# Minimal Surfaces with Only Horizontal Symmetries

Authors: Márcio Fabiano da Silva; Guillermo Antonio Lobos; Valério Ramos Batista
Journal: ISRN Geometry
 (2011)

---

## Abstract

The Schwarz reflection principle states that a minimal surfaceS 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.


---

## Body

## 1. Introduction

During theClay  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:“Ahorizontal  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 thecone  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 inℝ3 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 inℝ3. 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 length23 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 surfacesC2 and L2,4 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 L2,4.(a) The surfacesC2; (b) the surfaces L2.
(a)(b)(a) Schwarz’s D-surfaces; (b) the surfacesL4.
(a)(b)The reader will notice that the surfacesC4, 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 inℝ3, 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 inℝ3. 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.
Fork=2,4, there exists a one-parameter family of triply periodic minimal surfaces in ℝ3, of which the members are called Lk, and for any of them the following holds. (a)
The quotient by its translation group has genus2k+1.(b)
The whole surface is generated by a fundamental piece, which is a surface with boundary inℝ3. 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.Sections3 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.
LetX: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,…,pr.Definition 2.2.
LetP={p1,…,pr} as in Theorem 2.1. An end is the image under X of a punctured neighbourhood Vp of a point p∈P such that P∩Vp=∅. We say that the surface has no  ends when P=∅.Theorem 2.3 (Weierstraß Representation).
LetR 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
(2.1)X(p):=Re∫p(ϕ1,ϕ2,ϕ3),where(ϕ1,ϕ2,ϕ3):=12(1g-g,ig+ig,2)dh,
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 Theorems2.1 and 2.3, the Weierstraß data (g,dh) extend meromorphically on R¯.The functiong is the stereographic projection of the Gauß map N:R→S2 of the minimal immersion X. It is a covering map of ℂ̂ and ∫SKdA=-4πdeg(g). These facts will be largely used throughout this work.


## 3. The SurfacesM̅ and the Functions z

Consider the surface indicated in Figure1(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 M̅. It is not difficult to conclude that it has genus 9. The fundamental domain of M̅ is the shaded region indicated in Figure 3(a).(a) The fundamental domain ofM; (b) the function z on M.
(a)(b)The surfaceM̅ is invariant under 180∘-rotations around the directions x⃗3 and x⃗2. These rotations we call rv and rh, respectively, (see Figure 3(a)). Based on this picture, one sees that the fixed points of rv are S, S′, L, L′, F, F′ and the images of S and S′ under the symmetries of M̅. They sum up 8 in total. The quotients by rv and rh we call ρv and ρh, respectively. The surface ρv(M̅) is still invariant under the rotation rh. In this case, the fixed points of rh will be ρv(A), ρv(A′) and their images under the symmetries of ρv(M̅). They sum to 8 in total. Because of that,
(3.1)χ(ρh(ρv(M̅)))=1-9+8/22+82=2.Let us definez:=ρh∘ρv:M̅→S2≈ℂ̂, such that z(S)=0, z(L)=1 and z(B)=i. The involutions of M̅ are induced by ρv and ρh on ℂ̂, and since all the involutions of ℂ̂ 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 M̅, 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∈ℂ̂ such that |x|<1 and Arg(x)∈(0,π/2). Consequently, z(A′)=-x̅ and one easily gets the other values of z at the images of A and A′ under the symmetries of M̅.


## 4. Theg-Function on M̅ in Terms of z

First of all, observe that Jorge-Meeks’ formula givesdeg(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,x̅,x-1,-x̅-1,ia,ia-1}, where a∈(0,1). From this point on we introduce the following notation:
(4.1)Z∶=z-z-1,X∶=x-1-x,A∶=a+a-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
(4.2)(g-1g)2=-icZ3⋅(Z-iA)2(Z-X)(Z+X̅),
where c is a positive constant. Now we define M̅ 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 M̅ 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
(4.3)c=AA2+2AIm⁡{X}+|X|2.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→z̅. In this case we will get g→ig̅ or g→-ig̅. Therefore
(4.4)(g+1g)2=-icZ3⋅(Z+iA)2(Z-X̅)(Z+X).At this point we are ready to prove thatM̅ has genus 9. The function z is a four-sheeted branched covering of the sphere. The values 0, ∞, ±1, ±x±1, ±x̅±1 correspond to the only branch points of z, all of them of order 2, and each of these values is taken twice on M̅. Therefore, from the Riemann-Hurwitz’s formula we have
(4.5)genus(M̅)=12⋅(2-1)⋅22-4+1=9.Now we are ready to find some relations that the parametersa, 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 M̅.


