On the Weierstrass-EnneperRepresentation of Minimal Surfaces

Albin IngelströmBachelor’s thesis2017:K15

Faculty of ScienceCentre for Mathematical SciencesMathematics

CE

NT

RU

MSC

IEN

TIA

RU

MM

AT

HE

MA

TIC

AR

UM

Abstract

The Weierstrass-Enneper representation of minimal surfaces gives an importantlink between differential geometry and complex analysis. In short, it tells us thatany regular minimal surface can be represented by two meromorphic functions ofone complex variable. The main focus of this paper is to prove this fact and to giveexamples of this representation for well known minimal surfaces.

We will assume some basic knowledge of complex analysis while some basic resultsfrom differential geometry will be stated in Chapter 1. This to remind the readerof these and to establish the notation that will be used in the rest of the thesis.In Chapter 2 the Weierstrass-Enneper representation and its preliminaries will bediscussed and in Chapter 3 we will look at some classical examples.

Throughout this work it has been my firm intention to give reference to the statedresults and credit to the work of others. All theorems, propositions, lemmas andexamples left unmarked are assumed to be too well known for a reference to be given.

Acknowledgments

This text would not have been if not for the people around me who have encour-aged me towards this goal. First I would like to thank my thesis supervisor, Sig-mundur Gudmundsson, for his guidance during the writing process and his teachingprior and parallel to this project. My parents have always given me their unlimitedsupport, and for that I am forever grateful. Finally, I would like to thank my partnerMalin and my dear friend Kenneth, for making my time here in Lund as good as itcould ever be.

Albin Ingelstrom

Contents

1 Minimal Surfaces 11.1 General Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Weierstrass-Enneper Representation 72.1 Preliminary Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Defining the Weierstrass-Enneper Representation . . . . . . . . . . . 10

3 Examples and Applications 133.1 The Catenoid and the Helicoid . . . . . . . . . . . . . . . . . . . . . . 133.2 Reflection Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The Enneper Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Scherk’s Doubly Periodic Surface . . . . . . . . . . . . . . . . . . . . 17

4 Summing Up 21

Bibliography 23

Chapter 1

Minimal Surfaces

To be able to discuss the properties of minimal surfaces and their relationship withmeromorphic functions we first need to establish some terminology. The notation issimilar to that used in [4], and complete proofs may be found there.

1.1 General Surface Theory

The partial derivatives of X with respect to u and v will be denoted by Xu and Xv,respectively, and a unit normal vector field to the surface by N . In this text we willbe concerned with regular parametrisations.

Definition 1.1. A differentiable map X : U ⊆ R2 → R3 is said to be a regularparametrised surface if

Xu(q)×Xv(q) 6= 0,

for every q ∈ U .

The coefficients of the first and second fundamental forms will be denoted by E,F, G, e, f and g, respectively. The following formulas serve as definitions of thesecoefficients and for a further discussion of the first and second fundamental formsthe reader is referred to [2] and [4].

Definition 1.2. Let X : U ⊆ R2 → R3 be a regular parametrised surface. Then itsfirst fundamental form is defined as

E du2 + 2F dudv +G dv2,

where

E =< Xu, Xu >, F =< Xu, Xv > and G =< Xv, Xv > .

Definition 1.3. Let X : U ⊆ R2 → R3 be a regular parametrised surface with theunit normal vector field N given by

N =Xu ×Xv

|Xu ×Xv|.

Then its second fundamental form is defined as

e du2 + 2f dudv + g dv2,

1

where

e =< Xuu, N >, f =< Xuv, N > and g =< Xvv, N > .

It is well-known that holomorphic functions are conformal i.e. angle-preserving.For all results presented here, we assume that the minimal surface is parametrisedby isothermal coordinates.

Definition 1.4. A regular parametrisation X : U ⊆ R2 → R3 of a surface is said tobe conformal if it preserves angles. This means that the surface is parametrised byisothermal coordinates i.e.

