Minimal Surfaces with Only Horizontal Symmetries

Abstract

The Schwarz reflection principle states that a minimal surface 𝑆 in ℝ³ is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-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:

“A horizontal symmetry is a reflection at a vertical plane or a rotation about a horizontal line. A vertical symmetry is a reflection at a horizontal plane or a rotation about a vertical line.”

With this terminology, he proved that such symmetries induce symmetries in the cone metrics determined by 𝑑ℎ, 𝑔𝑑ℎ, and 𝑑ℎ/𝑔 from a Weierstraß pair (𝑔,𝑑ℎ) 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 180°-rotations about straight lines contained in the surface, outside the triply periodic class it is easy to find complete embedded minimal surfaces in ℝ³ of which these symmetries are either only horizontal or only vertical. For instance, the Costa surface (see [5–7]) 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 ℝ³. 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 “TT-surfaces” 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 “TT-surfaces” are generated by an annulus, of which the boundary consists of two twisted equilateral triangles. For edge length 2√3 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 Karcher’s 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, Traizet’s method is not explicit (in the sense explained in [15]) whereas Fujimori-Weber’s 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 𝐶₂ and 𝐿_2,4 from [16, 17]. Any of those is generated by a fundamental piece, which is a surface with boundary in ℝ³ 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 𝐶₂ and 𝐿_2,4.

(a)

(b)

Figure 1 

(a) The surfaces 𝐶2; (b) the surfaces 𝐿2.

(a)

(b)

Figure 2 

(a) Schwarz’s D-surfaces; (b) the surfaces 𝐿4.

The reader will notice that the surfaces 𝐶₄, also described in [16, 17], were not mentioned beforehand. This is because, for them, the “piling up” 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 ℝ³, of which the members are called 𝐶₂, 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 ℝ³. 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 180°-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 𝑘=2,4, there exists a one-parameter family of triply periodic minimal surfaces in ℝ³, of which the members are called 𝐿_𝑘, and for any of them the following holds. (a)The quotient by its translation group has genus 2𝑘+1. (b)The whole surface is generated by a fundamental piece, which is a surface with boundary in ℝ³. The boundary consists of four line segments. The fundamental piece has a symmetry group generated by reflections in two vertical planes and 180° -rotations about two line segments. Each of these segments makes an angle of 𝜋/𝑘 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, 18–20].

Theorem 2.1. Let 𝑋∶𝑅→𝔼 be a complete isometric immersion of a Riemannian surface 𝑅 into a three-dimensional complete flat space 𝔼. If 𝑋 is minimal and the total Gaussian curvature ∫_𝑅𝐾𝑑𝐴 is finite, then 𝑅 is biholomorphic to a compact Riemann surface 𝑅 punched at finitely many points 𝑝₁,…,𝑝_𝑟.

Definition 2.2. Let 𝑃={𝑝₁,…,𝑝_𝑟} as in Theorem 2.1. An end is the image under 𝑋 of a punctured neighbourhood 𝑉_𝑝 of a point 𝑝∈𝑃 such that 𝑃∩𝑉_𝑝=∅. We say that the surface has no ends when 𝑃=∅.

Theorem 2.3 (Weierstraß Representation). Let 𝑅 be a Riemann surface, 𝑔 and 𝑑ℎ meromorphic function and 1-differential form on 𝑅, such that the zeros of 𝑑ℎ coincide with the poles and zeros of 𝑔. Suppose that 𝑋∶𝑅→𝔼, given by 𝑋(𝑝)∶=Re^𝑝𝜙₁,𝜙₂,𝜙₃,where𝜙₁,𝜙₂,𝜙₃∶=12𝑔1−𝑔,𝑔𝑖+𝑖𝑔,2𝑑ℎ,(2.1) is well defined. Then 𝑋 is a conformal minimal immersion. Conversely, every conformal minimal immersion 𝑋∶𝑅→𝔼 can be expressed as (2.1) for some meromorphic function 𝑔 and 1-form 𝑑ℎ.

