Scherk Surfaces in Heisenberg Groupswebdoc.sub.gwdg.de/ebook/dissts/Bochum/Allouss2008.pdf · SCHERK SURFACES IN HEISENBERG GROUPS BELAID ALLOUSS Abstract. In a recent paper Uwe Abresch

Scherk Surfaces in Heisenberg Groups

Inauguraldissertation

zur

Erlangung des Grades eines Doktorsder Naturwissenschaften

an der Fakultat fur Mathematikder Ruhr-Universitat Bochum

eingereicht im Januar 2008 von

Belaid Allouss

geboren am 09.09.1975 in Beni Chiker, Marokko

Bochum 2008

Tag der Disputation: 25.02.2008

Dekan: Prof. Dr. Hans Ulrich Simon

Gutachter: Prof. Dr. Uwe AbreschProf. Dr. Gerhard Knieper

SCHERK SURFACES IN HEISENBERG GROUPS

BELAID ALLOUSS

Abstract. In a recent paper Uwe Abresch and Harold Rosenberg de-scribed local Scherk graphs Σr in the Heisenberg group (Nil(3), gτ ) [3].Their plan was to use them as comparison surfaces in order to obtaincurvature bounds for global minimal graphs and finally solve the Bern-stein problem in (Nil(3), gτ ). Therefore, we will study a construction ofsuch surfaces by using a generalized Weierstraß representation in termsof spinors [31]. Resulting from the integrable system approach we derivea reduced Dirac equation and a modified Sinh-Gordon equation.

Ruhr-Universitat BochumFakultat fur MathematikUniversitatsstr. 150D–44780 Bochum, Germany

Key words and phrases. minimal surfaces, Scherk surfaces, Weierstraß representation,Heisenberg groups, Lie groups, spinors, Dirac equation.

Acknowledgement

I would like to express my profound respect and sincere gratitude to Prof.Dr. Uwe Abresch for the great opportunity by confiding this project tome. He has been very helpful and supportive during the development of mydoctoral thesis, providing me invaluable scientific assistance and encouragingme with many comments and inspiring discussions.

My deep gratitude to Mrs. Ramona Strankowski for her encouragement andadvice since the very beginning of this project.

I would also like to express my gratitude and appreciation to my friendsand colleagues: Hulya Cigerli, Meike Detering, Peter Kailuweit, Ralf Muno,Sani Noor, Markus Schwabe, Beate Strunk, Andre Thrun.

Finally, I would like to thank my parents and my siblings for their supportand motivating words even though they are still wondering what I was doing.

SCHERK SURFACES IN HEISENBERG GROUPS

Contents

Introduction 1

1. Preliminary Facts 5

1.1. Local Representation of Minimal Surfaces in R3 5

1.2. Global Representation 8

1.3. Dirac Operator and Spin Bundles 12

1.4. Holomorphic Bundles 17

1.5. Heisenberg Groups 21

2. Immersion of Scherk Surfaces in (Nil(3), gτ ) 24

2.1. Divisors and Coverings 29

2.2. First Concept 34

2.3. Second Concept 35

2.4. The Global Dirac Equation for Scherk Surfaces 40

3. A Modified Sinh-Gordon Equation 41

References 46

SCHERK SURFACES IN HEISENBERG GROUPS 1

Introduction

Scherk surfaces serve as important tools in the study of certain limitingminimal surface problems and in the study of harmonic diffeomorphismsof the hyperbolic space. Similar minimal surface problems are also studiedon other quadrilaterals in the Euclidean plane resp. hyperbolic plane. In2006, Harold Rosenberg and Pascal Collin used hyperbolic Scherk surfacesto construct a harmonic diffeomorphism from the complex plane onto thehyperbolic plane, using the Poincare disk model D. They were able todisprove the Schoen-Yau conjecture [11].There is no doubt that in the last decade a lot of new knowledge has beenachieved concerning 2-dimensional constant mean curvature (cmc) surfacesΣ immersed into homogeneous 3-manifolds [7, 13, 14, 16, 17, 30, 31, 32].For example the classical Jenkins-Serrin theorem for minimal graphs in R3

is crucial for the constructions of new minimal surfaces with non compactpolygonal boundary. An intriguing challenge was to look for an analogoustheorem resp. concept in general ambient spaces. Indeed, there has beenmade good progress for homogeneous 3-manifolds [12, 24, 27]. Anotherleading light for the present development is an invariant on 2-dimensionalcmc surfaces Σ. In 1951 Heinz Hopf introduced the well known quadraticHopf differential Q, emerging as a powerful tool to understand cmc surfacesconformally immersed into 3-dimensional space forms [19]. His essentialcontribution lies in the verification that the (2, 0)-part Q := π2,0(hΣ) of thesecond fundamental form hΣ is a holomorphic quadratic differential on suchsurfaces.In recent years an analogous result for 3-dimensional homogeneous targetspaces with an at least 4-dimensional isometry group was found by UweAbresch and Harold Rosenberg. In this context their first joint paper [2]deals with the introduction of a modified holomorphic quadratic differentialQ for cmc surfaces Σ immersed into product spaces S2×R and H2×R. Thisresult has inspired a lot of new research works on cmc surfaces in generalambient spaces [7, 13, 16, 32].The subsequent joint paper [3] describes the extension of this concept for theremaining homogeneous 3-manifolds. Moreover, as an example, they pointout the existence of global minimal surfaces in 3-dimensional Heisenberggroups (Nil(3), gτ ) which have similar properties like the doubly-periodicScherk surface in Euclidean 3-space. More precisely, it is in fact a Jenkins-Serrin graph in (Nil(3), gτ ). One can describe this surface as a graph overeach of the black squares in a suitable checkerboard tiling of the plane. It alsocontains the vertical lines over the vertices of this tiling. Their motivationfor this construction is to obtain comparison surfaces. As a consequence,the explicit solutions of such Scherk surfaces would help to find curvatureestimates for global minimal graphs and, in conclusion, to solve the generalBerstein problem in (Nil(3), gτ ) [4, 8, 10, 29].

The subject of my doctoral thesis is to derive and discuss several partialdifferential equations (PDE) for the unknown conformal immersion Fτ ofScherk surfaces Σ ⊂ CP1 into 3-dimensional Heisenberg groups (Nil(3), gτ )with left-invariant metric gτ = dx2 + dy2 + (dz − 1

2 τ(xdy − ydx))2. One

2 BELAID ALLOUSS

can think of a family of surfaces realized by an analytical deformation withparameter τ . Here we locally consider (Nil(3), gτ ) as a 1-dimensional fibra-tion over the flat two-space with the bundle curvature τ . The unit vectorfield tangent to this fibration is a Killing field. Now, the existence anduniqueness of such surfaces for all bundle curvatures τ is ensured by a gen-eralized Jenkins-Serrin theorem for homogeneous 3-manifolds [12]. SinceR3 = (Nil(3), g0), we begin to study these PDEs by considering the knowngeometric data from the classical immersion of this minimal surface in R3.In addition, as an application we use the modified Hopf differential Q to getmore information concerning the analytic extension of the solutions in theirsingularities.Historically seen, the classical Scherk surface in R3 appeared for the firsttime in a paper of Heinrich Ferdinand Scherk submitted to the Royal Dan-ish Academy of Sciences and Letters in Copenhagen in September 1833 [28].He studied all the solutions of the non-parametric minimal surface equation

(1 + ζ2v )ζuu − 2ζuζvζuv + (1 + ζ2

u)ζvv = 0

which are written in the form of graphs (u, v, ζ(u, v)) with ζ(u, v) = f(u) +g(v) and which suffice the constraints ζ(0, 0) = 0, ∇ζ(0, 0) = 0. Using thetheory of ordinary differential equations he solved the resulting initial valueproblem and obtained

ζ(u, v) = 1c ln cos(cu)

cos(cv) .

The solutions are only defined on the open squares

Ω := (u, v) ∈ R2 : |u+ k πc | <π2c , |v + lπc | <

π2c , k, l ∈ Z

where k and l are both either even or odd.The geometry of (Nil(3), gτ ) does not make this approved method available.Instead of this, we use the generalized Weierstraß representation for surfacesin 3-dimensional Lie groups introduced by Iskander Taimanov and DimitryBerdinsky in [7]. For our purposes, this representation is written in terms ofspinors ψ±τ defined over Σ. These spinors ψ±τ depend on the bundle curvatureτ and are interpreted as square roots of differentials respectively 1-formsdz ∈ T ∗Σ =: K2. The first step in this thesis is to find an appropiateidentification for these sections of line bundles K. Our first main resultstates.

Theorem. Let π : Σ → Σ be a two-sheeted covering map of Σ and L2 :=π∗(K2) the respective pull back of the space of 1-forms. Due to the geometryof Σ there exist meromorphic sections σ1, σ2 ∈ Γ(L) which determine thebundle L over Σ uniquely so that the divisors of σi are

Div(σ1) = −4∑i=1

(qi, 0) +2∑i=1

(pi∞), Div(σ2) = −4∑i=1

(qi, 0) +2∑i=1

(pi0)

and

σ21 = π∗θ1, σ2

2 = π∗θ3, σ2 = zσ1

where pi0, pi∞ are the preimages of 0, ∞ ∈ CP1 under the covering map π.


For the generalized Weierstraß representation resp. spinor representationwe get certain integrability conditions. Equivalently, ψ±τ satisfy a non-linearDirac equation. At this, the known Weierstraß data for the classical Scherksurface in R3 lead to a reduced system of differential equations. This resultis stated in our second theorem.

Theorem. Let Σ ⊂ CP1 # (Nil(3), gτ ) be a conformal immersion of Scherksurfaces into the 3-dimensional Heisenberg group (Nil(3), gτ ). Then theirspinor representation ψτ yields the following system of differential equations

∂zf−τ ⊗ dz = i

τ

2|zf+

τ |2 − |f−τ |2

z4 − 1· zf+

τ ⊗ dz,

∂z(zf+τ )⊗ dz = −iτ

2|zf+

τ |2 − |f−τ |2

z4 − 1· f−τ ⊗ dz,

with ψ+τ = zf+

τ ψ+

0 and ψ−τ = f−τ ψ−0 . Here, the spinor representation ψ±0 of the

classical Scherk surface immersed into R3 are interpreted as basis spinors.Furthermore, the complex-valued functions f±τ are holomorphic and boundedin Σ.

We give a brief outline of the methods in order to derive solutions for thisproblem. First, a basic approach to get more information one can takeadvantage of existing symmetries of the sought surface in (Nil(3), gτ ) byinserting them into the Dirac equation. Another way is the pertubationtheory, i.e. find an approximate solution to this problem by starting fromthe exact solution of the related problem in R3. Our non-linear systemdepends on the bundle curvature τ . Therefore, we get an expression forthe desired solution f±τ in terms of a power series with respect to a smallparameter τ that quantifies the deviation from the exactly solvable problem.

A further observation has been made with the help of the modified Hopfdifferential of surfaces in (Nil(3), gτ )

A = (ψ−τ ∂zψ+τ − ψ+

τ ∂zψ+τ ) + 2Hiτ

2H+iτ (ψ+τ )2(ψ−τ )2.

and the results in [7, 32]. The holomorphicity of A on cmc surfaces and thepotential of a more general Dirac equation leads to

Proposition. Consider the immersion Fτ of general surfaces in (Nil(3), gτ ).Then the potential of the resulting Dirac equation in terms of ψ±τ is

U(Nil(3),gτ ) := H2

(|ψ+τ |2 + |ψ−τ |2

)+ i τ4

(|ψ−τ |2 − |ψ+

τ |2)

Combining this with the modified Hopf differential A yields a system of socalled structure equations

∂zψτ =(

wz1τ e−wA

−i τ4 ew 0

)ψτ , ∂zψτ =

(0 i

τ4 ew

1τ e−w ¯A wz

)ψτ

with ψτ := t(ψ+τ , ψ

−τ ) and ew := |ψ−τ |2 − |ψ+

τ |2. Finally, by using the zero-curvature condition ∂z∂zψτ = ∂z∂zψτ a Sinh-Gordon equation for minimalsurfaces in (Nil(3), gτ ) is derived

wzz + 1τ2 e−2w|A|2 − τ2

16e2w = 0.

4 BELAID ALLOUSS

Solving this equation means to identify the potential U(Nil(3),gτ ) = i4W .

There is an important link between this equation and a certain hyperbolicGauss map G : Σ → H2 introduced by Isabel Fernandez and Pablo Mira[16]. They proved that the hyperbolic Gauss map is harmonic for cmc-1

2

surfaces conformally immersed into H2 × R. This map is equivalent to thepotential i

4W of the Dirac equation. However, the analytic continuationof our potential just fails in the zeroes of p(z) = z4 − 1 in C, while theharmonicity of the hyperbolic Gauss map fails in a larger range, namely in∂∞D of the Poincare disk model of H2.

The thesis is organized as follows. In section 1 we begin to revise the classicalsurface theory in R3. Only the basic facts that we require for our purposesare outlined there. After that we discuss the general properties of spinorbundles and Dirac operators in the Clifford algebra. Additionally, we recordmeromorphic sections σi of the spinor bundle which are derived from theconformal structure of the sought Scherk surfaces. They are defined onthe Riemann surface Σ, the two-sheeted covering of the 4-punctured sphereΣ ⊂ CP1 with certain zeroes as branching points. We learn more about thesesections by including involutions given by the symmetries of the surfaces.Then we review the facts about hermitian holomorphic bundles on complexmanifolds. Finally, for our main contributions we manage the set up for thenecessary computations. In other words, we deal with the Weierstraß dataof Scherk surfaces in R3, the comparison of the classical Heisenberg groupH3 known in the literature with the general one (Nil(3), gτ ), and we beginwith the construction of a left-invariant metric in Heisenberg groups. Thelast two sections are devoted to the main results which are derived by theabove mentioned methods.


1. Preliminary Facts

We begin with some basic facts concernig minimal surfaces Σ in R3. The fol-lowing subsections contain necessary preparations in order to formulate theEnneper-Weierstraß representation formula [15, 25]. As an application ofthis formula we write down the Weierstraß representation of the Scherk sur-face. Finally, we also calculate geometric quantities of this minimal surfacecorresponding to the given representation.

1.1. Local Representation of Minimal Surfaces in R3. We want tostudy properties of a surface which are independent of the choice of pa-rameters. In other words it is convenient to choose parameters which aredetermined by the effect that geometric properties of the surface are reflectedin the parameter plane. Therefore, we use conformal maps from the param-eter plane onto the surface. These maps keep the angles between curves inthe parameter plane equal to those between the correspondig curves on thesurface. The expression of this condition in terms of the fundamental formare the following

g11 = g22, g12 = 0 (1.1)or

gij = λ2δij , λ = λ(u) > 0. (1.2)

The parameters u, v of the map

X : D ⊂ R2 −→ Σ ⊂ R3

u 7−→ X(u)

with u = (u, v) ∈ D satisfying the above conditions are called isothermalparameters.The following results turn out to be very useful for further purposes.

Lemma 1.1. Let a regular surface Σ be defined by X(u) ∈ C2(D,R3) whereu, v are isothermal parameters. Then

4X = 2λ2H

where H is the mean curvature vector.

The immediate consequence of this lemma is the following

Lemma 1.2. Let X(u) ∈ C2(D,R3) define a regular surface Σ in isother-mal parameters. The necessary and sufficient condition for the coordinatefunctions xi(u, v) to be harmonic is that Σ is a minimal surface.