E =< Xu, Xu >=< Xv, Xv >= G and F =< Xu, Xv >= 0.

It is a classical result that, locally, every surface can be parametrised by isothermalcoordinates. A complete proof of this can be found in [5].

We now define the important notion of the area of a surface.

Definition 1.5. Let X : U ⊆ R2 → R3 be a regular parametrisation of a surface.Then the area A of the subset X(D) of X(U) is defined by

A(X(D)) =

∫D

√EG− F 2 · dA.

The mean curvature H of a surface is equal to the mean of its principal curvatures.The first and second fundamental forms give us a useful formula for calculating Hwithout determining the principal curvatures. This standard formula is stated herewithout a proof.

Theorem 1.6. Let X : U ⊆ R2 → R3 be a regular parametrised surface. Then itsmean curvature H satisfies the following equation

H =eG− 2fF + gE

2(EG− F 2),

where E, F, G, e, f and g are the coefficients of the first and second fundamentalforms, respectively.

1.2 A Variational Problem

Definition 1.7. A regular surface in R3 is said to be minimal if its mean curvatureis constantly equal to zero.

To explain why a surface with vanishing mean curvature is called a minimalsurface we need the following notion of a normal variation.

Definition 1.8. A normal variation of a regular parametrised surface X, determinedby the function h : D → R, is a map ϕ : D × (−ε, ε)→ R3 such that

ϕ(u, v, t) = X(u, v) + t · h(u, v) ·N(u, v),

where ε ∈ R and ε > 0.

2

We now employ the notion of a normal variation to show that a minimal surfaceis a critical point of the area functional. The following argument may be found in[1].

Let X : U ⊆ R2 → R3 be a parametrisation of a regular surface, let D be anopen subset of U and let h : D → R be a continuous function which is differentiableon D. Then for any fixed small t ∈ R the map

X t(u, v) = ϕ(u, v, t) = X(u, v) + t · h(u, v) ·N(u, v)

also parametrises a surface. Calculating the partial derivatives gives us

X tu = Xu + thNu + thuN,

X tv = Xv + thNv + thvN.

Then calculating the first fundamental form for each member of this family of sur-faces X t yields

Et =< X tu, X

tu >

=< Xu, Xu > +2th < Xu, Nu > +2thu < Xu, N >

+ t2h2 < Nu, Nu > +2t2hhu < Nu, N > +t2h2u < N,N >

= E − 2the+ t2(h2 < Nu, Nu > +h2u).

Similarly, we get

F t =F − 2thf + t2(h2 < Nu, Nv > +huhv),

Gt =G− 2thg + t2(h2 < Nv, Nv > +h2v).

We then obtain

EtGt − (F t)2 = EG− F 2 − 2th(Eg − 2Ff +Ge) +R,

where every term of R is a multiple of t2 so that R/t → 0 as t → 0. By using thegeneral formula

H =Eg − 2Ff +Ge

2(EG− F 2)

we then obtain

EtGt − (F t)2 =(EG− F 2)(1− 4thH) +R

=(EG− F 2)(1− 4thH +R

EG− F 2).

Calculating the area for each member X t of this family of surfaces and differentiatingwith respect to the family parameter t gives us

A(t) =

∫D

√EtGt − (F t)2 · dA

=

∫D

√(EG− F 2)(1− 4thH +

R

EG− F 2) · dA.

This shows that

A′(0) =−∫D

2hH√EG− F 2 · dA.

These results provide us with the necessary tools to prove the following:

3

Proposition 1.9. A regular parametrised surface X : U ⊆ R2 → R3 is minimal ifand only if, for every differentiable function h : D → R and every open subset D ofU , the area functional

A(t) =

∫D

√EtGt − (F t)2 · dA

has a critical point at t=0.

Proof. If X is minimal then H = 0 which implies that

A′(0) = −∫D

2hH√EG− F 2dA = 0.

