a)*The quotient by its translation group has genus 9.*(b)*The whole surface is generated by a fundamental piece, which is a surface with boundary in . The boundary consists of four curves, each contained in a vertical plane. The fundamental piece has a symmetry group generated by reflections in two vertical planes and 180Β°-rotations about two line segments. *(c)*By successive reflections with respect to planes bounding the fundamental domain, and successive vertical translations, one obtains the triply periodic surface. *

Theorem 1.2. *For , there exists a one-parameter family of triply periodic minimal surfaces in , of which the members are called , and for any of them the following holds. *(a)*The quotient by its translation group has genus . *(b)*The whole surface is generated by a fundamental piece, which is a surface with boundary in . The boundary consists of four line segments. The fundamental piece has a symmetry group generated by reflections in two vertical planes and 180Β° -rotations about two line segments. Each of these segments makes an angle of with the boundary. *(c)*By successive rotations about the boundary of the fundamental piece, and successive vertical translations, one obtains the triply periodic surface. *

Sections 3 to 7 are devoted to the proof of Theorem 1.1. The proof of Theorem 1.2 follows very similar arguments, and we briefly discuss it in Section 8.

#### 2. Preliminaries

In this section we state some basic definitions and theorems. Throughout this work, surfaces are considered connected and regular. Details can be found in [6, 7, 18β20].

Theorem 2.1. *Let be a complete isometric immersion of a Riemannian surface into a three-dimensional complete flat space . If is minimal and the total Gaussian curvature is finite, then is biholomorphic to a compact Riemann surface punched at finitely many points .*

*Definition 2.2. *Let as in Theorem 2.1. An *end* is the image under of a punctured neighbourhood of a point such that . We say that the surface has *no ends* when .

Theorem 2.3 (WeierstraΓ Representation). *Let be a Riemann surface, and meromorphic function and 1-differential form on , such that the zeros of coincide with the poles and zeros of . Suppose that , given by
**
is well defined. Then is a conformal minimal immersion. Conversely, every conformal minimal immersion can be expressed as (2.1) for some meromorphic function and 1-form .*

*Definition 2.4. *The pair is the *WeierstraΓ data* and , , are the *WeierstraΓ forms* on of the minimal immersion .

Theorem 2.5. *Under the assumptions of Theorems 2.1 and 2.3, the WeierstraΓ data extend meromorphically on .*

The function is the stereographic projection of the GauΓ map of the minimal immersion . It is a covering map of and . These facts will be largely used throughout this work.

#### 3. The Surfaces and the Functions

Consider the surface indicated in Figure 1(a). A reflection in any of its vertical planar curves of the boundary leads to a fundamental piece which represents the quotient of a triply periodic surface by its translation group. We are going to denote this quotient by . It is not difficult to conclude that it has genus 9. The fundamental domain of is the shaded region indicated in Figure 3(a).

**(a)**

**(b)**

The surface is invariant under -rotations around the directions and . These rotations we call and , respectively, (see Figure 3(a)). Based on this picture, one sees that the fixed points of are , , , , , and the images of and under the symmetries of . They sum up 8 in total. The quotients by and we call and , respectively. The surface is still invariant under the rotation . In this case, the fixed points of will be , and their images under the symmetries of . They sum to 8 in total. Because of that,

Let us define , such that , and . The involutions of are induced by and on , and since all the involutions of are given by MΓΆbius transformations, we can conclude the following: , and . By applying the symmetries of , one easily reads off the other values of at the images (under these symmetries) of , , , , , , , and . Regarding the points and , we have such that and . Consequently, and one easily gets the other values of at the images of and under the symmetries of .

#### 4. The -Function on in Terms of

First of all, observe that Jorge-Meeksβ formula gives . Let us then consider Figure 3(b). We will have if and only if . Moreover, if and only if , where . From this point on we introduce the following notation:

By following Karcherβs method in [6, 7], Figure 3 represents the surfaces whose existence we want to prove. From this picture we read off the necessary conditions for Theorem 1.1 to be valid. Afterwards, these will *prove* Theorem 1.1. The first condition is an algebraic relation between and . Hence, based on Figure 3 and Karcherβs method, it is not difficult to conclude that
where is a positive constant. Now we define as a member of the family of compact Riemann surfaces given by the algebraic equation (4.2). Later on, we are going to verify that has genus 9 indeed. But first we derive some conditions on the variables , and in order to guarantee that at . This will be the case if

Since , one easily sees that is positive.

Now we analyse what happens to (4.2) under the map . In this case we will get or . Therefore

At this point we are ready to prove that has genus 9. The function is a four-sheeted branched covering of the sphere. The values 0, , , , correspond to the only branch points of , all of them of order 2, and each of these values is taken twice on . Therefore, from the Riemann-Hurwitzβs formula we have

Now we are ready to find some relations that the parameters , , and will have to satisfy. These relations will make (4.2) and (4.4) consistent with the values of on the symmetry curves and lines of .