## 5. Conditions on the Parametersa, c, and x

Consider the curvesS′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∈eiπ/4ℝ and on F′S one should have g∈e-iπ/4ℝ. Let us now verify under which conditions this will really happen.OnS′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:
(5.1)(g-g-1)2|z(t)=-icT3⋅(T-iA)2(T2-2iIm⁡{X}⋅T-|X|2).Therefore,(5.2)Re{(g-g-1)2}=-2cT2⋅(Im⁡{X}⋅T2+AT2-Im⁡{X}⋅A2-A|X|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
(5.3)c=1A+Im⁡{X},(5.4)A=-|X|2Im⁡{X}.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|eiθ 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
(5.5)Re2{X}>-2Im⁡{X}-Im2{X}.Figure4 illustrates the X-domain established by (5.5), and we recall that |x|<1 and θ∈(0,π/2).Figure 4
Definition domain of theX-variable satisfying (5.4).It is not difficult to prove that (5.5) is equivalent to the following inequality:
(5.6)|x|<12{sin⁡θ+1+3cos⁡2θ-2sin⁡θ(1+3cos⁡2θ-sin⁡θ)}.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 M̅.Table 1SymmetryInvolutionz-valuesg-valuesSB(g,z)→(g̅,-z̅)it,0<t<1g∈ℝ+BL(g,z)→(-g̅,1/z̅)eit,π/2>t>0g∈ℝ+LS′(g,z)→(ig̅,z̅)t,1>t>0g∈eiπ/4ℝS′B′(g,z)→(g̅,-z̅)it,0<t<1g∈ℝ-B′F′(g,z)→(-g̅,1/z̅)eit,π/2>t>0g∈ℝ-F′S(g,z)→(-ig̅,z̅)t,1>t>0g∈e-iπ/4ℝρh(g,z)→(-1/g,z)-x,x̅,x-1,-x̅-1g=±iNotice that the points(g,z)=(±i,-ia±1) do not come out as fixed points of ρh 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 Differentialdh on M̅