For the converse, let us assume that X is not minimal. Then there exists a pointq ∈ D such that H(q) 6= 0. Choose h such that h(q) = H(q) and h = 0 outside asmall neighbourhood of q. Then, for the area corresponding to this h, we have thatA′(0) < 0 which contradicts our assumption. Hence X must be minimal.

For a conformal regular parametrisation X of a surface, there exists a simplerelationship between the harmonicity of its components and the minimality of X.This will be used later on when looking at the relationship between holomorphicfunctions and minimal surfaces.

Theorem 1.10. Let X : U ⊆ R2 → R3 be a conformal regular parametrisation of asurface. Then H ≡ 0 if and only if the map X is harmonic i.e. Xuu +Xvv = 0.

Proof. By the general formula

H =Eg − 2Ff +Ge

2(EG− F 2)

and the fact that X is conformal, we get that H ≡ 0 if and only if e + g = 0. Bydefinition, this means that

< Xv, Nv >= − < Xu, Nu > .

But by the product rule, we have

< Xv, Nv > + < Xvv, N >=< Xv, N >v= 0,

or equivalently,

< Xv, Nv >= − < Xvv, N > .

Similarly, we have

< Xu, Nu >= − < Xuu, N > .

This immediately implies that H ≡ 0 if and only if

< Xvv, N > + < Xuu, N >= 0,

or equivalently,

< Xuu +Xvv, N >= 0.

4

Then, if Xuu +Xvv = 0 it is clear that H ≡ 0. Next, we assume that

< Xuu +Xvv, N >= 0

and will show that Xuu + Xvv = 0 holds. By using the fact that {Xv, Xu, N} is alocal orthogonal frame for R3 we only need to show that Xuu +Xvv is orthogonal toXu and Xv i.e.

< Xuu +Xvv, Xu >= 0 and < Xuu +Xvv, Xv >= 0.

Calculating the first scalar product yields

< Xuu +Xvv, Xu >=1

2< Xu, Xu >u + < Xu, Xv >v − < Xuv, Xv >

=1

2(< Xu, Xu > − < Xv, Xv >)u+ < Xu, Xv >v .

But since X is conformal this vanishes identically. By the symmetry of u and v, wealso have

< Xuu +Xvv, Xv >= 0.

This proves that X is harmonic.

5

6

Chapter 2

The Weierstrass-EnneperRepresentation

In this chapter we will consider the construction of minimal surfaces using complexanalysis. Our goal is to define the so called Weierstrass-Enneper representation. Todo this we need some theory concerning the function ϕ : U ⊆ R2 → C3 defined by

ϕ(z) = Xu − iXv =f(z)

2((1− g(z)2), i(1 + g(z)2), 2g(z)).

It will be shown that if f : C → C is a holomorphic function and g : C → C is ameromorphic function satisfying certain special conditions, then the map X : C →R3, whose partial derivatives are found in the formula above, will define a minimalsurface. To prove this we need the following two lemmas.

2.1 Preliminary Theory

Lemma 2.1. Let X : U ⊆ R2 → R3 be a conformal parametrisation of a regularsurface and let ϕ : U ⊆ R2 → C3 be the map given by

ϕ = Xu − iXv.

Then X is minimal if and only if each component of ϕ is holomorphic.

Proof. The result of Theorem 1.10 tells us that H ≡ 0 if and only if

Xuu +Xvv = 0

which is equivalent to the following

(Xu)u = (−Xv)v.

This is one of the Cauchy-Riemann equations. The other Cauchy-Riemann equationfollows directly from the regularity of the surface, since Xuv = Xvu is just anotherway of writing

(Xu)v = −(−Xv)u.

7

Lemma 2.2. Let X : U ⊆ R2 → R3 be a conformal parametrisation of a regularminimal surface. Then the holomorphic map ϕ : U ⊆ C→ C3, defined by

ϕ(u+ iv) = (ϕ1, ϕ2, ϕ3) = Xu − iXv,

satisfies the following two conditions

ϕ21 + ϕ2

2 + ϕ23 ≡ 0 and ϕ is never zero.