Definition 2.4. The pair (𝑔,𝑑ℎ) is the Weierstraß data and 𝜙₁, 𝜙₂, 𝜙₃ are the Weierstraß forms on 𝑅 of the minimal immersion 𝑋∶𝑅→𝑋(𝑅)⊂𝔼.

Theorem 2.5. Under the assumptions of Theorems 2.1 and 2.3, the Weierstraß data (𝑔,𝑑ℎ) extend meromorphically on 𝑅.

The function 𝑔 is the stereographic projection of the Gauß map 𝑁∶𝑅→𝑆² of the minimal immersion 𝑋. It is a covering map of ℂ and ∫_𝑆𝐾𝑑𝐴=−4𝜋deg(𝑔). These facts will be largely used throughout this work.

3. The Surfaces 𝑀 and the Functions 𝑧

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 𝑀 by its translation group. We are going to denote this quotient by 𝑀. It is not difficult to conclude that it has genus 9. The fundamental domain of 𝑀 is the shaded region indicated in Figure 3(a).

(a)

(b)

Figure 3 

(a) The fundamental domain of 𝑀; (b) the function 𝑧 on 𝑀.

The surface 𝑀 is invariant under 180^∘-rotations around the directions 𝑥₃ and 𝑥₂. These rotations we call 𝑟_𝑣 and 𝑟_ℎ, respectively, (see Figure 3(a)). Based on this picture, one sees that the fixed points of 𝑟_𝑣 are 𝑆, 𝑆^, 𝐿, 𝐿^, 𝐹, 𝐹^ and the images of 𝑆 and 𝑆^ under the symmetries of 𝑀. They sum up 8 in total. The quotients by 𝑟_𝑣 and 𝑟_ℎ we call 𝜌_𝑣 and 𝜌_ℎ, respectively. The surface 𝜌_𝑣(𝑀) is still invariant under the rotation 𝑟_ℎ. In this case, the fixed points of 𝑟_ℎ will be 𝜌_𝑣(𝐴), 𝜌_𝑣(𝐴^) and their images under the symmetries of 𝜌_𝑣(𝑀). They sum to 8 in total. Because of that, 𝜒𝜌_ℎ𝜌^𝑣𝑀=1−9+8/22+82=2.(3.1)

Let us define 𝑧∶=𝜌_ℎ∘𝜌_𝑣∶𝑀→𝑆²≈ℂ, such that 𝑧(𝑆)=0, 𝑧(𝐿)=1 and 𝑧(𝐵)=𝑖. The involutions of 𝑀 are induced by 𝜌_𝑣 and 𝜌_ℎ on ℂ, and since all the involutions of ℂ are given by Möbius transformations, we can conclude the following: 𝑧(𝑆^)=0, 𝑧(𝐹^)=−𝑧(𝐹)=−𝑧(𝐿^)=1 and 𝑧(𝐵^)=𝑖. By applying the symmetries of 𝑀, one easily reads off the other values of 𝑧 at the images (under these symmetries) of 𝑆, 𝑆^, 𝐿, 𝐿^, 𝐹, 𝐹^, 𝐵, and 𝐵^. Regarding the points 𝐴 and 𝐴^, we have 𝑧(𝐴)=𝑥∈ℂ such that |𝑥|<1 and Arg(𝑥)∈(0,𝜋/2). Consequently, 𝑧(𝐴^)=−𝑥 and one easily gets the other values of 𝑧 at the images of 𝐴 and 𝐴^ under the symmetries of 𝑀.

4. The 𝑔-Function on 𝑀 in Terms of 𝑧

First of all, observe that Jorge-Meeks’ formula gives deg(𝑔)=9−1=8. Let us then consider Figure 3(b). We will have 𝑔−𝑔⁻¹=∞ if and only if 𝑧−𝑧⁻¹∈{0,∞}. Moreover, 𝑔−𝑔⁻¹=0 if and only if 𝑧∈{−𝑥,𝑥,𝑥⁻¹,−𝑥⁻¹,𝑖𝑎,𝑖𝑎⁻¹}, where 𝑎∈(0,1). From this point on we introduce the following notation: 𝑍∶=𝑧−𝑧⁻¹,𝑋∶=𝑥⁻¹−𝑥,𝒜∶=𝑎+𝑎⁻¹.(4.1)