Let us examine the further connection between harmonic functions and mi-nimal surfaces. Therefore, we use the following notation like in [25]. Givena surface X(u), we consider the complex-valued functions

%k(z) =∂xk∂u− i∂xk

∂v; z = u+ iv. (1.3)

6 BELAID ALLOUSS

We have the identities3∑

k=1

%2k(z) =

∣∣∣∣∂X∂u∣∣∣∣2 − ∣∣∣∣∂X∂v

∣∣∣∣2 − 2i∂X

∂u· ∂X∂v

(1.4)

= g11 − g22 − 2ig12

and3∑

k=1

|%k(z)|2 =∣∣∣∣∂X∂u

∣∣∣∣2 +∣∣∣∣∂X∂v

∣∣∣∣2 = g11 + g22 (1.5)

¿From these identities we can conclude the following properties

i) %k(z) is holomorphic in z ⇐⇒ xk is harmonic in u, v.ii) u, v are isothermal parameters

⇐⇒3∑

k=1

%2k(z) ≡ 0 (1.6)

iii) if u, v are isothermal parameters, then Σ is regular

⇐⇒3∑

k=1

|%k(z)|2 6= 0. (1.7)

The preceding results lead to

Theorem 1.3. Let X(u) define a regular minimal surface, with isothermalparameters u, v. Then the functions %k(z) defined in (1.3) are holomorphic,and they satisfy (1.6) and (1.7). Conversely, let %k(z) be holomorphic func-tions of z which satisfy (1.6) and (1.7) in a simply connected domain D.Then there exists a regular minimal surface X(u) defined on D, such thatequations (1.3) are valid.

For our purposes we just discuss the 3-dimensional case for minimal surfaces.In fact, for this case there exists an explicit solution of the equation

%21 + %2

2 + %23 = 0. (1.8)

Lemma 1.4. Let D be a domain in the complex z-plane, g(z) an arbitrarymeromorphic function in D and f(z) an holomorphic function in D havingthe property that at each point where g(z) has a pole of order m, f(z) has azero of order at least 2m. Then the functions

%1 = f(1− g2), %2 = i · f(1 + g2), %3 = 2fg (1.9)

will be holomorphic in D and satisfy (1.8). Conversely, every triple of holo-morphic functions in D satisfying (1.8) may be represented in the form (1.9),except for %1 ≡ i%2, %3 ≡ 0.

Now we can formulate

Theorem 1.5. Every simply-connected minimal surface in R3 can be rep-resented in the form

xk(z) = <e∫ z

0%k(ζ)dζ

+ ck (1.10)


where the %k are defined by (1.9), the functions f and g having the propertiesstated in Lemma 1.4, the domain D being either the unit disk or the entireplane, and the integral being taken along an arbitrary path from the originto the point z. The surface will be regular iff f satisfies the further propertythat it vanishes only at the poles of g, and the order of its zeroes at such apoint is exactly twice the order of the pole of g.

We call f and g the Weierstraß data of the minimal surface. There is animportant link between the meromorphic function g and the Gauss map N .We again suppose that we have a conformal parametrization F : Σ # R3,where Σ is a Riemann surface immersed as a minimal surface into R3. Σ hasa natural orientation which F (Σ) inherits. Thus, there is a continuous choiceof unit normal at every point in F (Σ) which, together with the orientationof F (Σ), gives the standard right-hand orientation of R3. Each unit normalcorresponds to a point of S2 ⊂ R3 which is the local definition of the Gaussmap N : F (Σ)→ S2. If we identify S2 with the Riemann sphere C∪ ∞ ∼=CP1 through stereographic projection then the composition g = N F : Σ→S2 is a meromorphic function on Σ. As mentioned before, it turns out thatF (Σ) can then be recovered from g by the following integral, where µ is aholomorphic 1-form which depends on g:

F (z) =∫ z

0(1− g2, i(1 + g2), 2g)µ.

As an application, we study the Scherk surface in R3. This minimal surfacecan be described by the set of solutions to the equation

ex3 =cosx2

cosx1or equivalently x3 = ln

∣∣∣∣cosx2

cosx1

∣∣∣∣ . (1.11)

If one colours the x1x2-plane like a checkerboard where the squares haveside length π, with a black square centered at (0, 0), then z is defined for theinterior of the black squares. Note x3 is not defined when cosx2

cosx1= 0, ∞, or

is negative. However, at the lattice points (i.e. where cosx1 = cosx2 = 0)we take the closure by allowing limits of solutions, which will in fact give x3

all possible values at these points. The Scherk surface is a doubly-periodicsurface, since it is invariant under the group of isometries of R3 generatedby the translations (x1, x2, x3) → (x1 + 2π, x2, x3), (x1, x2, x3) → (x1, x2 +2π, x3) and the 180-rotations around the diagonals of the black squares.Moreover, for the local Scherk surface we have

f(z) :=2

1− z4, g(z) := z,

so that we can now calculate the integral

X(z) = <e∫ z

0

1−ζ2

i(1+ζ2)

2ζ

21−ζ4 dζ = <e

∫ z

0

21+ζ2

2i1−ζ2

4ζ

1−ζ4

dζ

= <e

2·arctan(z)

2i·arctanh(z)

ln(

1+z2

1−z2

) = <e

i·ln( i+zi−z )i·ln( 1+z

1−z )ln(

1+z2

1−z2

) =

− arctan(−2·<ez

1−|z|2

)− arctan

(2·=mz1−|z|2

)ln∣∣ 1+z2

1−z2

∣∣ .

8 BELAID ALLOUSS

We begin with the calculation of the spherical image N in terms of f , g.The tangent plane is generated by the vectors

∂X

∂u,

∂X

∂v, where

∂X

∂u− i∂X

∂v= (%1, %2, %3).

It follows that∂X

∂u× ∂X

∂v= =m (%2%3, %3%1, %1%2)

respectively

∂X

∂u× ∂X

∂v=|f |2(1 + |g|2)

4(2<e g, 2=m g, |g|2 − 1).

The euclidean norm of this vector is∣∣∣∣∂X∂u × ∂X

∂v

∣∣∣∣ =[|f |(1 + |g|2)

2

]2

= λ2.

Now the unit normal vector on the Scherk surface with standard orientationis

N =1

|g|2 + 1(2<e g, 2=m g, |g|2 − 1). (1.12)

Using the known Weierstraß data f and g, the unit normal vector is

N(z) =1

u2 + v2 + 1(2 · u, 2 · v, u2 + v2 − 1).

The full Scherk surface will be discussed at the end of the following subsec-tion.

1.2. Global Representation. Before dealing with the concept of minimalsurfaces in (Nil(3), gτ ) we will describe in this preliminary section some gen-eral facts about global properties of minimal surfaces in R3. Remember thata surface X : Ω → R3, with a region Ω ⊂ R2, is called minimal surface ifX has isothermal parametrization and is harmonic. In other words in Ω wehave

λ :=∣∣∂X∂u

∣∣ ≡ ∣∣∂X∂v ∣∣ , ⟨∂X∂u ,

∂X∂v

⟩≡ 0 and 4X ≡ 0.

In order to allow a bigger class of domains we generalize this concept. Sup-pose Σ is connected and orientable and let X = (x1, x2, x3) : Σ→ R3 be animmersion of class Ck. The following theorem ensures that each point p ∈ Σhas a neighborhood in which isothermal parameters (u, v) are defined.

Theorem 1.6. Let U ⊆ Σ be a simply connected open set and let ψ : U → R3

be an immersion of class Ck, k ≥ 2. Then, there exists a diffeomorphismν : U → U of class Ck such that ψ = ψ ν is a conformal map.

The metric induced on Σ by X can be presented locally by

ds2 = λ2|dz|2,where z = u + iv ∈ C. Obviously, the transition functions for overlappingcharts of such parameters are conformal maps. The fact that Σ is orientableenables a restriction to a family of isothermal parameters whose transitionfunctions preserve the orientation. Thus, a surface Σ endowed with such


a family of isothermal parameters is nothing else than a Riemann surface.More generally, a Riemann surface X is a 2-dimensional topological manifoldwith a complex structure. Here, a complex structure means an equivalenceclass of biholomorphically equivalent atlases on X. In order to extend theconcept of holomorphic maps to such surfaces we have

Definition 1.7. Suppose X and Y are Riemann surfaces. A continuousmap f : X → Y is called holomorphic, if for every pair of charts ψ1 : U1 → V1

on X and ψ2 : U2 → V2 on Y with f(U1) ⊂ U2, the map

ψ2 f ψ−11 : V1 → V2

is holomorphic in the usual sense.

Furthermore, in a Riemann surface we also consider locally the differentialoperators of the Wirtinger calculus

∂z = 12

(∂∂u − i

∂∂v

), ∂z = 1

2

(∂∂u + i ∂∂v

).

With this notation we have

4 = 4λ2∂z∂z, K = −4 log λ and 4X = 2HN,

where K and H are the usual Gauss resp. mean curvature of the immersionX, and N is the Gauss map. One can show that a conformal immersionX : Σ → R3 is minimal iff the vector-valued φ := ∂zX is holomorphic. Aswell this function φ defined locally in X has values in C3, and its image liesin the quadric in C3 given by

z21 + z2

2 + z23 = 0, (z1, z2, z3) ∈ C3.

The latter property is a direct consequence of using φ = (φ1, φ2, φ3) whereφk = 1

2(∂xk∂u − i∂xk∂v ). Furthermore, one can obtain |φ|2 = 2λ2 and therefore

|φ| > 0. Due to the dependence of φ on conformal changes of variables it isgenerally not possible to integrate this function along a smooth path in Σ.But, if we consider vector-valued differential forms on different charts likeω = φdz and ω = φdw, we have

ω = φdz = φ∂zwdz = φdw = ω,

with holomorphic change of variables w = w(z), ∂zw 6= 0, and the equationφ = ∂wX. Indeed, we now have a vector-valued differential form ω globallydefined on Σ. It is called isotropic if it solves the above quadric globally.

In the rest of this subsection we shall point out all the necessary informationin order to formulate the famous Enneper-Weierstraß representation formula[6, 25, 26]. As an application of this formula we will reobtain the Weierstraßrepresentation of the full Scherk surface. Finally, we also list geometricquantities of this minimal surface corresponding to the given representation.¿From the theory of complex analysis we need the following fundamental

Theorem 1.8. Let Σ be a connected surface, X ∈ C1(Σ,Rn) and ω := φdzthe differential form to X. Then for every p, q ∈ Σ one has the identity

X(p)−X(q) = <e∫ q

pω = <e

∫γω

10 BELAID ALLOUSS

where γ : [0, 1] → Σ is an arbitrary C1 integration path in Σ with γ(0) = pand γ(1) = q

Now, one can conclude

Corollary 1.9. Let X : Σ→ Rn be a minimal surface with connected Σ andω = φdz, then ω is holomorphic and isotropic. X is an immersion iff φ hasno zeroes in Σ. Moreover, the following equation

X(p)−X(q) = <e∫ q

pω

is valid.

On the other hand, suppose we have ω, then one can ask which sufficientconditions we need to construct minimal surfaces X. Recall that a period ofa differentiable 1-form ω is

∮γ ω where γ is a closed path. ω has no periods

if∮γ ω = 0 for all closed paths in Σ.

Theorem 1.10 (Weierstraß representation formula). Let Σ be a con-nected Riemann surface. Let ω be a vector-valued 1-form defined globally inΣ. Furthermore, suppose that ω is holomorphic and isotropic with ω = φdzand <e

∮γ ω = 0 for all closed C1 paths in Σ. Then for every p0 ∈ Σ the

image of

X : Σ→ Rn, p 7→ <e∫ p

p0

ω

is a minimal surface whose branching points are exactly the zeroes of φ. Ifφ has no zeroes in Σ, then X is an immersion.

More precisely, we get the following elegant representation formula. This isanother variant of the lemma 1.4.

Theorem 1.11 (Enneper-Weierstraß representation formula). Let Σbe a connected Riemann surface and µ : Σ → C ∪ ∞ be a meromorphicfunction. Furthermore, a holomorphic 1-form with local representation νdzis defined on Σ.

i) If the following condition holds: µ has a pole of order m in p ∈ Σ, thenν has a zero of order k ≥ 2m in p. It then follows that the vector-valued1-form ω with the local representation ω = Φdz = (φ1, φ2, φ3)dz and

φ1 := 12(1− µ2)ν, φ2 := i

2(1 + µ2)ν, φ3 := µν

is holomorphic and isotropic in Σ. If Φ has no real periods, then for everyp0 ∈ Σ the image of

X : Σ→ R3, p 7→ X(p) = <e∫ p

p0

ω

is a minimal surface.

ii) If the zeroes of ν are exactly the poles of µ, and if the following strongcondition holds: µ has a pole of order m in p ∈ Σ, iff ν has a zero of orderk = 2m in p. It then follows X is an immersion.


Now, let us turn back to the Scherk surface in R3. From the previoussubsection we have

X(z) = <e(i ln

i+ z

i− z, i ln

1 + z

1− z, ln

1 + z2

1− z2

).

Using the branch with ln 1 = 0, we get

X(z) =(−arg

i+ z

i− z,−arg

1 + z

1− z, ln∣∣∣∣1 + z2

1− z2

∣∣∣∣) . (1.13)

The restriction to the set z : |z| ≤ 1, z 6= ±1,±i yields

π

2≤ arg

i+ z

i− z, arg

1 + z

1− z≤ 3π

2.

This shows that the mapping z 7→ (x1(z), x2(z)) maps bijectively the diskz : |z| < 1 to the square having the side length π and centered at (−π,−π).It follows that

cosx2(z)cosx1(z)

=∣∣∣∣z2 + 1z2 − 1

∣∣∣∣ = ex3(z)

which proves that the representation X(z), |w| < 1, defined by (1.13), para-metrizes the Scherk surface given by (1.11). Moreover, if C1, . . . , C4 denotethe four open quartercircles on |z| = 1 between the points 1, i,−1,−i, andif L1, . . . , L4 are the parallels to the x3-axiz through the vertices P1, . . . , P4

of the above mentioned square, then X provides a bijective mapping of Cionto Li. The rays z = reiθ, 0 ≤ r ≤ 1, θ = (2k+1)π

4 , 0 ≤ k ≤ 3, k ∈ N0,are mapped onto straight lines in the x1x2-plane emanating from the center(−π,−π) and ending at the points P1, . . . , P4. The next fact is that the raysz = reiθ, 0 ≤ r < 1, θ = kπ

2 , 0 ≤ k ≤ 3, k ∈ N0 are mapped by (x1(z), x2(z))onto the straight halflines emanating from (−π,−π) which are parallel tothe x1-axis resp. to the x2-axis, while x3(z) is non-linear and tends to ±∞.

Applying Schwarz reflection principle for holomorphic functions and his sym-metry principle for minimal surfaces (see subsection 2.1), one can concludethat a reflection of z : |z| ≤ 1, z 6= ±1,±i at one of the circular arcsC1, . . . , C4 corresponds to a reflection of the surface X(z) at one of thestraight lines L1, . . . , L4. In other words, each of the four quarter disksB1, . . . , B4 excised from the disk z : |z| < 1 by the x1- and x2-axes corre-sponds to one of the four congruent subsquares Q1, . . . , Q4 of the square hav-ing (−π,−π) as one of their corner points, and the representation X mapsthe mirror image B∗i of Bi onto the part of Scherk surface obtained fromthe graph over the square Qi by reflection in the straight line Li. Now, thisdescription shows which part of the Scherk surface (1.11) is parametrizedby the representation X : Σ ⊂ CP1 → R3. If X is lifted from the 4-punctured plane to the corresponding universal covering surface, one ob-tains a parametrization of the full Scherk surface described in the previoussubsection. The complete Scherk surface contains also the missing straightlines Li. For detailed computations we refer to [15].