Moreover, any holomorphic map ϕ satisfying these conditions will give rise to aconformal parametrisation of a minimal surface X by the identity

ϕ(u+ iv) = Xu − iXv.

Proof. Assuming thatX = (X1, X2, X3) is a conformal parametrisation of a minimalsurface, we see that

ϕ21 + ϕ2

2 + ϕ23 =

3∑k=1

((Xku)2 − (Xk

v )2 − 2i ·Xku ·Xk

v )

= |Xu|2 − |Xv|2 − 2i < Xu, Xv > .

This vanishes since X is conformal. Further, ϕ vanishes only if both Xu and Xv arezero, but this can never occur since X is a regular parametrisation.

For the converse, we assume that the map ϕ satisfies the given conditions andwill prove that X is conformal and minimal. Now define

X(u, v) = Re

∫γ

ϕ(ξ)dξ,

where γ is a curve in a simply connected domain U ⊆ C from the starting pointz0 = u0+iv0 to an arbitrary z = u+iv. Note that X is the real part of a holomorphicmap. Then calculating the partial derivatives and using the fact that

∂f

∂u=df

dzand

∂f

∂v= i

df

dz,

for a holomorphic function f , we have

Xu =∂

∂uRe(∫

γ

ϕ(ξ)dξ)

= Re( ddz

∫γ

ϕ(ξ)dξ)

= Reϕ(z),

Xv =∂

∂vRe(∫

γ

ϕ(ξ)dξ)

= Re( ddzi

∫γ

ϕ(ξ)dξ)

= − Imϕ(z).

This implies that the real part of ϕ is Xu and the imaginary part of ϕ is −Xv, hence

ϕ(u+ iv) = Xu(u, v)− iXv(u, v).

Since every component of ϕ is holomorphic, we see from Lemma 2.1 that X is aparametrisation of a minimal surface. To show that X is conformal we use the factthat

ϕ21 + ϕ2

2 + ϕ23 = |Xu|2 − |Xv|2 − 2i < Xu, Xv >= 0.

8

This directly implies that

F =< Xu, Xv >= 0 and E =< Xu, Xu >=< Xv, Xv >= G.

The second condition, namely that ϕ is nowhere zero, implies that Xu(w) and Xv(w)do not simultaneously vanish for any w ∈ U . Together with the identity

< Xu, Xu >=< Xv, Xv >

this implies that neither Xu nor Xv is zero anywhere. Hence X is regular.

Theorem 2.3. Let f : U ⊆ C→ C be a holomorphic function and g : U ⊆ C→ C ameromorphic function such that fg2 is holomorphic. Further, assume that if ξ ∈ Uis a pole of order n of g then ξ is a zero of order 2n of f , and these are the onlyzeroes of f . Then the map

ϕ(z) =f(z)

2((1− g(z)2), i(1 + g(z)2), 2g(z)) (∗)

satisfies the conditions for ϕ of Lemma 2.2. Further, for every such ϕ there exist aholomorphic map f and a meromorphic map g such that (∗) holds.

Proof. Given a map ϕ such that (∗) holds, we have that

ϕ21 + ϕ2

2 + ϕ23 =

1

4f(z)2(1− g(z)2)2 − 1

4f(z)2(1 + g(z)2)2 + f(z)2g(z)2

= −f(z)2g(z)2 + f(z)2g(z)2

= 0.

If ϕ were to equal zero we would have

0 = ϕ(z)

=f(z)

2((1− g(z)2), i(1 + g(z)2), 2g(z)).

The restrictions of the zeroes and poles of f and g, respectively, imply that

f(z)g(z)2 6= 0.

So for a fixed z the first and second coordinates can not both equal zero. Hence, ϕ isnowhere zero. Next, we assume that ϕ is a holomorphic map satisfying ϕ2

1+ϕ22+ϕ2

3 ≡0, ϕ is never zero and let

f(z) = ϕ1(z)− iϕ2(z) and g(z) =ϕ3(z)

ϕ1(z)− iϕ2(z).