By following Karcher’s 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 𝑔 and 𝑧. Hence, based on Figure 3 and Karcher’s method, it is not difficult to conclude that 𝑔−𝑔1²=−𝑖𝑐𝑍³⋅(𝑍−𝑖𝒜)²(𝑍−𝑋)𝑍+𝑋,(4.2) where 𝑐 is a positive constant. Now we define 𝑀 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 𝑀 has genus 9 indeed. But first we derive some conditions on the variables 𝑎,𝑥, and 𝑐 in order to guarantee that 𝑔²=−1 at 𝑧=−𝑖𝑎^±1. This will be the case if 𝑐=𝒜𝒜²+2𝒜Im{𝑋}+||𝑋||².(4.3)

Since |𝑋|²=Im²{𝑋}+Re²{𝑋}, one easily sees that 𝑐 is positive.

Now we analyse what happens to (4.2) under the map 𝑧→𝑧. In this case we will get 𝑔→𝑖𝑔 or 𝑔→−𝑖𝑔. Therefore 𝑔+𝑔1²=−𝑖𝑐𝑍³⋅(𝑍+𝑖𝒜)²𝑍−𝑋(𝑍+𝑋).(4.4)

At this point we are ready to prove that 𝑀 has genus 9. The function 𝑧 is a four-sheeted branched covering of the sphere. The values 0, ∞, ±1, ±𝑥^±1, ±𝑥^±1 correspond to the only branch points of 𝑧, all of them of order 2, and each of these values is taken twice on 𝑀. Therefore, from the Riemann-Hurwitz’s formula we have genus𝑀=12⋅(2−1)⋅22−4+1=9.(4.5)

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

5. Conditions on the Parameters 𝑎, 𝑐, and 𝑥

Consider the curves 𝑆^𝐿 and 𝐹^𝑆 represented in Figure 3. The same picture shows how we have positioned our coordinate system. On the curve 𝑆^𝐿, we expect that 𝑔∈𝑒^𝑖𝜋/4ℝ and on 𝐹^𝑆 one should have 𝑔∈𝑒^{−𝑖𝜋/4}ℝ. Let us now verify under which conditions this will really happen.

On 𝑆^𝐿, we ought to have Re{(𝑔−𝑔⁻¹)²}=−2. By taking 𝑧(𝑡)=𝑡, 0<𝑡<1, defining 𝑇∶=𝑡−𝑡⁻¹, and applying it to (4.2) we get the following equality: 𝑔−𝑔⁻¹²|||_𝑧(𝑡)=−𝑖𝑐𝑇³⋅(𝑇−𝑖𝒜)²𝑇²−2𝑖Im{𝑋}⋅𝑇−||𝑋||².(5.1)

Therefore, Re𝑔−𝑔⁻¹²=−2𝑐𝑇²⋅Im{𝑋}⋅𝑇²+𝒜𝑇²−Im{𝑋}⋅𝒜²−𝒜||𝑋||²,(5.2) on the curve 𝑧(𝑡). Since we want Re{(𝑔−𝑔⁻¹)²}=−2 on this curve, (5.2) will then give rise to the following conditions 𝑐=1𝒜=𝒜−+Im||𝑋||{₂𝑋},(5.3)Im{𝑋}.(5.4)

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 𝑎∈(0,1), then 𝒜>2, and by taking 𝑥=|𝑥|𝑒^𝑖𝜃 one clearly sees that Im{𝑋}<0 for 𝜃∈(0,𝜋/2). From (5.4) we finally get the following restriction for the 𝑥-variable Re²{𝑋}>−2Im{𝑋}−Im²{𝑋}.(5.5)

Figure 4 illustrates the 𝑋-domain established by (5.5), and we recall that |𝑥|<1 and 𝜃∈(0,𝜋/2).

Figure 4 

Definition domain of the 𝑋-variable satisfying (5.4).

It is not difficult to prove that (5.5) is equivalent to the following inequality: |𝑥|<12sin𝜃+√1+3cos²𝜃−2sin𝜃√1+3cos²𝜃−sin𝜃.(5.6)

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, Re{(𝑔−𝑔⁻¹)²}=2 on 𝐹^𝑆, 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 𝑀.