12 BELAID ALLOUSS

1.3. Dirac Operator and Spin Bundles. In order to use basic tools fromthe spin geometry let us recall the main facts in Clifford algebra and complexspinor bundles [18].

Definition 1.12. The Clifford algebra Cn is the real associative algebra withunit which is generated by Rn with the relations

xy + yx = −2〈x, y〉.

In other words Cn is generated by the canonical basis e1, . . . , en with therelations

ekek + elek = −2δkl, k, l = 1, . . . , n

Thus, we have

1 ∈ R and ek1ek2 · . . . · ekm , 1 ≤ k1 < k2 . . . < km ≤ nas a basis of Cn with dim Cn = 2n. Furthermore, R and Rn are seen as linearsubspaces of Cn.

Cn is a Z2-graded algebra, i.e.

(i) Cn = C0n ⊕ C1

n.(ii) C0

n · C0n ⊂ C0

n, C0n · C1

n ⊂ C1n, C1

n · C0n ⊂ C1

n, C1n · C1

n ⊂ C0n. In particular,

C0n is a subalgebra of Cn.

The group Pin(n) ⊂ Cn, called Pin group, is generated multiplicatively bythe elements of Sn−1 ⊂ Rn. A further definition is Spin(n) := Pin(n) ∩ C0

n,called Spin group, which consists of all elements of Pin(n) with an evennumber of factors. Moreover, we have a surjective group homomorphismρ : Pin(n) → O(n) with ρ−1(SO(n)) = Spin(n) and ker(ρ) = ±1. Forexplicit calculations we use the helpful

Lemma 1.13. Let Cn ⊗ C be complexification of the Clifford algebra Cn.

(i) For n = 2m even, Cn ⊗ C is isomorph to

M(2m,C) ∼= M(2,C)⊗ . . .⊗M(2,C)︸︷︷︸m−times

.

(ii) For n = 2m+ 1 odd, Cn ⊗ C is isomorph to M(2m,C)⊕M(2m,C).

This lemma motivates the definition of the representation space

∆n := C2m ∼= C2 ⊗ . . .⊗ C2 n = 2m, 2m+ 1

where the elements of ∆n are called complex n-spinors. Using this, we nowhave a module over Cn ⊗C because there exists an algebra isomorphism κnwhich is called the Spin representation of Cn ⊗ C with

κn : Cn ⊗ C −→ End(∆n) for n even, resp.

κn : Cn ⊗ C −→ End(∆n)⊕ End(∆n)pr1→ End(∆n) for n odd.

We define the restriction of the Spin representation κn on Spin(n)

κ := κn|Spin(n) : Spin(n)→ GL(∆n)


The representation κn can be used to define a multiplication of vectors fromRn with complex n-spinors

Definition 1.14. The Clifford multiplication µ : Rn ⊗ ∆n → ∆ is definedby

x · ψ := µ(x⊗ ψ) = κn(x)ψ x ∈ Rn, ψ ∈ ∆n.

It turns out that this multiplication has a crucial property for the construc-tion of Dirac operators on Riemannian manifolds.

Lemma 1.15. The Clifford multiplication is equivariant w.r.t. the spinrepresentation, i.e. for all u ∈ Spin(n), x ∈ Rn and ψ ∈ ∆n we have

κ(u)(x · ψ) = (ρ(u)x) · (κ(u)ψ).

Before extending the list of important definitions, some more preliminariesare needed. One can prove that Spin(n) is a Lie subgroup of C∗n. Therefore,we can define a principal fibre bundle π : P → M with a given Lie groupSpin(n) operating on P and a Riemannian manifold M as the base manifold.On the other hand, we have the denoted group representation κ togetherwith the Clifford multiplication µ. The link between tangent spaces andspinors on Riemannian manifolds is as follows. We use the covering mapρ : Spin(n) → SO(n) and consider a n-dimensional, oriented, Riemannianmanifold (M, g). As an application we have the first definition for the con-struction of spinor bundle on Riemannian manifolds.

Definition 1.16. Let SO(M) be the SO(n)-principal fibre bundle, i.e. themanifold of all orthonormal frames on TM with fixed orientation. A Spinstructure on (M, g) is a pair (P, F ) consisting of a Spin(n)-principal fibrebundle P and a map F : P → SO(M) such that

(i) πSO(M) F = πP(ii) F (pu) = F (p)ρ(u) ∀ p ∈ P, u ∈ Spin(n).

Notice that not every Riemannian manifold (M, g) has a spin structure.More precisely, we have following result

Proposition 1.17. An oriented Riemannian manifold (M, g) admits a spinstructure, iff the 2nd Stiefel-Whitney class w2(M) ∈ H2(M ; Z2) disappears.In this particular case the set of all isomorphy classes of spin structures on(M, g) is an affine space over the 1st cohomology group H1(M ; Z2) on Mwith values in Z2.

Finally we have

Definition 1.18. Let π : P → M be a principal bundle with the Lie groupSpin(n) as the fibre, (M, g) a base manifold and κ : Spin(n) → GL(∆n) thegroup representation of Spin(n). Using the defined Clifford multiplication µ,the associated vector bundle is

π : (P ×∆n) /Spin(n)→M

14 BELAID ALLOUSS

In particular, it is the quotient space (P ×∆n) /Spin(n) with the equivalencerelation

(p, ψ) ∼κ (pu, κ(u−1)ψ) ∀ (p, ψ) ∈ P ×∆n and u ∈ Spin(n).

This associated vector bundle is called spinor bundle of (M, g) w.r.t. thespin structure (P, F ) and denoted by P ×κ ∆n or simple S.

We consider a spinor bundle S over the 4-punctured Riemann sphere Σ =CP1 \ z|z4 − 1 = 0 ⊂ CP1, i.e. S = P ×κ ∆n with a Spin(2)-principalbundle π : P → Σ. We have

S = S+ ⊕ S−,

S+ ∼= K, S− ∼= K

where K and K are seen as holomorphic line bundles over Σ. Now thecomplex cotangent space denotes

T ∗Σ⊗ C = T ∗CΣ = T (1,0)Σ⊕ T (0,1)Σ

with T (1,0)Σ := im(12(1− iJ)) ∼= K2

and T (0,1)Σ := im(12(1 + iJ)) ∼= K2,

where J is the 90-rotation. We call J a complex structure on T ∗Σ

Remark. The real part of the projection TCΣ→ T ∗Σ identifies the subbun-dle (T (1,0)Σ, i) with (T ∗Σ, J) and respectively (T (0,1)Σ, i) with (T ∗Σ,−J).We are thus led to the commutative diagrams

T (1,0)Σ ·i−−−−→ T (1,0)Σ

<e

y y<e

T ∗Σ ·J−−−−→ T ∗Σand respectively.

Using the usual Levi-Civita connection ∇ on Σ, we can construct a covariantderivative ∇ : Γ(S) → Γ(S ⊗ T ∗Σ) on S as follows: Let X ∈ Γ(TΣ), ψ ∈Γ(S) and s : U → P a section in the Spin(2)-principal bundle P of the spinstructure (P, F ) with U ⊂ Σ, then we define ψs : U → ∆n by

ψ(x) = [s(x), ψs(x)] ∈ P ×κ ∆n.

Writing F s = (e1, e2) ∈ Γ(SO(Σ)), we set

∇Xψ := [s, dXψs] +14

2∑j=1

ej · ∇Xej · ψ

on U ⊂ Σ. With this so called spinor derivative we are ready to introducethe Dirac operator on Riemannian manifolds. Identifying the tangent bundleTΣ with the cotangent bundle T ∗Σ by

v ∈ TxΣ 7→ gx(v, ·) ∈ T ∗xΣ

we receive an isomorphism

g : T ∗Σ⊗ S→ TΣ⊗ S.


Definition 1.19. The Dirac operator D : Γ(S)→ Γ(S) is a sequence of themaps

Γ(S) ∇−→ Γ(T ∗Σ⊗ S)g−→ Γ(TΣ⊗ S)

µ−→ Γ(S)

Remark. The Dirac operator

D : σ 7→2∑j=1

εj∇ejσ, σ ∈ Γ(S)

can be split over sections of holomorphic lines bundles S± in the followingway

D+ : Γ(S+)→ Γ(S+ ⊗ λ),

D− : Γ(S−)→ Γ(S− ⊗ λ),

where λ is a trivial real line bundle over Σ with a scalar product whichcorresponds with the conformal weight of the Riemmanian metric

λC := λ⊗ C = K ⊗ K.These facts are based on the property of the covariant derivative which isthe map

∇ : Γ(S)→ Γ(S⊗ T ∗Σ).This identifies - not only metrically but also conformally correct - the maps

D+ = ∂z : Γ(S+) = Γ(K) −→ Γ(K ⊗ K2)

= Γ(K ⊗ (K ⊗ K)︸︷︷︸λC

)

= Γ(S− ⊗ λC),

D− = ∂z : Γ(S−) = Γ(K) −→ Γ(K ⊗K2)

= Γ(K ⊗ (K ⊗ K)

= Γ(S+ ⊗ λC).

For the rest of this subsection we deal with the comparison of Dirac oper-ators with respect to different metrics g and g on Σ. Here, they differ in afunctional multiplier, i.e. g := e2ug. g is called a conformal metric. Noticethat in dimension 2 all metrics are locally conformally flat. In particular,the so-called conform curvature tensor is identically equal zero in that case.For the Christoffel symbol, we have

Γ(X,Y ) = ∇XY −∇XY = dXu · Y + dY u ·X − g(X,Y ) · gradg u

with X,Y ∈ TΣ and ∇, ∇ the respective connections on Σ. The relationfor the Gauss curvature is given by

Kg = e−2u(−∆g u+Kg).

We now turn to the identification of the Clifford algebra bundles C(T ∗Σ, g)and C(T ∗Σ, g) as algebra bundles and afterwards to the suitable identifica-tion of the respective representation bundles S = S+ ⊕ S−. Therefore, wechoose a bundle isometry Φ: (T ∗Σ, g) → (T ∗Σ, g) with Φ(ξ) = eu · ξ, ξ is

16 BELAID ALLOUSS

a 1-form. Note that g(ξ, ξ) = e2ug(ξ, ξ) = g(Φ(ξ),Φ(ξ)). This isometryinduces a Clifford algebra isomorphism C(T ∗Σ, g)→ C(T ∗Σ, g), but it is notparallel, i.e. Φ∗∇ 6= ∇. One can be convinced of this fact by the followingidentity:

(Φ∗∇)Xξ = Φ−1(∇(Φ(ξ)) = e−u · ∇X(euξ) = ∇Xξ + dXu · ξ.

Finally, with the relation

(∇X) · Y = dX(ξ · Y )− ξ(∇XY ) = ∇Xξ · Y − ξ(Γ(X,Y ))

we get

(Φ∗∇)Xξ = ∇Xξ − ξ(X) · du+ ξ(gradg u) · g(X, .).

The relation for the induced connections on S with

(Φ∗∇)X(ξ · σ) = ((Φ∗∇)Xξ) · σ + ξ · (Φ∗∇)Xσ

resp. ∇X(ξ · σ) = (∇Xξ) · σ + ξ · ∇Xσ, σ ∈ S,

is given by the averaging formula

(Φ∗∇)Xσ = 14

∑ξ∈1,ε1,ε2,ε1ε2

[−(ξ−1 · (Φ∗∇)Xξ) · σ + ξ−1 · ∇X(ξ · σ)

]= ∇Xσ − 1

4

∑ξ∈1,ε1,ε2,ε1ε2

[(ξ−1 · (Φ∗∇ − ∇)Xξ) · σ

]

= ∇Xσ + 14

2∑j=1

[εj(−εj(X) · du+ εj(gradg u) · g(X, .)) · σ

]= ∇Xσ + 1

4(−X# · du+ du ·X#)

with X# := g(X, .). In conclusion, we receive the equation for the Diracoperator as follows

(Φ∗D)σ = Φ−1 · D(Φσ) =2∑j=1

Φ−1(εj · ∇ej (Φσ)) = e−u2∑j=1

εj · (Φ∗D)ejσ

= e−u(Dσ + 1

4

2∑j=1

εj(−e#

j · du+ du · e#

j )σ)

= e−u(Dσ − 1

4

2∑j=1

εj · [εj , du] · σ)

= e−u(Dσ + 12du · σ). (1.14)

To get rid of the second term on the right hand side of (1.14) we start a newattempt. Now, we identify the spinor bundles by virtue of

Φ: S(T ∗Σ, g) −→ S(T ∗Σ, g).

σ 7−→ e12uσ

Note that the identification of the Clifford algebra bundles C(T ∗Σ, g) ⊂S(T ∗Σ, g)⊗ S∗(T ∗Σ, g) and C(T ∗Σ, g) ⊂ S(T ∗Σ, g)⊗ S∗(T ∗Σ, g) is given by


Φ: ξ 7→ euξ, while the identification of S∗(T ∗Σ, g) and S∗(T ∗Σ, g) is repre-sented by the map Φ: σ∗ 7→ e

12uσ∗. Another aspect for our considerations is

that the glueing cocycle for a complex line bundle has values in C∗ = R∗×S1,and the glueing cocycle for the spin bundle is

ϑS : 7→ cos(12ϑ(x)) + sin(1

2ϑ(x)) · ε1ε2,

if ϑT : 7→(

cosϑ(x) − sinϑ(x)

sinϑ(x) cosϑ(x)

).

Therefore, the following is obvious: If we glue T ∗Σ by a map h(x) ∈ C∗,then we should generate the spinor bundle by glueing it with

hS : x 7→√h(x) =

√|h(x)| · ϑS(x)

=√|h(x)| · exp(1

2ϑ(x) · ε1ε2).

This means a holomorphic relation between the glueing cocycles h and hS ofT ∗CΣ resp. S(T ∗Σ), namely h2

S = h. Moreover, S(T ∗Σ) receives ∂z-operatorfrom T ∗CΣ. In the end, the compensation of the second term in (1.14) canbe realized by a logarithmic derivative so that we get

(Φ∗D)(Φ∗σ) = euΦ∗Dσ.

A consequence of these results is that for S+ resp. S− the Dirac operatorcan indeed be written as

Dϕ =(

0 i∂z−i∂z 0

)ϕ ϕ := (ϕ+, ϕ−), ϕ± ∈ S±.

Here, we take into account that S+ resp. S− are endowed with a con-form chart by the glueing map

√h(x) and

√h(x). These charts guaran-

tee the existence of the above operators ∂z : Γ(S+) 7→ Γ(S+ ⊗ T (0,1)Σ) and∂z : Γ(S−) 7→ Γ(S− ⊗ T (1,0)Σ).