Since the surfaceM 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(M̅). 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,±x̅±1}) and a pole of order 3 at the points z-1({∞}). Let the symbol ~ indicate that two meromorphic functions on M̅ differ by a nonzero proportional constant. It is not difficult to conclude that
(6.1)(dhdz)2~(z2-1)2(z2-x2)(z2-x-2)(z2-x̅2)(z2-x̅-2).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
(6.2)(Z-X)(Z+X̅)(Z-X̅)(Z+X)=-Z6c2(Z2+A2)2(g2-1g2)2,
which allows us to define
(6.3)(Z2-X2)(Z2-X̅2)∶=iZ3c(Z2+A2)(g2-1g2).Now we apply (6.3) to (6.1) and obtain
(6.4)dh=Z(Z2-X2)(Z2-X̅2)⋅dzz.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 2Symmetryz-valuesg∈dh(ż)∈SBit,0<t<1ℝ+iℝBLeit,π/2>t>0ℝ+ℝLS′t,1>t>0eiπ/4ℝℝS′B′it,0<t<1ℝ-iℝB′F′eit,π/2>t>0ℝ-ℝF′St,1>t>0e-iπ/4ℝℝFrom Table2 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 ofℝ3, 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 symmetryρh really exists in ℝ3.(2)
After an orthonormal projection of the fundamental domain in the directionx3, 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 pathP→A→P′ on M̅ 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:
(7.1)(x1,x2,x3)|(g,z)→(-1/g,z)=Re∫P=(g0,z0)A=(-i,x)(ϕ1,ϕ2,ϕ3)=Re∫P′=(-1/g0,z0)A=(-i,x)(ϕ1,-ϕ2,ϕ3)=Re∫P=(g0,z0)A=(-i,x)(-ϕ1,ϕ2,-ϕ3)=(-x1,x2,-x3).The pathP→A→P′ on M̅.
(a)(b)Therefore, our minimal surface isreally invariant under 180°-rotations around the x2-axis. This proves the existence of the symmetry ρh of our initial assumptions.Now we are ready to deal with the second condition. Consider Figure5 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
(7.2)Re∫SBϕ2=Re∫BLϕ1.OnSB we can take Z(t)=it, 2<t<∞. This implies that dz/z=-dt/t2-4. From (4.4) and (6.4) we have
(7.3)ϕ2|Z(t)=it=c1/2(t+A)t1/2(t2-2Im⁡{X}⋅t+|X|2)1/2⋅dtt2-4.OnBL we can take z(t)=eit, 0<t<π/2. From (4.2) and (6.4) it follows that
(7.4)ϕ1|z(t)=eit=12⋅c1/2(A-2sin⁡t)(4sin2t+4Im⁡{X}⋅sin⁡t+|X|2)1/2⋅dtsin⁡t.Now defineI1∶=(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:
(7.5)I1=∫01A-2u2(4u4+4Im⁡{X}⋅u2+|X|2)1/2⋅du1-u4,(7.6)I2=∫012+Au2(4-4Im⁡{X}⋅u2+|X|2u4)1/2⋅du1-u4.The next proposition will solve the period problem given by (7.2).Proposition 7.1.
For any fixed positive value ofRe{X} one has that the following limit exists and is positive
(7.7)lim⁡Im⁡{X}→0(-Im⁡{X})⋅(I1-I2).
For Im{X}=-1 one has that lim𝒜→2(I1-I2) exists and is negative.Proof.
By recalling (5.4), a simple reckoning will show that
(7.8)lim⁡Im⁡{X}→0(-Im⁡{X})⋅I1=∫01Re2{X}(4u4+Re2{X})1/2⋅du1-u4,lim⁡Im⁡{X}→0(-Im⁡{X})⋅I2=∫01Re2{X}⋅u2(4+Re2{X}⋅u4)1/2⋅du1-u4.
Since(7.9)u2(4+Re2{X}⋅u4)1/2<1(4u4+Re2{X})1/2
for every Re{X}>0 and u∈(0,1), from (7.8) the first assertion of Proposition 7.1 follows.
By fixingIm{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
(7.10)lim⁡A→2I1=2∫01(2u4-2u2+1)-1/2⋅(1-u21+u2)1/2du,(7.11)lim⁡A→2I2=2∫01(2+2u2+u4)-1/2⋅(1+u21-u2)1/2du.
The integrand of (7.10) can be rewritten as
(7.12)(u4+(1-u2)2)-1/2⋅(1-u21+u2)1/2=[u4(1-u2)2+1]-1/2⋅11-u4,
while one rewrites the integrand of (7.11) as
(7.13)(u4+(1+u2)2)-1/2⋅(1+u21-u2)1/2=[u4(1+u2)2+1]-1/2⋅11-u4.
Since(7.14)u4(1-u2)2>u4(1+u2)2
for every u∈(0,1), the last assertion of Proposition 7.1 follows.Proposition7.1 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 byMX, where X∈𝒞. Figure 6 shows that the projection of the unitary normal on a fundamental domain of MX is contained in the lower hemisphere of ℂ̂. 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 possiblex2-projection of the fundamental domain on x1Ox3.It is not difficult to prove that the contour of the shaded region in Figure7 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
(8.1)dh|Z(t)=t=tt4-Re{X2}t2+|X|4⋅dtt2+4.Hence, the stretchLS′ is also monotone. By using the symmetry ρh, 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 curveLS′ crosses the x3-axis or not. A little reflection will show that, if g does not take the value -eiπ/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 curve𝒞, 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 anX∈-i+(1,22) such that X∈𝒞 and MX is embedded.Proof.
We will prove thatg≠-eiπ/4 along LS′, for any X∈-i+(1,22). 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=-eiπ/4 for some T∈(-∞,0) if and only if
(8.2)(T2-A2)(T2-|X|2)=4AIm⁡{X}⋅T2.
Equation (8.2) will not be fulfilled by any T2∈(0,∞) providing
(8.3)|A2+|X|2+4AIm⁡{X}|<2A|X|,
or equivalently
(8.4)-Re{X}Im⁡{X}<22.
We have fixedIm{X}=-1, hence g≠-eiπ/4 along LS′ for any X∈-i+(1,22). Let us now verify that (I1-I2)|X=22-i>0. From (7.5) we have
(8.5)I1|X=22-i=∫019-2u2(4u4-4u2+9)1/2⋅du1-u4,
and from (7.6) it follows that
(8.6)I2|X=22-i=∫012+9u2(4+4u2+9u4)1/2⋅du1-u4.
But(8.7)9-2u2(4u4-4u2+9)1/2>3-23u2,∀u∈(0,1),
and if we define a:=1-11/17 it is possible to prove that
(8.8)2+9u2(4+4u2+9u4)1/2<au2-2au+1,∀u∈(0,1).
But(8.9)Ĩ1:=∫01(3-2u2/3)du1-u4=34B(14,12)-16B(34,12),Ĩ2:=∫01(au2-2au+1)du1-u4=a4B(34,12)-a2B(12,12)+14B(14,12).
Now we useB(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
(8.10)Ĩ1>Ĩ2.
Together with (8.5)–(8.9), (8.10) shows that (I1-I2)|X=22-i is positive.