Table 1 

Notice that the points (𝑔,𝑧)=(±𝑖,−𝑖𝑎^±1) do not come out as fixed points of 𝜌_ℎ in Table 1. This is because the germs of the function 𝑔 at these points are not the same (see [21] for details). This has to do with the fact that the power of (𝑍+𝑖𝒜) is a multiple of the power of (𝑔+𝑔⁻¹) in (4.4).

6. The Height Differential 𝑑ℎ on 𝑀

Since the surface 𝑀 has no ends, 𝑑ℎ must be a holomorphic differential form on it. The zeros of 𝑑ℎ are exactly at the points where 𝑔=0 or 𝑔=∞, and ord(𝑑ℎ)=|ord(𝑔)| at these points. They should sum up 16 in total, which is consistent with deg(𝑑ℎ)=−𝜒(𝑀). Let us now analyse the differential 𝑑𝑧. Based on Figure 3, one sees that 𝑑𝑧 has a simple zero at the points 𝑧⁻¹({0,±1,±𝑥^±1,±𝑥^±1}) and a pole of order 3 at the points 𝑧⁻¹({∞}). Let the symbol ~ indicate that two meromorphic functions on 𝑀 differ by a nonzero proportional constant. It is not difficult to conclude that 𝑑𝑑ℎ𝑧²∼𝑧²−1²𝑧²−𝑥²𝑧²−𝑥⁻²𝑧²−𝑥²𝑧²−𝑥⁻².(6.1)

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 𝑑ℎ in terms of 𝑧 and 𝑑𝑧. This square root exists indeed. By multiplying (4.2) and (4.4) it follows that (𝑍−𝑋)𝑍+𝑋𝑍−𝑋(𝑍+𝑋)=−𝑍⁶𝑐²𝑍²+𝒜²²𝑔²−1𝑔²²,(6.2) which allows us to define 𝑍²−𝑋²𝑍²−𝑋²∶=𝑐𝑍𝑖₂𝑍+³𝒜²𝑔²−1𝑔².(6.3)

Now we apply (6.3) to (6.1) and obtain 𝑑ℎ=𝑍𝑍²−𝑋²𝑍²−𝑋²⋅𝑑𝑧𝑧.(6.4)

At (6.4) the equality sign holds because we want Re{𝑑ℎ}=0 on the straight line segment 𝑆𝐵 (see Figure 3(a)). On this segment 𝑧 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 𝑀 really exist. From Table 1 and (6.4) we write down Table 2.

Table 2 

From Table 2 it follows that 𝑑𝑔/𝑔⋅𝑑ℎ is purely imaginary on 𝑆𝐵 and 𝑆^𝐵^. It is real on the other paths, confirming that 𝑀 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 ℝ³, 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 𝜌_ℎ really exists in ℝ³.(2)After an orthonormal projection of the fundamental domain in the direction 𝑥₃, we will have 𝑆=𝑆^ and 𝐵=𝐵^ (see Figure 5).

Figure 5 

𝑥3-projection of the fundamental domain with an open period.

The first condition is easy to prove. Take a path 𝑃→𝐴→𝑃^ on 𝑀 as indicated in Figure 6. Consider that 𝐴→𝑃^ with reversed orientation is the image of 𝑃→𝐴 under the involution (𝑔,𝑧)→(−1/𝑔,𝑧). Now we compute in ℝ³ what happens to the coordinates of our minimal surface: 𝑥₁,𝑥₂,𝑥₃||_{(𝑔,𝑧)→(−1/𝑔,𝑧)}==RRee^{𝐴=(−𝑖,𝑥)𝑃=(𝑔}_{0,𝑧0)𝐴=(−𝑖,𝑥)𝑃=(𝑔0,𝑧0)}𝜙₁,𝜙₂,𝜙₃=Re^{𝑃𝐴=(−𝑖,𝑥)}_{′=(−1/𝑔0,𝑧0)}𝜙₁,−𝜙₂,𝜙₃−𝜙₁,𝜙₂,−𝜙₃=−𝑥₁,𝑥₂,−𝑥₃.(7.1)

(a)

(b)

Figure 6 

The path 𝑃→𝐴→𝑃 on 𝑀.