1.4. Holomorphic Bundles. In this section we study the properties of the∂z-operator on sections of hermitian holomorphic bundles. In order to dothis we first collect basic facts about these bundles. Let us denote a hermit-ian holomorphic bundle over a complex manifold M with (K, 〈., .〉, ∂z) where〈., .〉 is a hermitian inner product. Per definition there exists a continuousmap π : K C−→ M which is called C-vector bundle of rank 1 if the followingconditions are satisfied:M has an open cover (Ui)i∈I and holomorphic charts

ϕi : K|Ui → Ui × C

such that for i, j ∈ I, the holomorphic transition charts ϕij for the associatedcharts ϕi and ϕj

ϕij : (Ui ∩ Uj)× C→ (Ui ∩ Uj)× C ⊂ Uj × C

are of the form

ϕij(x, v) = (x, gij(x) · v)

18 BELAID ALLOUSS

where gij : Ui ∩ Uj → GL(1,C) = C∗ is holomorphic over the fibre C. If welook to a section sj(x) ∈ K|Uj of this bundle with

ϕj sj(x) = hij · ϕj si(x),

we get

(∂zsj)(x) = hij · (∂zsi)(x).

This means from now on we can define a ∂z-operator

∂z : Γ(K)→ Γ(K ⊗ T (0,1)M)

by virtue of

ϕi(∂zsi) := ∂z(ϕi(si)).

This is possible because the above defined operator is independent of thechoice of charts in the intersection Ui ∩ Uj .

Remark. The holomorphic transition charts corresponding to the descrip-tion π : K C−→M guarantee the existence of a global ∂z-operator. Conversely,suppose there exist sufficiently many local holomorphic sections w.r.t. this∂z-operator. This means we can choose a basis for K so that the transitionmaps

gij : Ui ∩ Uj → GL(k,C)

are holomorphic.

Definition 1.20. We say a connection

∇ : Γ(K)→ Γ(K ⊗R TM) = Γ(K ⊗C (TM ⊗R C))

is holomorphic iff

(∂zs) ·X := 12(∇Xs+ i∇JXs)

=∇s ·(

12(id + iJ)X

).

In other words we have

∂z : Γ(K) ∇−→ Γ(K ⊗C (TM ⊗R C))12 (id+iJ)−−−−−−→ Γ(K ⊗C T

(0,1)M).

With this information we formulate

Proposition 1.21. Let π : K Ck−→ M be a holomorphic vector bundle ofrank k with a hermitian inner product 〈., .〉. Then there exists one and onlyone holomorphic connection ∇ which is compatible with this metric. Thisimplies the following properties:

i) ∇X(is) = i∇Xs.ii) ∇X〈s1, s2〉 = 〈s1,∇Xs2〉+ 〈∇Xs1, s2〉.iii) (∂zs) ·X = ∇s · 1

2(id + iJ)X.

We postpone the proof of this fact to the end of the present subsection andgather more information about the holomorphic connection ∇.


Bundle automorphisms over π : K →M induce a smaller moduli space. Thisis typically described by characterizing holomorphic bundles with the divi-sor of zeroes and poles of a global meromorphic section. This is determineduniquely up to modification by a principal divisor via the holomorphic bun-dle. For example, one can follow from the Weierstraß gap theorem thatfor holomorphic curves Σ the moduli space of holomorphic line bundlesπ : K C−→ Σ is finite-dimensional. We will discuss these facts nad necessarydefinitions in section 2Let us now look at the space of ∂z-operators on π : K → M regarded as aC∞-vector bundle. On the one hand, it is an affine space over Γ(EndC(K)⊗CT (0,1)M) by virtue of (∂zs) · X = (∂zs) · X + AXs, on the other hand, wehave for ∇

Proposition 1.22. The space of holomorphic connections ∇ on (π : K C−→M,∂z) is an affine space over Γ(EndC(K)⊗C T

(1,0)M).

Proof. We have ∇Xs = ∇Xs+A(X) · s and

(∂zs) ·X = 12(∇Xs+ i · ∇JXs) = 1

2(∇Xs+ i · ∇JXs).

From this follows, that the statement of the proposition is equivalent to

0 = (A(X) + i ·A(JX)) · s⇔ 0 = A · 12(id + iJ)

⇔ 0 =A π0,1 ⇔ A ∈ ker(π∗0,1).

We used the notation π0,1 := 12(id+iJ) : TM → TRM⊗C with the properties

im(π0,1) = T(1,0)M and im(π∗1,0) = T ∗(1,0)M = T (1,0)M . It is obvious that0 = π0,1 π1,0 = π1,0 π0,1 resp. im(π∗1,0) = ker(π∗0,1).

We are now ready to prove Proposition 1.21.

Proof. (Proposition 1.21) We look for a section A in the endomorphism fieldEndC(K)⊗C T

(0,1)M such that ∇ = ∇+A is metric, in other words

dX〈s1, s2〉 = 〈∇Xs1 +A(X) · s1, s2〉+ 〈s1,∇Xs2 +A(X) · s2〉.

We rewrite this equation to

〈A(X) · s1, s2〉+ 〈s1, A(X) · s2〉 = dX〈s1, s2〉−〈∇Xs1, s2〉 − 〈s1,∇Xs2〉.

(1.15)

Now, for the metric operator ∇ one can ask why it is not possible tochoose A as a section with values in the skew-hermitian endomorphismsover (K, 〈., .〉). However, this cannot be done, because A has to be an ele-ment in Γ(EndC(K)⊗C T

(0,1)M) which means

A(JX) = i ·A(X). (1.16)

We can make the following observation. On the one hand, we see thatequation (1.15) determines the hermitian part of A(X) and consequentlythe hermitian part of A(JX). But on the other hand, due to equation

20 BELAID ALLOUSS

(1.16), the skew-hermitian part of A(X) = −i ·A(JX) is determined by thehermitian part of A(JX). To be exact, we have

A(X) +A(X)∗ := B(X),

A(X)−A(X)∗ = −i · (A(JX) +A(JX)∗) = −i ·B(JX)

⇔ A(X) = B · 12(id− iJ) ·X

As an exemplary application we examine the special case: K := T ∗CP1 withthe chart C → CP1, z 7→ [1 : z]. Now the background metric 〈〈., .〉〉 yields|dz|2 = 1 and the Euclidean derivative d is the holomorphic connectionfor 〈〈., .〉〉. It is known that the natural metric in CP1 is given by 〈., .〉 =|h(z)|2·〈〈., .〉〉 with |h(z)| = 1

2(1+|z|2). The appropriate hermitian connection∇ = d+A is derived with the ansatz

A(X) +A(X)∗ = 2 · dX(ln|h|) ⇒ A(X) = dX ln|h| − i · dJX ln|h|.

Altogether, we have the result ∇Xs = dXs+(dX ln|h|−i ·dJX ln|h|) ·s whichcan be rewritten as ∇s = ds+ 2 · ∂z ln(1 + |z|2) · s = ds+ 2z

1+|z|2 · s.

The rest of this subsection is devoted to the procedure of determining holo-morphic line bundles K defined over surfaces Σ := CP1 \ qµ. In qν ∈ CP1,the divisor Div : CP1 → Z of meromorphic sections in K only take valuesin Z∗. We denote (σ) as the divisor of the section σ ∈ Γ(K). Let σ1 besome basic section in K, all other sections can be described as f · σ1 with aholomorphic function f . In order to determine the whole line bundle K, weneed basic sections σµ over neighbourhoods Uµ = B(qµ, ρµ), 0 < ρµ 1,and basis transformation matrices (i.e. basis transformation functions withvalues in C∗ which are isomorph to the glueing cocycle) hµ over Uµ∩Σ. Westudy hµ(z − qµ) for |z − qµ| 1. Now there are three cases to consider:

1) hµ is bounded over the punctured neighbourhood Uµ \ qµ. Thismeans qµ is a removable singularity. The continuation of σ1 by qµ isa holomorphic basic section.

2) For σµ(z−qµ) = (z−qµ)−kµ ·σ1(z−qµ), kµ > 0 we have the following:As σµ is a regular basic section, i.e. holomorphic and bounded, inqµ ∈ Uµ the equation

σ1(z − qµ)|z=qµ = 0

holds. In other words

z 7→ σ1(z − qµ) = (z − qµ)kµ · σµ(z − qµ)

where σ1 has a zero qµ of multiplicity kµ.3) In analogous manner, for σµ(z−qµ) = (z−qµ)−kµ ·σ1(z−qµ), kµ < 0

we have the equation

z 7→ σ1(z − qµ) = (z − qµ)kµ · σµ(z − qµ)

where σ1 has a pole qµ of order |kµ| = −kµ.


These cases correspond to the summands kµ · (qµ) in the divisor of linebundles constructed in this way. On the other hand, the above classificationof isolated points implies that the transformation matrices

hµ : Uµ \ qµ → C∗

can only have removable singularities or poles in qµ. For reasons of growth,the essential singularities are ruled out. The situation

hµ(z − qµ) = (z − qµ)µ · ϕµ(z − qµ), kµ ∈ Z

with holomorphic and bounded ϕµ : Uµ \ qµ → C∗ remains. Due to Rie-mann’s theorem on removable singularities, we can uniquely extend ϕµ to aholomorphic function z 7→ ϕµ(z− qµ) on the entire neighbourhood Uµ. Theidea is that we have to eliminate ϕµ from the equation by substituting σµwith ϕµσµ. Therefore, we get

ϕ1(z − qµ) = hµ(z − qµ) · σµ(z − qµ)

= (z − qµ)kµ · ϕµ(z − qµ)σµ(z − qµ)︸︷︷︸new basic section

.

1.5. Heisenberg Groups. This subsection is devoted to the classical andthe general 3-dimensional Heisenberg group, H3 and (Nil(3), gτ ). We showwhat is known about their group structure and the left-invariant vectorfields on these Lie groups. Additionally, the Lie algebra, the necessaryparametrization and the left-invariant metrics are achieved.

The standard representation of the classical 3-dimensional Heisenberg groupH3 is given in GL3(R) by 1 x z

0 1 y

0 0 1

,

with x, y, z ∈ R. It is a two-step nilpotent Lie group. This group is endowedwith a left-invariant metric g (in order to avoid confusions the metric gis denoted by 〈., .〉), which means that for any vectors ξ and η tangent to(H3, 〈., .〉) at h the inner product of their translations by g coincides withthe inner product of ξ and η

〈ξ, η〉 = 〈(Lg)∗ξ, (Lg)∗η〉, ξ, η ∈ Th(H3, 〈., .〉), g, h ∈ (H3, 〈., .〉).

The results of Figueroa, Mercuri and Pedrosa [17] show a complete classi-fication of the constant mean curvature (cmc) surfaces - including minimal- which are invariant with respect to 1-dimensional subgroups of the con-nected component of the isometry group of (H3, 〈., .〉).For the description of a left-invariant metric on H3 we need the Lie algebrastructure of the vector space of left-invariant vector fields which is isomorphto the tangent space TidH3 at the identity id of H3. The Lie algebra h3 ofH3 is given by matrices

V =

0 v1 v3

0 0 v2

0 0 0

,

22 BELAID ALLOUSS

with vi ∈ R. As a global parametrization, we use the exponential mapexpid: h3 → H3 at the identity of H3

expid(V ) = id + V + V 2

2 =

1 v1 v3+12v1v2

0 1 v2

0 0 1

.

These coordinates are also known as Riemannian normal coordinates. Theglobal parametrization has special properties at I ∈ H3 with F := expid

i) F (0) = I.ii) gij(0) = δij .iii) ∂gij

∂xi(0) = 0 and Γkik(0) = 0,

where ii) implies that (d expid)0 is a linear isometry. In what follows weidentify the tangent space to TidH3 at 0 ∈ TidH3 with TidH3 itself, TidH3 ≈T0(TidH3). This property of the metric can be transferred to the wholetangent space of H3 by using the before mentioned Lie algebra structure.Normal coordinates can be seen as a crucial point for easier calculations inthe Riemannian geometry. We identify the Lie algebra h3 with R3 byv1

v2

v3

↔0 v1 v3

0 0 v2

0 0 0

.

Now, how does the group structure of H3 look like? In other words, we askfor the left translation in the classical Heisenberg group. Let g, h ∈ H3, then

Lg(h) = g h =

1 g1 g3

0 1 g2

0 0 1

1 h1 h3

0 1 h2

0 0 1

=

1 g1+h1 g3+h3+g1h2

0 1 g2+h2

0 0 1

.

Using the exponential parametrization as well as the identification of theexponential coordinates with R3, the left translation changes to

LX1(X2) = X1X2 =

x1

y1

z1

x2

y2

z2

=

x1+x2

y1+y2

z1+z2+12 (x1y2−y1x2)

,

with the definitions exp(X1) =: g, exp(X2) =: h and exp(LX1(X2)) =: Lg(h)which we use from now on. In this context the exponential map is a grouphomomorphism between the two group structures (h3, ) and (H3, ).The Lie algebra bracket, in terms of the canonical basis e1, e2, e3 of R3, isgiven by

[e1, e2] = e3,

[ei, e3] = 0, i = 1, 2..

e1, e2, e3 is fixed as the orthonormal frame at the identity so that anorthonormal basis of left-invariant vector fields is given in exponential coor-dinates by

E1 = ∂∂x −

y2∂∂z , E2 = ∂

∂y + x2∂∂z , E3 = ∂

∂z


or w.r.t. the group structure of (H3, )

E′1 = ∂∂x′ , E′2 = ∂

∂y′ + x′ ∂∂z′ , E′3 = ∂∂z′ .

As a consequence the left-invariant metric g respectively g′ is given by

ds2 = dx2 + dy2 + (12ydx−

12xdy + dz)2

resp.

(ds′)2 = (dx′)2 + (dy′)2 + (dz′ − x′dy′)2.

Altogether we have the diagram

h3

LX1−−−−→ h3

expid

y yexpid

H3Lg−−−−→ H3

and for the differential maps

TX2h3

(LX1)∗−−−−→ TLX1

(X2)h3

(d expid)X2

y y(d expid)LX1X2

ThH3(Lg)∗−−−−→ TLg(h)H3

with arbitrary X1, X2 ∈ h3 and g, h ∈ H3.

The Heisenberg group (Nil(3), gτ ) is a simply connected, homogeneous3-manifold with 4-dimensional isometry groups which admits natural Rie-mannian submersions into R2 with 1-dimensional, totally-geodesic fibers. Inother words (Nil(3), gτ ) can be represented as line bundles of constant curva-ture τ over R2. As mentioned before, we are only interested in left-invariantmetrics. The 1-parameter family gτ which is indexed by bundle curvature τencompasses all these metrics.

Let us first explain some basic facts about the Heisenberg group (Nil(3), gτ ).For the left translation Lg : (Nil(3), gτ ) → (Nil(3), gτ ), h → hg, g, h ∈(Nil(3), gτ ) with x1

y1

z1

τ x2

y2

z2

=

x1+x2

y1+y2

z1+z2+1

2τ(x1y2−y1x2)

we use a left-invariant metric in (Nil(3), gτ ) which means that for any vectorsX and Y tangent to h ∈ (Nil(3), gτ ) the inner product of their translationsby g coincides with the inner product of X and Y

gτ |h(X,Y ) = gτ |Lgh((Lg)∗X, (Lg)∗Y ),

with

(Lg=t(x1,y1,z1))∗ =

1 0 0

0 1 0

0 τx1 1

.

24 BELAID ALLOUSS

The metric

gτ = dx2 + dy2 + (dz − τ2 (xdy − ydx))2,

or described as a matrix

gτ =

1+1

4τ2y2 − 1

4τ2xy

1

2τy

− 1

4τ2xy 1+

1

4τ2x2 − 1

2τx

1

2τy − 1

2τx 1

has this property. The left-invariant vector fields form a Lie algebra nτ on(Nil(3), gτ ). By means of nτ 3 X ∼←−−→ X|id ∈ Tid(Nil(3), gτ ) we identify thetangent space at the identity id of (Nil(3), gτ ) with the Lie algebra nτ . Inother words, the left-invariant metric is determined by the inner product oftangent vectors X ∈ (LF−1

τ)∗(TFτ (Nil(3), gτ )) at the identity of (Nil(3), gτ ).