## 9. The SurfacesL2,4

In order to prove Theorem1.2, 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 M̅ has genus 5, and Ox2 passes through point A. The piece is invariant under rv and rh, with quotient functions ρv and ρh, respectively.(a) The fundamental domain ofM; (b) the function z on M.
(a)(b)Since(9.1)χ(ρh(ρv(M̅)))=1-5+8/22+42=2,
we may define z:=ρh∘ρv:M̅→S2≈ℂ̂, 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 M̅, together with the behaviour of the unitary normal on symmetry curves and lines, we get
(9.2)(g+1g)2=1/a-a|x-a|2⋅(z-x)(z-x̅)(z+a)2z(1-z2).
Since there is a point in the segment FS at which g=-i, we should also have
(9.3)(g-1g)2=1/a-a|x-a|2⋅(z+x)(z+x̅)(z-a)2z(1-z2).
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
(9.4)dh=idz(z2-x2)(z2-x̅2),
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
(9.5)J1=∫01(t+a)dt|t+x|t(1-t2),J2=∫1∞(t-a)dt|t-x|t(t2-1).
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 surfacesL4, consider Figures 9(a) and 9(b). The fundamental piece M̅ has genus 9, and Ox2 passes through point A. The piece is invariant under rv and rh, with quotient functions ρv and ρh, 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,x̅,-x-1,x̅-1,ia,-ia-1}, where a∈(0,1).(a) The fundamental domain ofM; (b) the function z on M.
(a)(b)From this point on we redefine the following:(9.6)Z:=z-1+z,X:=x-1+x,A:=a-1-a.Based on Figure9 it is not difficult to conclude that
(9.7)(g-1g)2=icZ3⋅(Z+iA)2(Z+X)(Z-X̅),
where c is given by (4.3) again. Moreover, (9.7) is equivalent to
(9.8)(g+1g)2=icZ3⋅(Z-iA)2(Z+X̅)(Z-X).
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.IntegralsI1 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 ofRe{X}one has that the following limit exists and is positive:
(9.9)lim⁡Im⁡{X}→0(-Im⁡{X})⋅(2I1-I2).
For Im{X}=-1 one has that lim𝒜→0(I1-2I2) exists and is negative.The proof of Proposition9.1 is quite similar to the proof of Proposition 7.1, and so we will omit it here. The arguments for the embeddedness of L2,4 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.



---

*Source: 943462-2011-07-19.xml*