Therefore, our minimal surface is really invariant under 180°-rotations around the 𝑥₂-axis. This proves the existence of the symmetry 𝜌_ℎ of our initial assumptions.

Now we are ready to deal with the second condition. Consider Figure 5 with the segments 𝑆𝐵 and 𝐵𝐿 on it. The period will be zero if and only if these segments have the same length, or equivalently Re_𝑆𝐵𝜙₂=Re_𝐵𝐿𝜙₁.(7.2)

On 𝑆𝐵 we can take 𝑍(𝑡)=𝑖𝑡, 2<𝑡<∞. This implies that 𝑑𝑧/𝑧=−𝑑𝑡/√𝑡²−4. From (4.4) and (6.4) we have 𝜙₂||_{𝑍(𝑡)=𝑖𝑡}=𝑐^1/2(𝑡+𝒜)𝑡^1/2𝑡²−2Im{𝑋}⋅𝑡+||𝑋||²^1/2⋅𝑑𝑡√𝑡²−4.(7.3)

On 𝐵𝐿 we can take 𝑧(𝑡)=𝑒^𝑖𝑡, 0<𝑡<𝜋/2. From (4.2) and (6.4) it follows that 𝜙₁||_{𝑧(𝑡)=𝑒𝑖𝑡}=1√2⋅𝑐^1/2(𝒜−2sin𝑡)4sin²𝑡+4Im{𝑋}⋅sin𝑡+||𝑋||²^1/2⋅𝑑𝑡√sin𝑡.(7.4)

Now define 𝐼₁∶=(1/√2𝑐)∫_𝐵𝐿𝜙₁ and 𝐼₂∶=(1/√2𝑐)∫_𝑆𝐵𝜙₂. For 𝐼₁ apply the change of variables 𝑢²=sin𝑡 and for 𝐼₂, 𝑡=2𝑢⁻². A simple reckoning will lead to the following equalities: 𝐼₁=¹⁰𝒜−2𝑢²4𝑢⁴+4Im{𝑋}⋅𝑢²+||𝑋||²^1/2⋅𝑑𝑢√1−𝑢⁴𝐼,(7.5)₂=¹⁰2+𝒜𝑢²4−4Im{𝑋}⋅𝑢²+||𝑋||²𝑢⁴^1/2⋅𝑑𝑢√1−𝑢⁴.(7.6)

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

Proposition 7.1. For any fixed positive value of Re{𝑋} one has that the following limit exists and is positive lim_Im{𝑋}→0(−Im{𝑋})⋅𝐼₁−𝐼₂.(7.7) For Im{𝑋}=−1 one has that lim_𝒜→2(𝐼₁−𝐼₂) exists and is negative.