All further computations of the next section are seen in this light.

The Levi-Civita connection is given by

∇XY = ∂XY + BXY

= ∂XY + (B ·X)× Y with B := τ2

1 0 0

0 1 0

0 0 −1

(1.17)

where ∂ stands for the left-invariant connection. Let us choose an orthonor-mal basis e1, e2, e3 for the inner product on the Lie algebra nτ . Then we getthe following formulas

∇e1e2 = −∇e2e1 = τ2e3, ∇e1e3 = ∇e3e1 = − τ

2e2

∇e2e3 = ∇e3e2 = τ2e1, ∇e1e1 = ∇e2e2 = ∇e3e3 = 0

with the resulting commutation relations

[e1, e2] = τe3, [e1, e3] = [e2, e3] = 0.

2. Immersion of Scherk Surfaces in (Nil(3), gτ )

In 1966, Jenkins and Serrin proved an existence and uniqueness theoremfor minimal graphs in R3 bounded by straight lines which are non compact[20]. They yielded necessary and sufficient conditions to solve the Dirichletproblem in a compact convex domain bounded by a polygon. Along differentstraight segments of the boundary the graph take values ±∞ and continuousdata. In other words we have

Theorem 2.1 (Jenkins, Serrin). Let D be a bounded convex domainwhose boundary contains two sets of open straight segments A1, . . . , Ak andB1, . . . , Bl with the property that no two segments Ai and no two segmentsBi have a common endpoint. The remaining portion of the boundary con-sists of endpoints of the segments Ai and Bi, and open arcs C1, . . . , Cm.Consider the Dirichlet problem:


Determine a minimal graph in D which assumes the value+∞ on each Ai, −∞ on each Bi and assigned continuousdata on each of the open arcs Ci.

Let P denote a simple closed polygon whose vertices are chosen from amongthe end points of the segments Ai and Bj. Let α, β be, respectively, the totallength of the segments Ai and Bj which are part of P . Finally, let γ denotethe perimeter of P . Then, if the family of arcs Ci is not empty, the Dirich-let problem stated above is solvable iff 2α < γ and 2β < γ. Furthermore, thesolution is unique if it exists.

In recent years, analogous results were discovered for minimal graphs inthe homogeneous 3-manifolds H2 × R and S2 × R [24, 27]. Moreover, ageneralized Jenkins-Serrin theorem in homogeneous 3-manifolds will appearin [12]. Equipped with these results, we are encouraged to study Scherksurfaces Σ immersed into the 3-dimensional Heisenberg group

Fτ : Σ# (Nil(3), gτ ).

One can easily formulate the constraint of minimality with respect to F ∗τ gτ ,but this is not possible at the moment, because we do not know the immer-sion Fτ . We will discuss two ways out of this situation.

The first approach is to endow Σ with a background metric g0 and to studyimmersions Fτ of minimal surfaces Σ with F ∗τ gτ = g0. In that case, we areconfronted with the problem to solve a non-linear, elliptic PDE for Fτ withan over-determined non-linear side condition.

We will give a short overview of this approach. First we need a classicalresult for minimal graphs: Suppose that f : Ω ⊂ R2 → R is a C2 function.The graph of f

Graphf = (x1, x2, f(x1, x2))|(x1, x2) ∈ Ωhas area

area(Graphf ) =∫

Ω

√1 + |∇f |2.

One can prove that the graph of f is a critical point for an area functionaliff f satisfies the divergence form equation

div(

∇f√1+|∇f |2

)= 0.

This equation is equivalent to

0 = ∆f − 〈grad f,Hess(f) · grad f〉1 + |df |2

⇒ 0 = ∆f + 〈J · df,Hess(f) · J · df〉 with J =(

0 −1

1 0

).

We will refer to it as the minimal surface equation. To use this for ourpurposes we need the Ehresmann connection ω. As mentioned in subsection1.5 we have a fibre bundle Π: (Nil(3), gτ )→ R2. Let

V := ker[dΠ: T (Nil(3), gτ )→ Π∗TR2

]

26 BELAID ALLOUSS

be the vertical bundle consisting of vectors tangent to the fibres (Nil(3), gτ ),so that the fibre of V at g ∈ (Nil(3), gτ ) is Tg(EΠ(g)). An Ehresmann con-nection ω on (Nil(3), gτ ) is a smooth subbundle H of T (Nil(3), gτ ), calledthe horizontal bundle of the connection which is complementary to V inthe sense that it defines a direct sum decomposition T (Nil(3), gτ ) = H ⊕ V .Now, the bundle curvature identity for (Nil(3), gτ ) is given by

0 = τω + da

where a := 〈Y, .〉. Here, Y has the decomposition Y = Y0 + grad f . Notethat in the Euclidean space the term Y is exactly grad f , and ω = df .Furthermore we have for (Nil(3), gτ )

Y0|x := −12τ · J · x and a0(z) = −1

2τ〈J · x, z〉, x, z ∈ R2.

The differential of a0 yields

da0(z1, z2) = dz1a0(z2)− dz2a0(z1)

= −12τ(〈J · z1, z2〉 − 〈J · z2, z1〉)

= −τ〈J · z1, z2〉= −τω(z1, z2), zi ∈ R2.

In other words we have da0 = −τω. This means that for a1 := 〈Y − Y0, .〉the identity da1 = 0 holds. Thus, it is a1 = df with an appropriate functionf : Ω → R. The minimal surface equation in (Nil(3), gτ ) is generally givenby

0 = trace(dY ) + 〈J · Y, dY · J · Y 〉

with dY = dY0 + d2f and dY0 = d(−12τ · J · x) = −1

2τ · J . More precisely:

0 = ∆f + 〈J · df + 12τx,Hess(f) · (J · df + 1

2τx)〉.

Now finding Scherk surfaces in (Nil(3), gτ ) means to minimize

Iτ :=∫(−π2 ,−

π2 )2

√1 + |df − 1

2τ · J · x|2

with f = f0 +ϕ where f0 = ln cosx2− ln cosx1 and du0 = (tanx1,− tanx2).Altogether we have to minimize

Iτ (u0 + ϕ)

=∫

(−π2 ,−π2 )2

√1 + (tanx1 + 1

2τx2 + ∂x1ϕ)2 + (tanx2 + 12τx1 − ∂x2ϕ)2.

Straight computations leads to a first order pertubation equation

0 =[1 + (∂x2ϕ− tanx2 − 1

2τx1)2] (∂2x1ϕ+ 1

cos2 x1

)+[1 + (∂x1ϕ+ tanx1 + 1

2τx2)2] (∂2x2ϕ− 1

cos2 x2

).

We leave this approach. Instead, we will follow a second one which is de-scribed as follows: We again endow Σ with a background metric g0 andexamine conformal immersions Fτ . In other words we have F ∗τ gτ = e2ug0 :=e2udzdz with a differentiable function u : Σ→ R and a conformal parameter


z = x+ iy in a domain of Σ. The tangent vectors ∂zFτ and ∂zFτ meet thefollowing equations

gτ (∂zFτ , ∂zFτ ) = gτ (∂zFτ , ∂zFτ ) = 0 and gτ (∂zFτ , ∂zFτ ) = 12e

2u. (2.1)

A main argument which justifies this method is the use of the Teichmullertheory. Riemann surfaces which are topologically classified by the genus, arenot very interesting research objects, except for manifolds with higher di-mensions. Therefore, we need more structure. The set of complex structuresresp. conformal classes on a given orientable surface, modulo biholomorphicequivalence, itself forms a complex algebraic variety called moduli space. Letus briefly record some general definitions and properties.

Definition 2.2. Let Σ be a topological and compact manifold with genus G,supporting two complex structures Σ1, Σ2. We define

Σ1 ∼M Σ2,

if there exists a biholomorphic map M : Σ1 → Σ2. This is an equivalencerelation. The space MΣG of all equivalence classes is called moduli space.

In the same way, one can define the moduli space MΣG,N for puncturedsurfaces, and the moduli space for non-compact surfaces or surfaces withN boundaries components. Topologically there is just one underlying sur-face for each genus G. Two Riemann surfaces of genus G are consideredequivalent if there is a complex analytic homeomorphism between themwith complex analytic inverse, i.e. biholomorphic equivalent. For genus 0,there is only one equivalence class. Related to the moduli space, there ex-ists a space TΣ which preserves more information about the surface. Moreprecisely, the surface Σ or its underlying topological structure provides amarking Σ→ Σi of each Riemann surface Σi represented in TΣ. Whereas amoduli space identifies all surfaces which are isomorphic, TΣ only identifiesthose surfaces which are isomorphic via a biholomorphic map that is isotopicto the identity. We call TΣ a Teichmuller space.

Due to the already discussed ring of holomorphic differentials 〈θ1, θ2, θ3〉,the sought Scherk surface Σ can be represented by an algebraic variety viaa 1-canonical embedding

Σ→ CP2

z 7→ [θ1 : θ2 : θ3] = [1 : z : z2]

The image lies in the Zariski closure CΣ := [w0 : w1 : w2]|w0w2 − w21 =

0 ⊂ CP2. Moreover CΣ is smooth and biholomorphic to itself in CP1. Thus,the destinguished metrics are algebraically conformal to the pull back of theFubini-Study metric in CP2. In fact, we get the metric

g = (1+|z|2)2

|1−z4|2 |dz|2

where the conformal factor is precisely the denominator in the expressionof g. This metric is invariant under the 180-rotations around horizontaldiagonals of the Scherk surface, i.e. z 7→ ± i

z . For our case, we considermetrics of the same equivalence class with g = h(z)2 · g and

28 BELAID ALLOUSS

i) h(z) is real and bounded in (0,∞),ii) h(z) is invariant under z 7→ ± i

z and z 7→ −z.

Another destinguished structure in the light of the Teichmuller theory arethe four isolated singularities qi of Σ. Here we need additional criteria onsurfaces. In particular, for complete minimal surfaces Σ with finite volumeand bounded Gauss curvature K and Euler characteristic χ(Σ) we have

Theorem 2.3 (Osserman). For embedded and complete minimal surfacesΣ with finite total curvature

∫ΣKdA the equation

2πχ(Σ)−∫

ΣKdA = 2πk

holds where k is the number of the ends. In the case of immersed minimalsurfaces one has to take multiplicities at the ends into account.

Without the conditions of a finite volume and a bounded curvature, we havethe following

Theorem 2.4 (Cohn, Vossen). If (Σ, g) is a complete Riemannian man-ifold of finite topological type and with absolutely integrable Gauss curvatureK, then the inequality

2πχ(Σ) ≥∫

ΣKdA

holds. In particular, we have∫

ΣKdA ≤ 2π if M is non-compact.

Let us study the conformal type for the fundamental domain of the fulldoubly-periodic Scherk surface. In other words, we consider only the basispart Σ of this surface defined over some equilateral quadrilateral, a rhombus,in the x1x2-plane with the perpendicular diagonals d1 and d2, |di| = 2ri.We devide this local surface in four similar parts Σ1/4, each of which canbe described as follows: Let P0 be the center of the rhombus and P1, P2

the respective vertices. Take a right-angled triangle P0P1P2 with the half-diagonals P0Pi, |P0Pi| = ri. For the solution Fτ on P0P1P2 we know thatFτ = 0 on the half-diagonals and tends to ±∞ on [P1, P2]. In other words wehave a surface bounded by the half-diagonals and the straight lines which areparallels to the z-axis through the vertices. Furthermore, we know that thetotal curvature is finite. Note that for surfaces with genus G and boundarycomponents B the equation for the Euler characteristic reads χ(Σ) = 2 −2G−B. The theorem of Gauss-Bonnet yields

1 = χ(Σ1/4) = 12π

(∫Σ1/4

KdA+∫∂Σ1/4

kg ds)

= 12π

(∫Σ1/4

KdA+ 5π2

)⇔

∫Σ1/4

KdA = −π2 .


As we know, for minimal surfaces one has K ≤ 0. Now the full Scherksurface has infinitive total curvature and the normals to Σ attain all di-rections an infinite number of times. Thus, the complete minimal surfaceis of conformally hyperbolic type [23]. If we endow Σ with a hyperbolicmetric, we can identify the universal cover of Σ with the hyperbolic planeH2, the model space with constant curvature K ≡ −1. For H2, we use thePoincare model, i.e. the disk D = z ∈ C||z| < 1 endowed with the metricds2 = 4|dz|2

(1−|z|2)2 . Now, the group of orientation preserving isometries in H2

is PSL(2,R) = SL(2,R)/±id, where PSL(2,R) operates in D by virtue ofthe rational linear transformation(

a b

c d

)z = az+b

cz+d

Let Σ be a quadrilateral embedded in D with the flat vertical annuli ly-ing as vertices in ∂∞D. This quadrilateral defines a fundamental domainfor the isometric action of the fundamental group π1(Σ) in D via decktransformation. The group action is given by the injective homomorphismΘ : π1(Σ) → PSL(2,R). In other words we have a bijection between thecross-ratios of qi with (q1, q2, q3, q4) ∈ C \ 0, 1. Vice versa, the quotientD/Θ(π1(Σ)) is diffeomorph to Σ. The hyperbolic metric ds2 in D defines ahyperbolic metric in D/Θ(π1(Σ)). Therefore, we have the same metric inΣ. Now, the moduli space of all hyperbolic metrics is the quotient of theTeichmuller space of Σ modulo the group of homotopy classes of diffeomor-phisms in Σ. All told, we mention that the complex dimension of a modulispace for a N -punctured Riemann surface Σ is given by dimCMΣ0,N

= N−3.For the Scherk surface we have dimCMΣ0,4 = 1, i.e. the set of conformalclasses is a 1-dimensional moduli space. Besides, there is an important linkbetween the space Q(g) of holomorphic quadratic differentials and Riemannsurfaces Σ with a conformal structure g and a genus G ≥ 2. The so calledTeichmuller theorem states

Theorem 2.5. TG is diffeomorphic to Q(g) on an arbitrary Riemann sur-face (Σ, g) ∈ TG.

Now, we can begin to study holomorphic spinors for the immersion Fτ whichare defined over a certain covering of Σ.

2.1. Divisors and Coverings. The conformal structure for the Scherk sur-face is given by dz

z4−1with a double zero at z =∞. For simplifying notation

from now on qi, i ∈ 1, . . . , 4 are the zeroes of z4 − 1 = 0. They representthe branching points of Σ. The divisor of zeroes and poles and the degreeare as follows

Div(dz

z4 − 1) = 2(∞)−

( 4∑i=1

qi),

deg(dz

z4 − 1) = 2− 4 · 1 = −2.

30 BELAID ALLOUSS

It is known that the second fundamental form Qdz2

(z4−1)2

of the surface has

zeroes at the ends which are asymptotic to vertical planes. Therefore, weknow the Hopf differential Q itself has in qi at most simple poles, and else-where none because of the holomorphicity of Q. The degree of Q is −4 andthe divisor

Div(Q) = −( 4∑i=1

qi).

In other words we have

Q =dz2

z4 − 1up to a factor c 6= 0.