Then f is a holomorphic function and g is the quotient of holomorphic functions. Ifthe denominator in g is identically zero, we instead let

g(z) =ϕ3(z)

ϕ1(z) + iϕ2(z),

and proceed with the proof in a similar way. Now the denominator of g is notidentically zero, hence g is meromorphic. Further, the relation

ϕ21 + ϕ2

2 + ϕ23 ≡ 0

9

implies that

(ϕ1 + iϕ2)(ϕ1 − iϕ2) = −ϕ23

which, in terms of f and g, becomes

(ϕ1 + iϕ2) = −fg2.

This last equation, along with the definitions of f and g, show that

ϕ(z) =f(z)

2((1− g(z)2), i(1 + g(z)2), 2g(z)).

This proves the statement of the theorem.

This relationship between the functions f , g and the minimality of X by theequation

ϕ(z) = Xu − iXv =f(z)

2((1− g(z)2), i(1 + g(z)2), 2g(z)).

give us the necessary tools for defining the Weierstrass-Enneper representation ofminimal surfaces.

2.2 Defining the Weierstrass-Enneper Representation

Definition 2.4. Let U be a simply connected open subset of the complex planeC ∼= R2 and γ be a curve contained in U from a fixed point z0 to the arbitraryz = u + iv. Let f be a holomorphic function, not constantly zero, and g be ameromorphic function such that fg2 is holomorphic. Further, assume that at everypole of g of order n f has a zero of order 2n and that f has no other zeroes. Then

X(u, v) = x0 + Re(∫

γ

f(ξ)

2(1− g(ξ)2, i(1 + g(ξ)2), 2g(ξ))dξ

),

where x0 is a point in R3, is a regular parametrisation of a minimal surface. Further-more, for every nonplanar minimal surface there exist such a representation. Thisis called the Weierstrass-Enneper representation of minimal surfaces.

The Weierstrass-Enneper representation allows us to find regular and conformalparametrisations of minimal surfaces by simply integrating holomorphic functions.Another very interesting fact is that the meromorphic function g, occurring in therepresentation formula, has a close relationship with the normal vector field N ofthe surface. In fact, as shown in the following theorem, the normal vector field N isequal to the inverse of the stereographic projection of g.

Theorem 2.5. Let C be the extended complex plane and σ−1 : C→ S2 ⊂ R3 be theinverse of the stereographic projection from the north pole given by

σ−1(z) =1

1 + |z|2(z + z,−i(z − z), |z|2 − 1).

Then one of the unit normal vector fields N is the inverse of the stereographicprojection of the function g in the Weierstrass-Enneper representation of a minimalsurface i.e.

N = σ−1(g) =1

1 + |g|2(g + g,−i(g − g), |g|2 − 1).

10

Proof. It follows directly from the proof of Lemma 2.2 that

Xu = Reϕ and Xv = − Imϕ.

We now use this fact and calculate the cross product of these partial derivatives

Xu ×Xv = Reϕ× (− Imϕ)

= −(Reϕ2 Imϕ3 − Reϕ3 Imϕ2,Reϕ3 Imϕ1 − Reϕ1 Imϕ3,

Reϕ1 Imϕ2 − Reϕ2 Imϕ1)

= −(Imϕ2ϕ3, Imϕ3ϕ1, Imϕ1ϕ2).

In f and g the first coordinate becomes

− Imϕ2ϕ3 = − Im (if

2(1 + g2)fg)

= − Im (i|f |2

2(g + |g|2g))

= −|f |2

2Re (g + |g|2g)

= −|f |2

2(1 + |g|2) Re g.

A similar calculation for the other coordinates gives us the following

Xu ×Xv = −|f |2

4(1 + |g|2)(2 Re g, 2 Im g, |g|2 − 1)

= −|f |2

4(1 + |g|2)2 1

1 + |g|2(g + g,−i(g − g), |g|2 − 1)

= −|f |2

4(1 + |g|2)2σ−1(g).