Proof. By recalling (5.4), a simple reckoning will show that lim_Im{𝑋}→0(−Im{𝑋})⋅𝐼₁=¹⁰Re²{𝑋}4𝑢⁴+Re²{𝑋}^1/2⋅𝑑𝑢√1−𝑢⁴lim_Im{𝑋}→0,(−Im{𝑋})⋅𝐼₂=¹⁰Re²{𝑋}⋅𝑢²4+Re²{𝑋}⋅𝑢⁴^1/2⋅𝑑𝑢√1−𝑢⁴.(7.8)
Since 𝑢²4+Re²{𝑋}⋅𝑢⁴^1/2<14𝑢⁴+Re²{𝑋}^1/2(7.9) for every Re{𝑋}>0 and 𝑢∈(0,1), from (7.8) the first assertion of Proposition 7.1 follows.
By fixing Im{𝑋}=−1 and recalling (5.5), the convergence 𝒜→2 is equivalent to Re{𝑋}→1. This means that 𝑋 approaches the point 1−𝑖 indicated in Figure 4. An easy calculation will give us lim_𝒜→2𝐼₁=√2¹⁰2𝑢⁴−2𝑢²+1^−1/2⋅1−𝑢²1+𝑢²^1/2lim_𝒜→2𝑑𝑢,(7.10)𝐼₂=√2¹⁰2+2𝑢²+𝑢⁴^−1/2⋅1+𝑢²1−𝑢²^1/2𝑑𝑢.(7.11)
The integrand of (7.10) can be rewritten as 𝑢⁴+1−𝑢²²^−1/2⋅11−+𝑢𝑢²²^1/2=𝑢⁴1−𝑢²²+1^−1/2⋅1√1−𝑢⁴,(7.12) while one rewrites the integrand of (7.11) as 𝑢⁴+1+𝑢²²^−1/2⋅11+−𝑢𝑢²²^1/2=𝑢⁴1+𝑢²²+1^−1/2⋅1√1−𝑢⁴.(7.13)
Since 𝑢⁴1−𝑢²²>𝑢⁴1+𝑢²²(7.14) for every 𝑢∈(0,1), 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 𝑋 which belong to a curve 𝒞 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 𝑀_𝑋, where 𝑋∈𝒞. Figure 6 shows that the projection of the unitary normal on a fundamental domain of 𝑀_𝑋 is contained in the lower hemisphere of ℂ. This means that (𝑥₁,𝑥₃) is an immersion of 𝒟∶={𝑧∈ℂ∶|𝑧|<1and0<Arg(𝑧)<𝜋/2} in ℝ². Figure 7 shows a possible image of this map in ℝ²:

Figure 7 

A possible 𝑥2-projection of the fundamental domain on 𝑥1𝑂𝑥3.

It is not difficult to prove that the contour of the shaded region in Figure 7 is a monotone curve. The 𝑥₁-coordinate of the curve 𝐵𝐿 is given by the integral of −𝜙₁ as in (7.4). The integrand is clearly positive, hence this stretch is monotone. Regarding 𝐿𝑆^, where we can take 𝑍(𝑡)=𝑡, 0>𝑡>−∞, a simple reckoning gives us 𝑑ℎ||_{𝑍(𝑡)=𝑡}=𝑡𝑡⁴−Re𝑋²𝑡²+||𝑋||⁴⋅𝑑𝑡√𝑡²+4.(8.1)

Hence, the stretch 𝐿𝑆^ is also monotone. By using the symmetry 𝜌_ℎ, it follows that the whole contour indicated in Figure 7 is a monotone curve. Since the third coordinate of 𝐵𝐿 is increasing, the projections 𝐵𝐿 and 𝐿𝑆^ will intersect only at the point 𝐿. Nevertheless, it can happen that the projection 𝐿𝑆^ crosses 𝐵^𝐹^. 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 𝐿𝑆^ crosses the 𝑥₃-axis or not. A little reflection will show that, if 𝑔 does not take the value −𝑒^𝑖𝜋/4 along 𝐿𝑆^, then this curve does not intersect the vertical axis. Consequently, its projection will not intersect 𝐵^𝐹^. 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 𝒞, 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 𝑋∈−𝑖+(1,2√2) such that 𝑋∈𝒞 and 𝑀_𝑋 is embedded.