As already discussed in the previous subsection 1.4, in complex analysis itis common to define functions by first specifying the function on a small do-main only, and then extending it by analytic continuation. In practice, thiscontinuation is often done by first establishing some functional equation onthe small domain and then using this equation to extend the domain. Here,we list some facts which we need for our computations: Div(Q) determinesK4 as a holomorphic bundle over Σ ⊂ CP1 and guarantees that K4 can beuniquely continued as a holomorphic bundle over the whole CP1. Further-more, from the classical surface theory we have the symmetry principle forminimal surfaces discovered by H. A. Schwarz

Theorem 2.6.

i) Every straight line contained in a minimal surface is an axis of sym-metry of the surface.

ii) If a minimal surface intersects some plane E perpendicularly, thenE is a plane of symmetry of the surface.

and his reflection principle for holomorphic functions: For a region G ⊂ Cdefine G∗ := z|z ∈ G. If G is a symmetric region, that is G = G∗,then we define G+ := z ∈ G|=m z > 0, G− := z ∈ G|=m z < 0 andG0 := z ∈ G|=m z = 0.

Theorem 2.7. Let G ∈ C be a region such that G = G∗ and supposef : G+ ∪G0 : → C is a continuous function that is holomorphic on G+ andreal for x ∈ G0. Then there exists a holomorphic g : G → C such thatg(z) = f(z) for z ∈ G+ ∪G0.

In fact, any analytic curve which has a neighborhood biholomorphic to astraight line can be reflected across. The basic example is the boundary ofthe unit circle which is mapped to the real axis by z 7→ iz+1

z+i . For the soughtsurface Σ we know the qualitative information concerning its symmetriesin the Heisenberg group. We have the 180 rotations around the geodesicsy = x resp. y = −x in the xy-plane and around the vertical ends. In the4-punctured Riemann sphere, that means these involutions correspond tothe group which is generated by the involutions

ι1 : z 7→ iz, ι2 : z 7→ −iz


and the Schwarz reflection along S1\qi ⊂ Σ with ι3 : z 7→ 1z . It is necessary

to use orientation preserving involutions. In this case we have

τ1 = ι1ι3 = ι3ι1, τ2 = ι2ι3 = ι3ι2 and τ3 := τ1τ2 = ι1ι2,

with τ1 : z 7→ iz , τ2 : z 7→ − i

z and τ3 : z 7→ −z. These maps operate naturallyon TΣ. Moreover, they also operate on all powers of T (1,0)Σ, especially on

K2 ∼= T (1,0)Σ and K4 ∼= T (2,0)Σ.

Therefore, we have

Lemma 2.8. The Hopf differential Q = dz2

z4−1∈ Γ(K4) is invariant under

the above mentioned symmetries.

Proof. Just set the respective involutions in the holomorphic quadratic dif-ferential, then the easy computations lead to the desired property. Note thatthis invariance also follows from the fact that Q is a geometric object.

For our purposes we need the ring 〈θ1, . . . , θn〉 of holomorphic differentials θkwhich have the right growth conditions in direction to the branching pointsqi. They are defined as follows

θ1 := dzz4−1

, θ2 := zθ1 = zdzz4−1

, θ3 := z2θ1 = z2dzz4−1

.

As requested, they have simple poles in qi and are holomorphic on Σ. Thedivisors are

Div(θ1) = −4∑i=1

(qi) + 2(∞),

Div(θ2) = −4∑i=1

(qi) + (0) + (∞),

Div(θ2) = −4∑i=1

(qi) + 2(0).

Obviously, we have Q = (z4−1)θ21 = θ2

3−θ21. The next step is to characterize

the half spin bundle K as a holomorphic bundle itself. Our first result statesthe following

Theorem 2.9. Let π : Σ→ Σ be a two-sheeted covering map of Σ and L2 :=π∗(K2) the respective pull back of the space of 1-forms. Due to the geometryof Σ there exist meromorphic sections σ1, σ2 ∈ Γ(L) which determine thebundle L over Σ uniquely so that

Div(σ1) = −4∑i=1

(qi, 0) +2∑i=1

(pi∞), Div(σ2) = −4∑i=1

(qi, 0) +2∑i=1

(pi0)

and

σ21 = π∗θ1, σ2

2 = π∗θ3, σ2 = zσ1

where pi0, pi∞ are the preimages of 0, ∞ ∈ CP1 under the covering map π.

32 BELAID ALLOUSS

Proof. First of all, the symmetry group operates on the vector space spannedby θ1, θ2, θ3 as follows

τ∗1 θ1 = iθ3 τ∗2 θ1 = −iθ3 τ∗3 θ1 = −θ1

τ∗1 θ2 = −θ2 τ∗2 θ2 = −θ2 τ∗3 θ2 = θ2

τ∗1 θ3 = −iθ1 τ∗2 θ3 = iθ1 τ∗3 θ3 = −θ3.

The evident identity (τ∗j )2 = id already follows from the fact that τj itself isan involution, and that this involution operates naturally on T (1,0)Σ ∼= K2.Now, the half spin bundle K contains the square root of the differentialsθi. As we know, the square root creates monodromies around the points0, ∞, qi. In other words we have a multi-valued function which produces asign change for the spinors by walking around the respective points alongclosed paths. To avoid this sign ambiguity it is convenient to go to theRiemann surface Σ, the two-sheeted covering of the 4-punctured sphere withthe branching points qi. Now this only resolves the square root at the fourends. By not using θ2 we are also able to avoid trouble at the points 0 and∞ An easy computation shows

χ(Σ) = 2χ(CP1)− ]z ∈ C|z4 − 1 = 0= 2 · 2− 4 = 0,

thus the covering is recognized as a torus. This torus is given by the quartic

2w2 = z4 − 1,

or that is to say we have to identify two affine curves

S1 := (zµ, wµ) ∈ C2|2w2µ = z4

µ − 1, µ = 1, 2

over C∗ × C by virtue of the following transition map

τ1 : (z1, w1) 7→ (z2, w2) = ( iz1, iw1

z21

).

The glueing map τ1 extends to a symmetry on the whole torus. Moreover, inaccordance to the involutions τ2 and τ3 we get further symmetries τj : C∗ ×C→ C∗ × C on the torus as an analytic continuation

τ2(z, w) 7→ (−iz ,iwz2 ) and τ3(z, w) 7→ (−z, w).

In other words τj are still involutions. Let us denote the natural two-sheetedcovering Σ := (z, w) ∈ C2|2w2 = z4 − 1 ⊂ T 2 of Σ as

π : Σ −→ Σ

(z, w) 7−→ z.

In addition let

p10 = (0,

√2

2 i) ∈ S1,

p20 = (0,−

√2

2 i) ∈ S1

be the preimages of 0 ∈ CP1 and

p1∞ = (0, i) ∈ S2,

p2∞ = (0,−i) ∈ S2


the preimages of ∞ ∈ CP1. For the covering map π we need the pull backof the 1-forms θj ∈ K2 where the sections π∗θj ∈ Γ(L2), have the divisors:

Div(π∗θ1) = −2( 4∑i=1

(qi, 0))

+ 2(p1∞) + 2(p2

∞),

Div(π∗θ2) = −2( 4∑i=1

(qi, 0))

+ (p10) + (p2

0) + (p1∞) + (p2

∞),

Div(π∗θ3) = −2( 4∑i=1

(qi, 0))

+ 2(p10) + 2(p2

0).

Now, the properties can be easily derived from the above equations. More-over, the symmetries τ1, τ2 induce operations on the spinor bundle. In otherwords, they induce them on L2 which are determined uniquely only up to asign. With respect to these operations, the vector space spanned by σ1, σ2

is invariant. From the equations σ21 = π∗θ1 and σ2 = zσ1 we immediately

obtain the identities

[τ∗1 (σ1)]2 = τ∗1 (σ21) = τ∗1 (π∗θ1) = π∗τ∗1 θ1

= iπ∗θ3 = iz2π∗θ1 = i(zσ1)2

⇒ [τ∗1 (σ1)]2 = iσ22

⇒ τ∗1 (σ1) = ± 1√2(1 + i)σ2

and consequently

τ∗1 (σ2) = τ∗1 (zσ1) = iz τ∗1 (σ1)

= ± 1√2(1 + i) izσ2 = ∓ 1√

2(1− i)σ1.

Note that the Euler characteristic of L2 is

χ(L2) = deg(Div(π∗θj))

= 2χ(K2)

= 2 deg(Div(θj)) = −4.

Remark. Reviewing the above computations, we see that

(τ∗1 )2(σ1) = ± 1√2(1 + i)τ∗1 (σ2) = −σ1.

Be aware that this is not a contradiction to the symmetry property, becausethe operation of τ∗1 on L is unique only up to a sign. Hence, it is onlyfunctorial modulo a sign. Putting aside the fact that τ1 is an involution, wejust have the guarantee whether (τ∗1 )2 is either +id or −id on L. The reasontherefore is the non local construction of the spinor bundles which is relateddeeply with the sign ambiguity of the complex square root.

34 BELAID ALLOUSS

Additionally, the following identities for further meromorphic sections Γ(L)and Γ(L2) hold

Γ(L) with Div ≥ −( 4∑i=1

(qi, 0))

= spanσ1, σ2

and Γ(L2) with Div ≥ −2

( 4∑i=1

(qi, 0))

= spanπ∗θ1, π∗θ2, π

∗θ3, ⊕ C · dzw= spanσ2

1, σ1σ2, σ22 ⊕ C · dzw .

Note that dzw describes the tangent bundle of CP1.

Now, there are two concepts of describing the above mentioned second ap-proach. Both have advantages and disadvantages.

2.2. First Concept. The differential map dFτ resp. (LF−1τ

)∗dFτ meetsthree conditions.

In order to use results from the classical theory of minimal surfaces wecan consider the immersed minimal surface Σ as a holomorphic curve inthe complexified Heisenberg group (Nil(3), gτ ). Hence, with the followingdecomposition of ∂zFτ and ∂zFτ in the above basis

∂zFτ =3∑

k=1

Zkek, ∂zFτ =3∑

k=1

Zkek (2.2)

the conformality relation in nτ yields the first condition, the isotropy relation

Z21 + Z2

2 + Z23 = 0, |Z1|2 + |Z2|2 + |Z3|2 = 1

2e2u (2.3)

for the holomorphic curve. In this theory we call Σ an isotropic quadric inthe complexified Heisenberg group (Nil(3), gτ ).

Since the Levi-Civita connection is torsion-free we get the second condition,the compatibilty condition,

∇∂zFτ∂zFτ −∇∂zFτ∂zFτ = 0.

For the third condition, we list some facts. Let N ∈ (Nil(3), gτ ) be the usualnormal vector of the surface Σ at the origin. Using the real notation we canwrite down the first and second fundamental form as

I = e2u

(1 0

0 1

)and II = e2u

(h11 h12

h21 h22

)where we denote

h11 = gτ (∇x∂xFτ , N), h12 = gτ (∇x∂yFτ , N),

h21 = gτ (∇y∂xFτ , N), h22 = gτ (∇y∂yFτ , N).


Now the principal curvatures are the eigenvalues of the matrix e−2u II. Wesee that, for the conformal immersion Fτ ,

2H = h11 + h22 = e−2ugτ (∇x∂xFτ +∇y∂yFτ , N)

with the mean curvature H. Moreover, in addition with the torsion-freeconnection ∇ we have

∇x∂xFτ +∇y∂yFτ ⊥ ∂xFτ , ∂yFτ .This is easily seen by differentiating the identities (2.1). However, we nowget the third condition, the condition for constant mean curvature surfaces,

∇x∂xFτ +∇y∂yFτ = 2He2uN

or written in complex notation

∇∂zFτ∂zFτ +∇∂zFτ∂zFτ = He2uN.

Due to (1.17), we finally have

∂zzFτ − ∂zzFτ + B∂zFτ∂zFτ − B∂zFτ∂zFτ = 0, (2.4)

∂zzFτ + ∂zzFτ + B∂zFτ∂zFτ + B∂zFτ∂zFτ = He2uN. (2.5)

With the above notations (2.2) the equations (2.4) and (2.5) can now berewritten as ∑

j

(∂zZj − ∂zZj)ej +∑j,k

(ZjZk − ZjZk)∇ejek = 0 (2.6)

and∑j

(∂zZj + ∂zZj)ej +∑j,k

(ZjZk + ZjZk)∇ejek =

2iH[(Z2Z3 − Z2Z3)e1 + (Z3Z1 − Z3Z1)e2 + (Z1Z2 − Z1Z2)e3]. (2.7)

Note that B cancels because of the conformal invariance of the formulas(2.6) and (2.7).

Remarks. Considering the geometric data of the Scherk surface, this con-cept is invariant under translation and all the known symmetries are pre-served. Nevertheless, the isotropy in (2.3) is a non-linear side condition.

2.3. Second Concept. Let us try another ansatz. Again, we use the dif-ferential map ∂Fτ , and the isotropic quadric (2.3) is the starting point foranother approach. This quadric implies that a substitution of ∂zFτ in termsof ψτ := t(ψ+

τ , ψ−τ ) is possible

Z1 = i2 [(ψ−τ )2 + (ψ+

τ )2], Z2 = 12 [(ψ−τ )2 − (ψ+

τ )2], Z3 = ψ+τ ψ−τ . (2.8)

In general, we can interpret the above identities as a parametrization fromC2 to the null quadric in C3. For further observations, it is convenientto understand ψτ and its variants as spinors, i.e. sections of the half spinbundles, defined on the 4-punctured sphere CP1. Let us deal with the Scherksurface Σ immersed into R3 (τ= 0) where we always regard R3 = =mH aspurely imaginary quaternions. We learn from the subsection 1.1 that theclassical Weierstraß data of this surface are

f = 2z4−1

, g = z.

36 BELAID ALLOUSS

We want to reformulate the Weierstraß representation of the surface Σ byusing the following equation

F0(z) = <e∫ z

0

i2 [(ψ+

0 )2+(ψ−0 )2]12 [(ψ+

0 )2−(ψ−0 )2]ψ+

0 ψ−0

dζ. (2.9)

Here, the integrand is a C3-valued one-form on Σ or that is to say theholomorphic differential dF0 ∈ Γ(Ω), Ω := T ∗CΣ ⊗R C, of the holomorphiccontinued immersion F0 written in terms of half spin bundles ψ±0 .

Let us go deeper into the theory of spinors and conformal immersions [5, 21,22]. Every conformal immersion F0 of a Riemannian surface into R3 definesa spin structure on the Riemannian surface. Furthermore this spin structurecharacterizes uniquely the regular homotopy class F of this immersion. Inthis theory the square root

√dF0 of the differential of F0 is a section in

the associated spin bundle K, with K⊗2 = T ∗Σ. Nevertheless, K is still acomplex line bundle.Now a spin bundle on a Riemannian surface is a quartenionic line bundle K,and a choice of a spin bundle is equivalent to choosing a square root bundleof conformal C3-valued one-form on Σ. There is a known theorem whichsays

Theorem 2.10. A spin bundle on a Riemannian surface Σ is a quaternionicline bundle K on Σ with a chosen endomorphism J ∈ EndH(K) and anontrivial quaternionic-hermitian, fiber-preserving pairing

(., .) : K ×K → T ∗Σ⊗H,

so that J2 = −1, and for every two spinors, ψ, φ ∈ K based at the samepoint p we have

(φ, ψ)(JX) = (Jφ, ψ)(X) = (φ, Jψ)(X)

for every X ∈ TpΣ. Here JX denotes the action of the complex structureon the vector X.

For every spinor ψ ∈ K the form (ψ,ψ) = ω is imaginary quaternionicvalued, and is a conformal C3-valued one-form on Σ. This fact is importantfor the reconstruction of Fτ . Every spinor ψ defines a nonnegative half-density |ψ|2 := ‖(ψ,ψ)‖ where ‖.‖ is the norm in C3.