#### 5. Conditions on the Parameters , , and

Consider the curves and represented in Figure 3. The same picture shows how we have positioned our coordinate system. On the curve , we expect that and on one should have . Let us now verify under which conditions this will really happen.

On , we ought to have . By taking , , defining , and applying it to (4.2) we get the following equality:

Therefore, on the curve . Since we want on this curve, (5.2) will then give rise to the following conditions

Equation (5.3) can be deduced from (4.3) and (5.4) by a simple calculation. Equation (5.4) will restrict the definition domain of our parameters. Since , then , and by taking one clearly sees that for . From (5.4) we finally get the following restriction for the -variable

Figure 4 illustrates the -domain established by (5.5), and we recall that and .

It is not difficult to prove that (5.5) is equivalent to the following inequality:

Of course, the right-hand side of (5.6) is one of the two roots of a second-degree equation. One easily proves that the other root is bigger than 1. Its inverse is exactly the right-hand side of (5.6), and this shows that it is positive and smaller than 1.

Regarding our remaining restriction, namely, on , it is not difficult to verify that it leads to the same conditions (5.3) and (5.4). Therefore, we are now ready to write down Table 1, which summarizes some special involutions of .

Notice that the points do not come out as fixed points of in Table 1. This is because the *germs* of the function at these points *are not* the same (see [21] for details). This has to do with the fact that the power of is a *multiple* of the power of in (4.4).

#### 6. The Height Differential on

Since the surface has no ends, must be a holomorphic differential form on it. The zeros of are exactly at the points where or , and at these points. They should sum up 16 in total, which is consistent with . Let us now analyse the differential . Based on Figure 3, one sees that has a simple zero at the points and a pole of order 3 at the points . Let the symbol ~ indicate that two meromorphic functions on differ by a nonzero proportional constant. It is not difficult to conclude that

If we had a well-defined square root of the function at the right-hand side of (6.1), then we could get an explicit formula for in terms of and . This square root exists indeed. By multiplying (4.2) and (4.4) it follows that which allows us to define

Now we apply (6.3) to (6.1) and obtain

At (6.4) the equality sign holds because we want on the straight line segment (see Figure 3(a)). On this segment is purely imaginary and then we can fix both sides of (6.4) to be equal. Let us now verify if the symmetry curves and lines of really exist. From Table 1 and (6.4) we write down Table 2.

From Table 2 it follows that is purely imaginary on and . It is real on the other paths, confirming that will have the expected symmetry curves and lines.

#### 7. Solution of the Period Problems

The analysis of the period problems can be reduced to the analysis of the fundamental domain of our minimal immersion. If this fundamental domain is contained in a rectangular prism of , and if the boundary of the former is contained in the border of the latter, we will have that the fundamental piece of our minimal surface will be free of periods.

In order to obtain such a prism, a little reflection will show us that the following two conditions will be enough.(1)The symmetry really exists in .(2)After an orthonormal projection of the fundamental domain in the direction , we will have and (see Figure 5).

The first condition is easy to prove. Take a path on as indicated in Figure 6. Consider that with reversed orientation is the image of under the involution . Now we compute in what happens to the coordinates of our minimal surface:

**(a)**

**(b)**

Therefore, our minimal surface is *really* invariant under 180Β°-rotations around the -axis. This proves the existence of the symmetry of our initial assumptions.

Now we are ready to deal with the second condition. Consider Figure 5 with the segments and on it. The period will be zero if and only if these segments have the same length, or equivalently

On we can take , . This implies that . From (4.4) and (6.4) we have

On we can take , . From (4.2) and (6.4) it follows that

Now define and . For apply the change of variables and for , . A simple reckoning will lead to the following equalities:

The next proposition will solve the period problem given by (7.2).

Proposition 7.1. *For any fixed positive value of one has that the following limit exists and is positive
**
For one has that exists and is negative.*

*Proof. *By recalling (5.4), a simple reckoning will show that

Since
for every and , from (7.8) the first assertion of Proposition 7.1 follows.

By fixing and recalling (5.5), the convergence is equivalent to . This means that approaches the point indicated in Figure 4. An easy calculation will give us

The integrand of (7.10) can be rewritten as
while one rewrites the integrand of (7.11) as

Since
for every , the last assertion of Proposition 7.1 follows.

Proposition 7.1 provides a family of triply periodic surfaces of which a member is depicted in Figure 1(a). By looking at Figure 4, this family can be represented by the values of which belong to a curve contained in the shaded region. All members of this family will have *only* three periods, as suggested by Figure 1(a). Nevertheless, a priori there might be some nonembedded members, but it will not be the case. This is the subject of our next section.