Proof. We will prove that 𝑔≠−𝑒^𝑖𝜋/4 along 𝐿𝑆^, for any 𝑋∈−𝑖+(1,2√2). Moreover, (𝐼₁−𝐼₂)|_{√𝑋=22−𝑖} will be positive. These two facts together with Proposition 7.1 will conclude Proposition 8.1.
By recalling (5.1), we would have 𝑔=−𝑒^𝑖𝜋/4 for some 𝑇∈(−∞,0) if and only if 𝑇²−𝒜²𝑇²−||𝑋||²=4𝒜Im{𝑋}⋅𝑇².(8.2)
Equation (8.2) will not be fulfilled by any 𝑇²∈(0,∞) providing |||𝒜₂+||𝑋||²+4𝒜Im{𝑋}|||<2𝒜||𝑋||,(8.3) or equivalently −Re{𝑋}Im{𝑋}<2√2.(8.4)
We have fixed Im{𝑋}=−1, hence 𝑔≠−𝑒^𝑖𝜋/4 along 𝐿𝑆^ for any 𝑋∈−𝑖+(1,2√2). Let us now verify that (𝐼₁−𝐼₂)|_{√𝑋=22−𝑖}>0. From (7.5) we have 𝐼₁||_{√𝑋=22−𝑖}=¹⁰9−2𝑢²4𝑢⁴−4𝑢²+9^1/2⋅𝑑𝑢√1−𝑢⁴,(8.5) and from (7.6) it follows that 𝐼₂||_{√𝑋=22−𝑖}=¹⁰2+9𝑢²4+4𝑢²+9𝑢⁴^1/2⋅𝑑𝑢√1−𝑢⁴.(8.6)
But 9−2𝑢²4𝑢⁴−4𝑢²+9^1/2>3−23𝑢²,∀𝑢∈(0,1),(8.7) and if we define 𝑎∶=1−11/√17 it is possible to prove that 2+9𝑢²4+4𝑢²+9𝑢⁴^1/2<𝑎𝑢²−2𝑎𝑢+1,∀𝑢∈(0,1).(8.8)
But 𝐼₁∶=¹⁰3−2𝑢²/3𝑑𝑢√1−𝑢⁴=3𝐼₂4𝐵41,12−61𝐵43,12,∶=¹⁰𝑎𝑢²−2𝑎𝑢+1𝑑𝑢√1−𝑢⁴=𝑎4𝐵34,12−𝑎2𝐵12,12+41𝐵41,21.(8.9)
Now we use 𝐵(𝑚,𝑛)=Γ(𝑚)Γ(𝑛)/Γ(𝑚+𝑛), Γ(1/4)=3,625600…, Γ(1/2)=√𝜋 and Γ(3/4)=1,225417… in order to conclude that 𝐼₁>𝐼₂.(8.10)
Together with (8.5)–(8.9), (8.10) shows that (𝐼₁−𝐼₂)|_{√𝑋=22−𝑖} is positive.

9. The Surfaces 𝐿_2,4

In order to prove Theorem 1.2, one follows very similar ideas already explained in Sections 3 to 7. For the surfaces 𝐿₂, consider Figures 8(a) and 8(b). The fundamental piece 𝑀 has genus 5, and 𝑂𝑥₂ passes through point 𝐴. The piece is invariant under 𝑟_𝑣 and 𝑟_ℎ, with quotient functions 𝜌_𝑣 and 𝜌_ℎ, respectively.

(a)

(b)

Figure 8 

(a) The fundamental domain of 𝑀; (b) the function 𝑧 on 𝑀.

Since 𝜒𝜌_ℎ𝜌^𝑣𝑀=1−5+8/22+42=2,(9.1) we may define 𝑧∶=𝜌_ℎ∘𝜌_𝑣∶𝑀→𝑆²≈ℂ, such that 𝑧(𝑆)=0,𝑧(𝐵)=1 and 𝑧(𝐿)=∞. The symmetries imply 𝑧(𝑆^)=0,𝑧(𝐵^)=1, and 𝑧(𝐿^)=∞ whereas 𝑧(𝐴) is a certain complex 𝑥 in the first quadrant. Moreover, there is a point in the segment 𝐵𝑆 at which 𝑔=1. After analysing the divisors of 𝑧 and 𝑔 on 𝑀, together with the behaviour of the unitary normal on symmetry curves and lines, we get 𝑔+𝑔1²=1/𝑎−𝑎|𝑥−𝑎|²⋅(𝑧−𝑥)𝑧−𝑥(𝑧+𝑎)²𝑧1−𝑧².(9.2) Since there is a point in the segment 𝐹𝑆 at which 𝑔=−𝑖, we should also have 𝑔−𝑔1²=1/𝑎−𝑎|𝑥−𝑎|²⋅(𝑧+𝑥)𝑧+𝑥(𝑧−𝑎)²𝑧1−𝑧².(9.3) In order to have equivalence between (9.2) and (9.3), a necessary and sufficient condition is 𝒜=𝑎+𝑎⁻¹=(|𝑥|²+1)/Re{𝑥}. Now, it is easy to get 𝑑ℎ=𝑖𝑑𝑧𝑧²−𝑥²𝑧²−𝑥²,(9.4) 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 𝐽₁=𝐽₂, where 𝐽₁=¹⁰(𝑡+𝑎)𝑑𝑡𝐽₂|𝑡+𝑥|𝑡1−𝑡²,=^∞1(𝑡−𝑎)𝑑𝑡|𝑡−𝑥|𝑡𝑡²−1.(9.5) The change 𝑡↦1/𝑡 for 𝐽₂ makes clear that 𝐽₁<𝐽₂ (𝐽₁>𝐽₂) providing 𝑅₁<𝑅₂ (𝑅₁>𝑅₂), where 𝑅₁=(𝑡+𝑎)/(1−𝑎𝑡) and 𝑅₂=|(𝑡+𝑥)/(1−𝑥𝑡)|, 0<𝑡<1. On the one hand, for a fixed 𝑟=Re{𝑥}>1, if Im{𝑥}→0 then 𝑎→1/𝑟, and consequently 𝑅₁<𝑅₂. On the other hand, by fixing Im{𝑥} and letting Re{𝑥}→0, then 𝑎→0 and so 𝑅₁>𝑅₂. In this case, notice that the singularity at 𝑡=1 of both integrands in (9.5) is easily removable with a change of variables. This means, no matter if we have 𝑅₁|_𝑡=1=𝑅₂|_𝑡=1, it still holds 𝐽₁>𝐽₂.