A choice of a spin bundle is equivalent to a choice of a spin structure, that is,a holomorphic square root of the canonical bundle T (1,0)Σ of Σ. Note thatfor ω we identify the vector space C∞(Σ, T ∗RΣ ⊗ C) with C∞(Σ, (T (1,0)Σ ⊕T (0,1)Σ)⊗ C). In other words we have

C∞(Σ, T (1,0)Σ⊗ C) := =m(1 + iJ)ω |ω ∈ C∞(Σ, T ∗RΣ⊗ C).

When we look at the Dirac equation of the previous section, the operator Dis the conformal Dirac operator with

D : C∞(Σ,K ⊕ K)→ C∞(Σ, (K ⊕ K)⊗R T∗Σ)


Do not forget that in this generalized case T ∗Σ is a subset in the dual spaceof the Lie algebra nτ , spanned by the complex (Nil(3), gτ )-valued one-forms∂zFτdz and ∂zFτdz.

Now let us look to the explicit formulas of (2.6) and (2.7) in nτ .

∂zZ1 − ∂zZ1 = 0

∂zZ2 − ∂zZ2 = 0

∂zZ3 − ∂zZ3 + τ(Z1Z2 − Z1Z2) = 0

∂zZ1 + ∂zZ1 + τ(Z2Z3 + Z2Z3) = 2iH(Z2Z3 − Z2Z3)

∂zZ2 + ∂zZ2 − τ(Z1Z3 + Z1Z3) = 2iH(Z3Z1 − Z3Z1)

∂zZ3 + ∂zZ3 = 2iH(Z1Z2 − Z1Z2)

(2.10)

Note that we deal with minimal surfaces, i.e. the mean curvature H is equalzero. In other words the left hand side of (2.7) vanishes. Inspired by the theresults in [7], we state our second result for Scherk surfaces in (Nil(3), gτ ).

Theorem 2.11. Let Σ ⊂ CP1 # (Nil(3), gτ ) be a conformal immersion ofScherk surfaces into the 3-dimensional Heisenberg group (Nil(3), gτ ). Thentheir spinor representation ψτ yields the following system of differentialequations

∂zf−τ ⊗ dz = i

τ

2|zf+

τ |2 − |f−τ |2

z4 − 1· zf+

τ ⊗ dz,

∂z(zf+τ )⊗ dz = −iτ

2|zf+

τ |2 − |f−τ |2

z4 − 1· f−τ ⊗ dz,

(2.11)

with ψ+τ = zf+

τ ψ+

0 and ψ−τ = f−τ ψ−0 . Here, the spinor representation ψ±0 of the

classical Scherk surface immersed into R3 are interpreted as basis spinors.Furthermore, the complex-valued functions f±τ are holomorphic and boundedin Σ.

Proof. If we substitute (2.8) into the formulas (2.10), we obtain the follow-ing expressions for the first pair of equations and for the fourth and fifthequation:

−i[ψ−τ ∂zψ

−τ + ψ+

τ ∂zψ+τ

]− i[ψ−τ ∂zψ

−τ + ψ+

τ ∂zψ+τ

]= 0[

ψ−τ ∂zψ−τ − ψ+

τ ∂zψ+τ

]−[ψ−τ ∂zψ

−τ − ψ+

τ ∂zψ+τ

]= 0

− i2

[∂z(ψ−τ )2 + ∂z(ψ+

τ )2]

+ i2

[∂z(ψ−τ )2 + ∂z(ψ+

τ )2]

+τ(

12

[(ψ−τ )2 − (ψ+

τ )2]ψ+τ ψ−τ + 1

2

[(ψ−τ )2 − (ψ+

τ )2]ψ+τ ψ−τ

)= 0

12

[∂z(ψ−τ )2 − ∂z(ψ+

τ )2]

+ 12

[∂z(ψ−τ )2 − ∂z(ψ+

τ )2]

−τ(i2

[(ψ−τ )2 + (ψ+

τ )2]ψ+τ ψ−τ − i

2

[(ψ−τ )2 + (ψ+

τ )2]ψ+τ ψ−τ

)= 0.

(2.12)

Multiplying the first equation with i and summing up with the second oneyields

∂z(ψ−τ )2 + ∂z(ψ+τ )2 = 0. (2.13)

We redo this for the last pair of equations in (2.12) and get

∂z(ψ−τ )2 − ∂z(ψ+τ )2 + iτ ψ+

τ ψ−τ

(|ψ−τ |2 − |ψ+

τ |2)

= 0. (2.14)

38 BELAID ALLOUSS

If we demand that ψ+τ ψ−τ 6= 0 we can rewrite these two equations in the

following way

i∂zψ−τ − τ

4ψ+τ

(|ψ−τ |2 − |ψ+

τ |2)

= 0

−i∂zψ+τ − τ

4ψ−τ

(|ψ−τ |2 − |ψ+

τ |2)

= 0(2.15)

or written in the form of a non-linear Dirac equation

Dψτ = τ4

(|ψ−τ |2 − |ψ+

τ |2)ψτ . (2.16)

¿From the classical Weierstraß data of the Scherk surface we get the basisspinors

ψ+

0 = ± 1w · η, ψ−0 = ± 1

w · η, with η2 = dz and 2w2 = z4 − 1

For our purposes we write the sought spinor as

ψτ =(zf+τ ψ

+

0f−τ ψ

−0

)where the basis spinors ψ±0 are defined on the two-sheeted covering Σ of the4-punctured sphere Σ in order to avoid the monodromy around the zeroesqi. It follows that the complex-valued pair of functions zf+

τ and f−τ definedon Σ itself are bounded and have no monodromy. In other words, we havethe following diagram of sections in which the solutions are defined

Γ(Σ,K ⊕ K) Γ(Σ, (K ⊕ K)⊗KK)

Γ(Σ,C⊕ C) Γ(Σ, T (0,1)Σ⊕ T (1,0)Σ)

Γ(Σ, L⊕ L) Γ(Σ, (L⊕ L)⊗ LL)

.............................................................................................................................................................................................................................................................................................................................. ............

(0 ∂z

−∂z 0

)..................................................................................................................................................................................................................................................................................................................

elliptic, Index=2

....................................................................................................................................................................................................................................................................................................................

elliptic

......................................................................................................................................................................................................................................................................................................................................

elliptic

....................................................................................................................................................................................................................................................................................................................

←−These maps yield additional pointsof indeterminacy, because the bun-dles are not isomorph!

−→

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

....................

............

(.⊗ η, .⊗ η)

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

....................

............

(.⊗ η, .⊗ η)

................................................................................................................................................................................................................................................................................................................................ ............D

.................................................................................................................................................................................................................................................................................................................................................. ............D.............................................................................................................................

π|Σ

.............................................................................................................................

π|Σ

.............................................................................................................................

2:1

.............................................................................................................................

2:1

The generalized system of the Dirac equation can be formulated like

∂zψ−τ ⊗ dz = i τ4ψ

+τ

(|ψ+τ |2 − |ψ−τ |2

),

∂zψ+τ ⊗ dz = −i τ4ψ

−τ

(|ψ+τ |2 − |ψ−τ |2

).

With the above notation this system translates into[∂zf

−τ · 1

w + f−τ · ∂z( 1w )]η ⊗ dz = i τ4

|zf+τ |2−|f−τ |2|w|2 · 1

w · zf+τ η ⊗ η ⊗ η,[

∂z(zf+τ ) · 1

w + zf+τ · ∂z( 1

w )]η ⊗ dz = −i τ4

|zf+τ |2−|f−τ |2|w|2 · 1

w · f−τ η ⊗ η ⊗ η.

Note that the zeroes and poles of the defined spinors ψ+τ and ψ−τ are fixed.

Now the above system can easily be reduced to the equations in (2.11).

We have to point out that several versions of the Dirac equation (2.16) havebeen also observed by other researchers [1, 9].


Remarks. The advantages of this concept are that the non-linear constraintvanishes, and the Dirac operator yields elliptic regularity, i.e. the solutionsare as good as it is allowed by the right hand side of the equation (2.16)and by the Dirichlet boundary condition. Furthermore, we have a link tointegrable systems, and this ansatz deals with less variables than the firstconcept. Nevertheless, there is no chance to lift all known symmetries of theScherk surface into the spin bundles.

A basic ODE technique to get further information for the solution of theDirac equation (2.16) is the pertubation theory with respect to the bundlecurvature τ . It leads to an expression resp. approximation for the desiredsolution in terms of a power series for a small pertubation of τ that quantifiesthe deviation from the exactly solvable problem

ψτ =∞∑j=0

τ j · ψτ, j .

The leading term in this power series is the solution of the exactly solvableproblem in R3, while further terms describe the deviation in the solution,due to the deviation from the initial problem. Formally, we get a system ofequations

Dψτ, j+1 =∑

j1,j2,j3≥0j1+j2+j3=j

(〈ψ−τ, j1 , ψ

−τ, j2〉 − 〈ψ+

τ, j1, ψ+

τ, j2〉)· ψτ, j3 .

Now, we apply these considerations to the system (2.11) which is defined onΓ(Σ, T (0,1)Σ⊕ T (1,0)Σ). We insert the respective power series

f−τ =∞∑j=0

τ j · aj , zf+τ =

∞∑j=0

τ j · zbj

into these equations and get

∂zaj+1 =i

21

z4 − 1

∑j1,j2,j3≥0j1+j2+j3=j

(zbj1zbj2 − aj1 aj2

)zbj3 ,

∂z(zbj+1) = − i2

1z4 − 1

∑j1,j2,j3≥0j1+j2+j3=j

(zbj1zbj2 − aj1 aj2

)aj3 .

(2.17)

Here, the initial values are f±0 ≡ 1 resp. a0 ≡ 1 and b0 ≡ 1, in other wordsthe exact solutions for the Scherk surface in R3. For the time being, theremaining coefficient functions aj , bj can only defined in the following set ofintegrable functions ⋂

p<2

Lp1 ∩⋂q<∞Lq.

This is due to additional points of indeterminacy by the map (. ⊗ η, . ⊗ η).Let us study the explicit solutions of some further terms of f−τ and zf+

τ . For

40 BELAID ALLOUSS

j = 0 we get the solutions

a1 =i

8

4∑j=1

qj(z − qj) ln(|z − qj |2),

zb1 =i

8

4∑j=1

q2j (z − qj) ln(|z − qj |2).

We have paid our attention that the adapted holomorphic resp. antiholo-morphic integration constants meet the desired non-monodromy around thesingularities qi. Differentiating them, we exactly retrieve the equations in(2.17) for j = 0

∂za1 =i

4

4∑j=1

qjz − qjz − qj

=i

2z(zz − 1)z4 − 1

,

∂z(zb1) =i

4

4∑j=1

q2j

z − qjz − qj

= − i2zz − 1z4 − 1

.

Moreover, we are able to verify the boundedness and the additional points ofindeterminacy for the above solutions. For ζi := z−qi we have the inequalityζi(ln|ζi|2) ≤ cα|ζ|α with α > 0 and |ζi| 1. This clarifies the boundedness.Rewriting the solutions as follows

a1 =i

4

[z

(ln∣∣∣∣z − 1z + 1

∣∣∣∣+ i · ln∣∣∣∣z − iz + i

∣∣∣∣)+ ln∣∣∣∣z2 + 1z2 − 1

∣∣∣∣] ,b1 = − i

4

[1z

(ln∣∣∣∣ z − 1z + 1

∣∣∣∣+ i · ln∣∣∣∣ z + i

z − i

∣∣∣∣)− ln∣∣∣∣ z2 − 1z2 + 1

∣∣∣∣] ,we see that the limits for this pair of functions lim

|z|→0resp. lim

|z|→∞are not

determined. Note that a better regularity of these solutions are obstructedby the indeterminacy points at 0 and ∞.

2.4. The Global Dirac Equation for Scherk Surfaces. Let us extendequation (2.11) for the whole surface Σ endowed with the metric

g =(

1 + |z|2

|1− z4|

)2

|dz|2

which reflects the geometry and shape of the surface. For easier computationwe use g = e2u|dz|2 with u = ln(1 + |z|2)− ln |1− z4| so that we obtain forthe Christoffel symbol

Γ(X,Y ) = (dXu) · Y + (dY u) ·X − 〈X,Y 〉 gradu.


Let us denote the Dirac operator w.r.t. the metric g with D(Σ,g). The newDirac eqaution is derived as follows: On the left-hand side of (2.11) we get

D(Σ,g)ψ =2∑j=1

ej · ∇ejψ

=2∑j=1

ej

(dejψ +

14

2∑k=1

ek · ∇ejek · ψ)

=2∑j=1

ej

(dejψ +

14

2∑k=1

ek

[(deju) · ek

+ (deku) · ej − 〈ej , ek〉 gradu]ψ)

=2∑j=1

ej · dejψ.

(2.18)

We conclude that the left-hand side is nothing else than the original one of(2.11). This is due to the conformal invariance of the Dirac equation

3. A Modified Sinh-Gordon Equation

To get more information concerning the solution for the spinors we usethe generalized Hopf differential which was introduced by Uwe Abresch andHarold Rosenberg [2, 3]. In this context we follow the calculation of DimitryBerdinsky and Iskander Taimanov in [7, 32], keeping track of the bundlecurvature τ as a parameter of the equation.

So far for surfaces in (Nil(3), gτ ) we know the potentials of the Dirac equation

U(Nil(3),gτ ) = V(Nil(3),gτ ) := H2

(|ψ+τ |2 + |ψ−τ |2

)+ i τ4

(|ψ−τ |2 − |ψ+

τ |2). (3.1)

In order to calculate the quadratic Hopf differential A = 〈 ∇∂z∂zFτ , N〉 we usethe normal vector of the surface which is translated to the Lie algebra nτ byleft multiplication with LF−1

τ. We get

N = e−u[i(ψ+τ ψ−τ − ψ+

τ ψ−τ )e1 − (ψ+

τ ψ−τ + ψ+

τ ψ−τ )e2 + (|ψ−τ |2 − |ψ+

τ |2)e3]

which leads to the identity

A = (ψ−τ ∂zψ+τ − ψ+

τ ∂zψ+τ ) + iτ(ψ+

τ )2(ψ−τ )2.

Furthermore the Weingarten equations expressed in terms of ψ±τ are theDirac equation and the system

∂zψ+τ = uzψ

+τ +Ae−uψ−τ − i τ2 (ψ+

τ )2ψ−τ ,

∂zψ−τ = uzψ

−τ − Ae−uψ+

τ − i τ2 ψ+τ (ψ−τ )2.

(3.2)

By means of the zero-curvature conditions

∂z∂zψτ = ∂z∂zψτ

42 BELAID ALLOUSS

one can get the Codazzi equations

uzz − |A|2e−2u +H2

4e2u =

4τ2 − 116

e2u − τ2|Z3|2,

∂z

(A+

τ2Z23

2H + iτ

)=

12Hze

2u + ∂z

(1

2H + iτ

)τ2Z2

3 .

(3.3)

From the results above one can follow

Corollary 3.1. The modified quadratic differential

Adz2 =(A+

τ2Z23

2H + iτ

)dz2 (3.4)

is holomorphic for constant mean curvature surfaces in (Nil(3), gτ ).

Moreover we have

Proposition 3.2 (Abresch). If the differential Adz2 is holomorphic thenthe surface in (Nil(3), gτ ) has constant mean curvature.