#### 8. Embeddedness of the Triply Periodic Surfaces

From now on we will denote our triply periodic surfaces by , where . Figure 6 shows that the projection of the unitary normal on a fundamental domain of is contained in the lower hemisphere of . This means that is an immersion of in . Figure 7 shows a possible image of this map in :

It is not difficult to prove that the contour of the shaded region in Figure 7 is a monotone curve. The -coordinate of the curve is given by the integral of as in (7.4). The integrand is clearly positive, hence this stretch is monotone. Regarding , where we can take , , a simple reckoning gives us

Hence, the stretch is also monotone. By using the symmetry , it follows that the whole contour indicated in Figure 7 is a monotone curve. Since the third coordinate of is increasing, the projections and will intersect only at the point . Nevertheless, it can happen that the projection crosses . If we prove that this is not the case, the contour will have no self-intersections. The shaded region will then be simply connected, and we will conclude that the fundamental domain is a graph, hence embedded.

But even so, it can happen that the expanded triply periodic surface will not be embedded. We do not know whether the curve crosses the -axis or not. A little reflection will show that, if does not take the value along , then this curve does not intersect the vertical axis. Consequently, its projection will not intersect . In this case, since the triply periodic surface is expanded horizontally by reflections only, and vertically by rotations only, the whole surface will then be embedded.

By using the maximum principle, if we find an embedded member of our family in the curve , the whole family will then consist of embedded surfaces. The following proposition gives us such a member and will conclude this section.

Proposition 8.1. *There is an such that and is embedded.*

*Proof. *We will prove that along , for any . Moreover, will be positive. These two facts together with Proposition 7.1 will conclude Proposition 8.1.

By recalling (5.1), we would have for some if and only if

Equation (8.2) will not be fulfilled by any providing
or equivalently

We have fixed , hence along for any . Let us now verify that . From (7.5) we have
and from (7.6) it follows that

But
and if we define it is possible to prove that

But

Now we use , , and in order to conclude that

Together with (8.5)β(8.9), (8.10) shows that is positive.

#### 9. The Surfaces

In order to prove Theorem 1.2, one follows very similar ideas already explained in Sections 3 to 7. For the surfaces , consider Figures 8(a) and 8(b). The fundamental piece has genus 5, and passes through point . The piece is invariant under and , with quotient functions and , respectively.

**(a)**

**(b)**

Since we may define , such that and . The symmetries imply , and whereas is a certain complex in the first quadrant. Moreover, there is a point in the segment at which . After analysing the divisors of and on , together with the behaviour of the unitary normal on symmetry curves and lines, we get Since there is a point in the segment at which , we should also have In order to have equivalence between (9.2) and (9.3), a necessary and sufficient condition is . Now, it is easy to get with a well-defined square root in the denominator. One checks the assumed symmetries the same way we did in Tables 1 and 2. The unique period problem is again (7.2), which can be visualised again by Figure 5. Therefore, (7.2) is equivalent to , where The change for makes clear that () providing (), where and , . On the one hand, for a fixed , if then , and consequently . On the other hand, by fixing and letting , then and so . In this case, notice that the singularity at of both integrands in (9.5) is easily removable with a change of variables. This means, no matter if we have , it still holds .

For the surfaces , consider Figures 9(a) and 9(b). The fundamental piece has genus 9, and passes through point . The piece is invariant under and , with quotient functions and , respectively. We will have if and only if . Moreover, if and only if , where .

**(a)**

**(b)**

From this point on we redefine the following:

Based on Figure 9 it is not difficult to conclude that
where is given by (4.3) again. Moreover, (9.7) is equivalent to
Similar arguments as in Section 5 will give again (5.3) and (5.4), but unlike (5.5) there is *no* restriction now. Regarding , it still holds (6.4), but unlike Figure 5 the period problem is now illustrated by Figure 10.

Integrals and are again given by (7.5) and (7.6), but now the period is solved when . This will come with the following.

Proposition 9.1. *For any fixed positive value of one has that the following limit exists and is positive:
**
For one has that exists and is negative.*

The proof of Proposition 9.1 is quite similar to the proof of Proposition 7.1, and so we will omit it here. The arguments for the embeddedness of are even easier than the ones used in Section 8 for , because now the contours are given by *four* straight line segments and *two* curves, pairwise congruent.

#### Acknowledgments

For this present paper, V. R. Batista was supported by the Grants βBolsa de Produtividade CientΓficaβ from CNPqβConselho Nacional de Desenvolvimento CientΓfico e TecnolΓ³gico, and βBolsa de PΓ³s-Doutoradoβ FAPESP 2000/07090-5.