Here we see that N is a scalar multiple of σ−1(g), but since N is of unit length thisscalar has to be −1. Then

N = − Xu ×Xv

|Xu ×Xv|

=1

1 + |g|2(g + g,−i(g − g), |g|2 − 1).

This completes the proof.

11

12

Chapter 3

Examples and Applications

We will now look at how the Weierstrass-Enneper representation can be used tofind minimal surfaces by simply integrating complex functions. We will look at fourclassical minimal surfaces that were the first ones to be discovered beside the plane,see [1] and [2]. We will introduce their Weierstrass-Enneper representations andprove that this representation does in fact yield a well-known parametrisation of thecorresponding surfaces.

3.1 The Catenoid and the Helicoid

A historically important example of a minimal surface is the catenoid. This surfacecan be represented by the globally defined holomorphic functions f, g : C→ C with

f(z) = e−z and g(z) = ez.

Substituting these into the Weierstrass-Enneper representation formula and inte-grating from z0 = (0, 0) to the general point z = u+ iv we obtain

X(u, v) = x0 + Re(∫ z

0

f(ξ)

2(1− g(ξ)2, i(1 + g(ξ)2), 2g(ξ))dξ

)= x0 + Re

(∫ z

0

e−ξ

2(1− e2ξ, i(1 + e2ξ), 2eξ)dξ

)= x0 + Re

(∫ z

0

1

2(e−ξ − eξ, i(e−ξ + eξ), 1)dξ

)= Re

(1

2(−e−z − ez),− 1

2i(−e−z + ez), z

)= Re

(− cosh z, i sinh z, z

)= (− coshu cos v,− coshu sin v, u).

Apart from the plane, the catenoid is the only minimal surface of revolution, see[1], and locally there exists an isometry between the catenoid and the helicoid. More-over, the catenoid can be continuously deformed to a part of the helicoid througha family of minimal surfaces. This can be explained using the interesting notion ofan associate family.

13

Definition 3.1. Let X : U ⊆ R2 → R3 be a parametrised minimal surface, repre-sented by the functions f and g. Then the family of surfaces defined by

Xθ = Re(eiθ∫γ

f(ξ)

2(1− g(ξ)2, i(1 + g(ξ)2), 2g(ξ))dξ

)is called the associate family of X. Of particular interest is the surface parametrisedby Xπ/2 which is called the conjugate of X.

Notice that the surfaces of the same associate family can be represented by thesame function g, but would then differ by the function f with a rotation about theorigin. Then fθ = eiθf is still holomorphic and not constantly zero, hence Xθ is aminimal surface for every θ. The helicoid is the conjugate of the catenoid and cantherefore be represented by the functions

f(z) = ie−z and g(z) = ez.

3.2 Reflection Principles

With the notion of an associate family defined, we would like to introduce thereader to two reflection principles that apply to minimal surfaces. The proofs relyon Schwarz’s reflection principle for harmonic functions and may be found in [6].We will state the theorems and discuss an interesting fact about the different typesof reflections and how they relate to a surface and its conjugate.

Theorem 3.2. Let U be a subset of R2 which is symmetric with respect to the realaxis and define

I = {(x, y) ∈ U | y = 0},U+ = {(x, y) ∈ U | y > 0},U− = {(x, y) ∈ U | y < 0}.

Let X : U+ ∪ I → R3 be a conformal and regular parametrisation of a minimalsurface that is twice differentiable on U+ and continuous on U+ ∪ I. Further,assume that X maps I to a straight line L in R3. Then the minimal surface X canbe extended to be defined on the whole of U by a reflection about the line L. Thismeans that

X(u, v) = (X(u,−v))∗

for (u, v) ∈ U− where A∗ denotes the reflection image of A about the line L.