For the surfaces 𝐿₄, consider Figures 9(a) and 9(b). The fundamental piece 𝑀 has genus 9, and 𝑂𝑥₂ passes through point 𝐴. The piece is invariant under 𝑟_𝑣 and 𝑟_ℎ, with quotient functions 𝜌_𝑣 and 𝜌_ℎ, respectively. We will have 𝑔−𝑔⁻¹=∞ if and only if 𝑧+𝑧⁻¹∈{±𝑖,0,∞}. Moreover, 𝑔−𝑔⁻¹=0 if and only if 𝑧∈{−𝑥,𝑥,−𝑥⁻¹,𝑥⁻¹,𝑖𝑎,−𝑖𝑎⁻¹}, where 𝑎∈(0,1).

(a)

(b)

Figure 9 

(a) The fundamental domain of 𝑀; (b) the function 𝑧 on 𝑀.

From this point on we redefine the following: 𝑍∶=𝑧⁻¹+𝑧,𝑋∶=𝑥⁻¹+𝑥,𝒜∶=𝑎⁻¹−𝑎.(9.6)

Based on Figure 9 it is not difficult to conclude that 𝑔−𝑔1²=𝑖𝑐𝑍³⋅(𝑍+𝑖𝒜)²(𝑍+𝑋)𝑍−𝑋,(9.7) where 𝑐 is given by (4.3) again. Moreover, (9.7) is equivalent to 𝑔+𝑔1²=𝑖𝑐𝑍³⋅(𝑍−𝑖𝒜)²𝑍+𝑋(𝑍−𝑋).(9.8) Similar arguments as in Section 5 will give again (5.3) and (5.4), but unlike (5.5) there is no restriction now. Regarding 𝑑ℎ, it still holds (6.4), but unlike Figure 5 the period problem is now illustrated by Figure 10.

Figure 10 

𝑥3-projection of the fundamental domain with an open period.

Integrals 𝐼₁ and 𝐼₂ are again given by (7.5) and (7.6), but now the period is solved when 2𝐼₁=𝐼₂. This will come with the following.

Proposition 9.1. For any fixed positive value of Re{𝑋}one has that the following limit exists and is positive: lim_Im{𝑋}→0(−Im{𝑋})⋅2𝐼₁−𝐼₂.(9.9) For Im{𝑋}=−1 one has that lim_𝒜→0(I₁−2I₂) 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 𝐿_2,4 are even easier than the ones used in Section 8 for 𝐶₂, 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 “Bolsa de Produtividade Científica” from CNPq—Conselho Nacional de Desenvolvimento Científico e Tecnológico, and “Bolsa de Pós-Doutorado” FAPESP 2000/07090-5.