Proof. Copy the proof from [7] while considerating the additional bundlecurvature τ .

We are now ready to derive a new partial differential equation, a modifiedSinh-Gordon equation, from (3.1) and (3.4). First of all we differentiate thepotential U(Nil(3),gτ ) with respect to z

∂zU(Nil(3),gτ ) =2H+iτ4 [(∂zψ−τ )ψ−τ + ψ−τ ∂zψ

−τ

+ 2H−iτ4 [(∂zψ+

τ )ψ+τ + ψ+

τ ∂zψ+τ ] + 1

2Hz|ψτ |2

where ψτ := (ψ+τ , ψ

−τ ). Using the Dirac equation we get

=2H+iτ4 ψ−τ ∂zψ

−τ + 2H−iτ

4 ψ+τ ∂zψ

+τ ]

+ 2H+iτ4 ψ−τ

[−H

2 |ψτ |2 − i τ4

(|ψ−τ |2 − |ψ+

τ |2)]ψ+τ

+ 2H−iτ4 ψ+

τ

[H2 |ψτ |

2 − i τ4(|ψ−τ |2 − |ψ+

τ |2)]ψ−τ

+ Hz2 |ψτ |

2

=2H+iτ4 ψ−τ ∂zψ

−τ + 2H−iτ

4 ψ+τ ∂zψ

+τ − iτ H2 ψ

+τ ψ−τ |ψ−τ |2.

We rewrite this equation as2H+iτ

4 ψ−τ ∂zψ−τ + 2H−iτ

4 ψ+τ ∂zψ

+τ

−iτ H2 ψ+τ ψ−τ |ψ−τ |2 = ∂zU(Nil(3),gτ ) − Hz

2 |ψτ |2.

(3.5)

Multiplying equation (3.5) with ψ+τ and the modified Hopf differential

A = (ψ−τ ∂zψ+τ − ψ+

τ ∂zψ+τ ) + 2Hiτ

2H+iτ (ψ+τ )2(ψ−τ )2 (3.6)

with 14(2H + iτ)ψ−τ we get

(∂zU(Nil(3),gτ ) − Hz2 |ψτ |

2)ψ+τ + 1

4(2H + iτ)Aψ−τ=[

2H+iτ4 |ψ−τ |2 + 2H−iτ

4 |ψ+τ |2]∂zψ

+τ

= U(Nil(3),gτ ) · ∂zψ+τ .


If we restrict ourselves to minimal surfaces, H ≡ 0, in (Nil(3), gτ ), a newsystem of equations similar to (3.2) reveals

∂zψ+τ = wzψ

+τ + 1

τ e−wAψ−τ ,

∂zψ−τ = 1

τ e−w ¯Aψ+

τ + wzψ−τ

where we use the substitution ew := |ψ−τ |2 − |ψ+τ |2.

Remark. Note that this specialization comes as a singular limit of the for-mulas by Taimanov and Berdinsky. Thus the subsequent computations arereally a new result.

In other words the Weingarten equations for the immersion of minimal sur-faces in (Nil(3), gτ ) are rewritten in the following structure equations

∂zψτ =(

wz1τ e−wA

−i τ4 ew 0

)ψτ ,

∂zψτ =(

0 iτ4 ew

1τ e−w ¯A wz

)ψτ .

(3.7)

Now using the zero-curvature condition and (3.7) we obtain[∂z

(wz

1τ e−wA

−i τ4 ew 0

)− ∂z

(0 i

τ4 ew

1τ e−w ¯A wz

)+(

wz1τ e−wA

−i τ4 ew 0

)(0 i

τ4 ew

1τ e−w ¯A wz

)−(

0 iτ4 ew

1τ e−w ¯A wz

)(wz

1τ e−wA

−i τ4 ew 0

)]ψτ = 0

⇒[(

wzz1τ (e−wAz−wze−wA)

−i τ4wzew 0

)−(

0 iτ4wze

w

1τ (e−w ¯Az−wze−w ¯A) wzz

)+( 1τ2 e−2w|A|2 i

τ4wze

w+1τ wze

−wA

0τ2

16 e2w

)−(

τ2

16 e2w 0

1τ wze

−w ¯A−i τ4wzew 1

τ2 e−2w|A|2

)]ψτ =0

By considerating the holomorphicity of the modified Hopf differential, i.e.∂zA = 0, and the fact that ew = |ψ−τ |2 − |ψ+

τ |2 6= 0 we can summarize(wzz+

1τ2 e−2w|A|2− τ

2

16 e2w 0

0 wzz− 1τ2 e−2w|A|2+

τ2

16 e2w

)ψτ = 0.

As a consequence we have the following partial differential equation of thetype of a Sinh-Gordon equation with an additional term

wzz + 1τ2 e−2w|A|2 − τ2

16e2w = 0.

A handy substitution w := w + ln τ leads to14∆w − 1

16e2w − |A|2e−2w = 0

⇒ ∆w − sinh w cosh w + (4|A|2 − 14)e−2w = 0. (3.8)

To get more information about this equation we multiply (3.8) with e2w anduse the substitution tanα = W := ew, so that

44 BELAID ALLOUSS

(tan4 α− 1)|∇α|2 + (tan3 α+ tanα)∆α

−14(tan4 α− 1) + (4|A|2 − 1

4) = 0,

⇒tanα[2(tan3 α+ tanα)|∇α|2 + (tan2 u+ 1)∆α]

−(1 + tan2 α)2|∇α|2 − 14(tan4 α− 1) + (4|A|2 − 1

4) = 0,

⇒ W∆W − |∇W |2 − 14(W 4 − 16|A|2) = 0. (3.9)

Notice that W is nothing else than the potential U(Nil(3),gτ ) = V(Nil(3),gτ ) forminimal surfaces multiplied with −4i. With regard to the solution of thisequation we need analytic details concerning the potential. Therefore, wecollect some important results for our purpose which were achieved in re-cent years and give us a cross link to the above result. In spring 2005 BenoıtDaniel [13] extended the classical Lawson correspondence between cmc sur-faces in space forms to cmc surfaces in further 3-dimensional homogeneousspaces. He presented a generalized Lawson correspondence which includesthe existence of a bijective isometric correspondence between simply con-nected surfaces with constant mean curvature H = 1

2 (cmc-12) in H2×R and

simply connected minimal surfaces in the 3-dimensional Heisenberg group(Nil(3), gτ ). Note that this correspondence maps graphs to graphs.

Furthermore, in spring 2006 Isabel Fernandez and Pablo Mira [16] intro-duced a hyperbolic Gauss map into the Poincare disk for surfaces in H2 ×Rwith regular vertical projection. They proved that for surfaces with H = 1

2the hyperbolic Gauss map is harmonic. Vice versa, they showed that everynowhere conformal harmonic map from an open simply connected Riemannsurface Σ into the Poincare disk is the hyperbolic Gauss map of such sur-faces.

In their paper they describe the immersion of surfaces in H2 × R as fol-lows: The 3-dimensional homogeneous space H2×R can be presented in theLorentz-Minkowski 4-space L4 as

H2 × R = (x0, x1, x2, x3) ∈ L4| − x20 + x2

1 + x22 = −1, x0 > 0

where L4 is endowed with a Lorentzian metric. For simply connected sur-faces the immersion in H2 × R is denoted as

ϕ = (N,h) : Σ→ H2 × R.

Here N : Σ→ H2 is the vertical projection and h : Σ→ R the height functionof ϕ. For their purposes N has to be regular, i.e. dN is a linear isomorphismat every point. Furthermore η : Σ→ S3

1 ⊂ L4 denotes the unit normal vectorin H2 × R. S3

1 = (x0, x1, x2, x3) ∈ L4| − x20 + x2

1 + x22 + x2

3 = 1 is calledthe de Sitter 3-space. The pair η,N turns out to be a orthonormal frameof the Lorentzian normal bundle of ϕ in L4. The next step is to use thesplitting notation η = (N , u) : Σ→ L3×R where u : Σ→ [−1, 1] is called theangle function of ϕ. With u 6= 0 at every point which provides a canonicalorientation for such surfaces they considered a certain map ξ := η+N

u , taking


values in the intersection of the light cone in L4 with the horizontal affinehyperplane x3 = 1 of L4. In addition there is some G : Σ → H2 such thatξ = (G, 1). G is the above mentioned hyperbolic Gauss map of ϕ. It hassimilarity with the usual hyperbolic Gauss map for surfaces in H3. We noticethe following

Remark. The potential W of (3.9) for minimal surfaces in (Nil(3), gτ ) isnothing else than the Gauss map G of the Lawson corresponding cmc-1

2

surfaces in H2×R. Moreover, the fact that the analytical extension does notwork on the set z|W = 0 is equivalent to the non-harmonicity of G whenit is horizontal. G maps the set W = 0 to the boundary of H2.

Nevertheless, from our Dirac equation (2.11) we know that the potentialfails to be analytic only in the zeroes qi = ±1,±i, while the Gauss mapG fails to be harmonic in ∂∞D of the Poincare disk model.

References

[1] U. Abresch: Spinor representation of CMC surfaces, Lecture at Luminy, 1989.[2] U. Abresch, H. Rosenberg: The Hopf differential for constant mean curvature surfaces

in S2 × R and H2 × R, Acta Mathematica 193 (2004), 141–174.[3] U. Abresch, H. Rosenberg: Generalized Hopf Differentials, Mat. Contemp. 28 (2005),

1–28.[4] F. J. Almgren: Some interior regularity theorems for minimal surfaces and an exten-

sion of the Bernstein’s theorem, Ann. of Math. 85 (1966), 277–292.[5] M. Atiyah: Riemann surfaces and spin structures, Ann. Scient. Ecole Norm. Sup. 4

(1971), 47–62.[6] J. L. M. Barbosa, A. G. Colares: Minimal Surfaces in R3, Lecture Notes in Mathe-

matics 1195. Springer-Verlag, Berlin, 1986.[7] D. A. Berdinsky, I. A. Taimanov: Surfaces in 3-dimensional Lie Groups, Siberian

Math. Journal 46:6 (2005), 1005–1019.

[8] S. N. Bernstein: Uber ein geometrisches Theorem und seine Anwendung auf die par-tiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551–558.

[9] A. I. Bobenko: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases,Harmonic Maps and Integrable Systems. Vieweg, Braunschweig/Wiesbaden, 1994.

[10] E. Bombieri, E. DeGiorgi, E. Giusti: Minimal cones and the Bernstein problem, In-vent. Math. 7 (1969), 244–268.

[11] P. Collin, H. Rosenberg: Construction of harmonic diffeomorphisms and minimalgraphs. Preprint, http://arxiv.org/abs/math/0701547, 2007.

[12] P. Collin, H. Rosenberg: The Jenkins-Serrin theorem for minimal graphs in homoge-neous 3-manifolds. Preprint in preparation, 2007.

[13] B. Daniel: Isometric immersions into 3-dimensional homogeneous manifolds, Com-ment. Math. Helv. 82:1 (2007), 87–131.

[14] B. Daniel, L. Hauswirth Half-space theorem, embedded minimal annuli and minimalgraphs in the Heisenberg group. Preprint, http://arxiv.org/abs/0707.0831, 2007.

[15] U. Dierkes, S. Hildebrandt, A. Kuster, O. Wohlrab: Minimal Surfaces I, Grundlehrender mathematischen Wissenschaften 295. Springer-Verlag, Berlin, 1992.

[16] I. Fernandez, P. Mira: Harmonic maps and constant mean curvature surfaces in H2×R, American Journal of Mathematics 129:4 (2007), 1145–1181.

[17] C. B. Figueroa, F. Mercuri, R. H. L. Pedrosa: Invariant Surfaces of the HeisenbergGroups, Ann. Math. Pura Appl. 177 (1999), 173–194.

[18] T. Friedrich: Dirac-Operatoren in der Riemannschen Geometrie, Advanced Lecturesin Mathematics, Vieweg Verlagsgesellschaft, Braunschweig, 1997.

[19] H. Hopf: Differential geometry in the large, Lecture Notes in Mathematics, 1000.Springer-Verlag, Berlin, 1983.

[20] H. Jenkins, J. Serrin: Variational problems of minimal surface type II. Boundaryvalue problems for the minimal surface equation, Arch. Rational Mech. Anal. 21:4(1966), 321–342.

[21] G. Kamberov, P. Norman, F. Pedit, U. Pinkall: Quaternions, spinors, and surfacesContemporary Mathematics, 299. AMS, Providence, 2002.

[22] R. Kusner, N. Schmitt: The spinor representation of surfaces in space. Preprint,http://www.arxiv.org/abs/dg-ga/9610005, 1996.

[23] W. H. Meeks III: A survey of the geometric results in the classical theory of minimalsurfaces, Bol. Soc. Brasil. Mat. 12:1 (1981), 29–86.

[24] B. Nelli, H. Rosenberg: Minimal surfaces in H2 × R, Bull. Braz. Math. Soc. 33:2(2002), 26–292.

[25] R. Osserman: A Survey of Minimal Surfaces, Dover Publications Inc., 1986.[26] R. Osserman: Global properties of minimal surfaces in E3 and En, Ann. of Math.

80:2 (1964), 340–364.[27] A. L. Pınheiro: Teorema de Jenkins-Serrin em M2 × R, submitted to Bull. Braz.

Math. Soc., http://www.pg.im.ufrj.br/teses/Matematica/Doutorado/140.pdf.[28] H. F. Scherk: Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen,

Crelle J. 13 (1835), 185–208.


[29] J. Simons: Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968),6–105.

[30] I. A. Taimanov: Dirac Operators and Conformal Invariants of Tori in 3-Space, Pro-ceedings of the Steklov Institute of Mathematics 244 (2004), 233–263.

[31] I. A. Taimanov: Two-dimensional Dirac operator and the theory of surfaces RussianMath. Surveys 61:1 (2005), 79–159.

[32] I. A. Taimanov: Surfaces in three-dimensional Lie groups in terms of spinors.Preprint, http://arxiv.org/abs/0712.4139, 2007.

Lebenslauf

Personliche Daten

Name: Belaid Allouss

E-Mail: [email protected]

Geboren: 9. September 1975 in Beni Chiker, Marokko

Hochschulstudium

2003 – 2007 Bearbeitung der vorliegenden Dissertation an der Ruhr-UniversitatBochum. Betreuer: Prof. Dr. Uwe Abresch.

Sommerschule des Institut de Mathematique de Jussieuin Paris zum Thema:

”Surfaces Minimales et Problemes Variationnels”

von 30. Juni bis 8. Juli 2004.

2003 Diplom der Mathematik.

2001 – 2002 Diplomarbeit an der Ruhr-Universitat Bochum zum Thema:

”Ein auf Kantenresiduen basierender robuster Fehlerschatzerfur singular gestorte Reaktions-Diffusionsgleichungen”.

Betreuer: Prof. Dr. Rudiger Verfurth.

1995 – 2001 Studium der Mathematik an der Ruhr-Universitat Bochummit Nebenfach Physik.

Schulbildung

1986 – 1995 Pestalozzi-Gymnasium in Herne, Abitur.

1982 – 1986 Stadtische Gemeinschaftsgrundschule an der Dungelstraßein Herne.

Bochum, 7. Januar 2008

Belaid Allouss

Documents

Scherk Surfaces in Heisenberg Groupswebdoc.sub.gwdg.de/ebook/dissts/Bochum/Allouss2008.pdf · SCHERK SURFACES IN HEISENBERG GROUPS BELAID ALLOUSS Abstract. In a recent paper Uwe Abresch