Theorem 3.3. Let U , U+, U−, I and X be defined as in Theorem 3.2 but assumethat X maps I to a curve contained in a plane P in R3 such that P is perpendicularto X along X(I). Then the minimal surface X can be extended to be defined on thewhole of U by a reflection about the plane P . This means that

X(u, v) = (X(u,−v))∗

for (u, v) ∈ U−, where A∗ denotes the mirror image of A about the plane P .

14

Figure 3.1: The deformation of the catenoid into a part of the helicoid. The first picture is of thecatenoid, the only minimal surface of revolution apart from the plane. The last image is of thehelicoid which is the only ruled minimal surface apart from the plane, see [1].

15

The symmetries of minimal surfaces are summed up well in the following state-ment of H. Karcher, see [8]:

“If a planar geodesic resp. a straight line lies on a complete minimal sur-face, then reflection in the plane of the planar geodesic resp.180◦-rotation around the straight line is a congruence of the minimal sur-face.“

There is also an intimate relationship between planar geodesics on a surface andstraight lines on its conjugate. Looking at the deformation of the catenoid into thehelicoid, we can see that a meridian in the catenoid, which is a planar geodesic,turns into a straight line on the helicoid. This is in fact true in general. A planargeodesic of X will be deformed into a straight line contained in Xπ/2 and vice versa,see [3].

3.3 The Enneper Surface

Another interesting example of a minimal surface is Enneper’s surface. Its Weierstrass-Enneper representation is obtained by choosing

(f(w), g(w)) = (1, w).

This means that the conformal parametrisation X becomes

X(z) = Re(1

2

∫ z

0

((1− ξ2), i(1 + ξ2), 2ξ)dξ)

=1

2Re(z − z3

3, iz +

iz3

3, z2)

=1

2(u− u3

3+ uv2,−v +

v2

3− u2v, u2 − v2)

for z = u + iv. The Enneper surface is both complete and minimal but containsself intersections. Enneper’s surface is conjugate to itself i.e. Enneper’s surface andits conjugate are the same geometric objects. For a proof of this fact we refer thereader to [6].

16

Figure 3.2: A plot of Enneper’s surface, a self-intersecting minimal surface.

3.4 Scherk’s Doubly Periodic Surface

The famous doubly periodic Scherk surface, also called Scherk’s first surface, pro-vides us with another example of a historically important minimal surface. We willdiscuss the Weierstrass-Enneper representation of the surface and the existence ofvertical lines in the surface. A more detailed discussion of the surface can be foundin [6]. Scherk’s doubly periodic surface is defined by the equation

ez =cos y

cosx.

We will now prove that its Weierstrass-Enneper representation is given by the func-tions f : C \ {±1,±i} → C and g : C→ C where

(f(w), g(w)) =( 2

1− w4, w).

We first note that

f(1− g2) =2

1 + w2=

i

w + i− i

w − i,

i · f(1 + g2) =2i

1− w2=

i

w + 1− i

w − 1,

2fg =4w

1− w4=

2w

w2 + 1− 2w

w2 − 1.

17

Figure 3.3: A plot of a fundamental piece of Scherk’s first surface. The surface continues in acheckerboard fashion as it is periodic in both the x and y directions.

18

Integrating this gives us

X(w) = Re(i log

w + i

w − i, i log

w + 1

w − 1, log

w2 + 1

w2 − 1

),

which simplifies to

X(w) =(− arg

w + i

w − i,− arg

w + 1

w − 1, log

∣∣∣w2 + 1

w2 − 1

∣∣∣)when using the principal value of the logarithm function. Now, by using the followingidentities,

w + i

w − i=|w|2 − 1

|w − i|2+ i

w + w

|w − i|2,

w + 1

w − 1=|w|2 − 1

|w − 1|2+ i

w − w|w − 1|2

,

we find expressions for cosx and cos y. We have that

cosx = cos(− argw + i

w − i)

= cos(argw + i

w − i)

= cos(arg( |w − i||w + i|

· w + i

w − i

))

= Re( |w − i||w + i|

· w + i

w − i

)=|w − i||w + i|

Re(w + i

w − i

)=|w − i||w + i|

· |w|2 − i

|w − i|2

=|w|2 − 1

|w2 + 1|.

Similarly,

cos y = cos(− argw + 1

w − 1)

=|w − 1||w + 1|

· |z|2 − i

|w − 1|2

=|w|2 − 1

|w2 − 1|.

This implies that

cos y

cosx=

w2 + 1

|w2 − 1|= ez.

Hence, the maps f and g as above define Scherk’s doubly periodic surface.

19

Further, to show that Scherk’s doubly periodic surface contains straight linesparallel to the z-axis, we consider the coordinate functions (x(w), y(w), z(w)) underthe restriction w = eiθ for some θ ∈ (0, π

2). Then

x(eiθ) = − arg(eiθ + i

eiθ − i

)= − arg

(ieiθ + e−iθ

|eiθ − i|2)

= −π2.

Similarly, we have

y(eiθ) = − arg(eiθ + 1

eiθ − 1

)= − arg

(ie−iθ − eiθ

|eiθ − 1|2)

= −3π

2.

Hence, the x and y coordinates are constant while

z(eiθ) = log∣∣∣w2 + 1

w2 − 1

∣∣∣tends to ±∞ as θ tends to 0 and π

2, respectively. This means that X maps the

arc {eiθ ∈ C|0 < θ < π2} onto the straight line {(−π

2,−3π

2, s) ∈ R3|s ∈ R}. Hence,

Scherk’s doubly periodic surface contains straight lines, which is what we wanted toprove. Since the surface is doubly periodic it contains an infinite amount of lines ofsymmetry like this one.

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Chapter 4

Summing Up

We have discussed the theory needed to state the Weierstrass-Enneper representa-tion of minimal surfaces and we have defined the representation itself. We haveshown that there exists a simple relationship between the function g found in therepresentation and the normal vector field N of the minimal surface. We have thenlooked at four examples of minimal surfaces and defined the associate family of aminimal surface. We have also mentioned two reflection principles and how theyrelate to a minimal surface and its conjugate. There are numerous more examplesof interesting minimal surfaces with known Weierstrass-Enneper representations andwe have also seen only a fraction of how the representation can be used to analyse aminimal surface. The representation has been used to prove more deep results aboutminimal surfaces. For example, to illustrate the usefulness of the representation, thefollowing two theorems may be proven using the relationship between the function gand the vector field N . The theorems make no mention of the Weierstrass-Enneperrepresentation although the proofs rely on it.

Theorem 4.1. [6] The normal vector field N of a minimal surface X : C → R3

omits at most two points of the unit sphere unless X(C) is contained in a plane.

Theorem 4.2. [7] If X is a complete minimal surface in R3, then either X is aplane or the normals to X are everywhere dense in the sphere.

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22

Bibliography

[1] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall (1976).

[2] A. Pressley, Elementary Differential Geometry, Second Edition, Springer UndergraduateMathematics Series, Springer (2010).

[3] M. Weber, Classical Minimal Surfaces in Euclidean Space by Examples (2001).http://www.indiana.edu/ minimal/research/claynotes.pdf

[4] S. Gudmundsson, An Introduction to Gaussian Geometry, Lecture Notes in Mathematics,University of Lund (2017).www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf

[5] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish (1979).

[6] U. Dierkes, S. Hildebrandt, A. Kuster, O. Wohlrab Minimal Surfaces I, Boundary ValueProblems, Springer (1992).

[7] R. Osserman, A Survey of Minimal Surfaces, Dover (2002).

[8] H. Karcher, Construction of Minimal Surfaces, Tokyo (1989).http://www.math.uni-bonn.de/people/karcher/karcherTokyo.pdf

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Bachelor’s Theses in Mathematical Sciences 2017:K15ISSN 1654-6229

LUNFMA-4062-2017Mathematics

Centre for Mathematical SciencesLund University

Box 118, SE-221 00 Lund, Swedenhttp://www.maths.lth.se/