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Thesis

Backlund transformations for minimal surfaces

Per Back

LiTH-MAT-EX--2015/04--SE

Backlund transformations for minimal surfaces

Department of Mathematics, Linkoping University

Per Back

LiTH-MAT-EX--2015/04--SE

Master’s thesis: 30 hp

Level: A

Supervisor: Jens Hoppe,Department of Mathematics, Royal Institute of Technology

Examiner: Joakim Arnlind,Department of Mathematics, Linkoping University

Stockholm: September 2015

Abstract

In this thesis, we study a Backlund transformation for minimal surfaces – sur-faces with vanishing mean curvature – transforming a given minimal surfaceinto a possible infinity of new ones.

The transformation, also carrying with it mappings between solutions tothe elliptic Liouville equation, is first derived by using geometrical concepts,and then by using algebraic methods alone – the latter we have not been ableto find elsewhere. We end by exploiting the transformation in an example,transforming the catenoid into a family of new minimal surfaces.

Keywords: Backlund transformations, Liouville equation, minimal surfaces,Ribaucour transformations, Thybaut transformations.

Back, 2015. v

Acknowledgements

I would like to thank my supervisor Jens Hoppe for having me as a student,invaluable comments on the manuscript, inspirational talks, and peculiar ad-ventures. I would also like to thank Joakim Arnlind for taking on the role asan examiner without blinking, and for great help and discussions regarding thethesis. My greatest gratitude also goes to my opponent Daria Burdakova forcomments on the manuscript, Eric Wolter for input on the Matlab script andall the talks, Aleksandr Zheltukhin for a long and fruitful discussion on the sub-ject, Hans Lundmark for pointing out a mistake in the formulation of one of thetheorems, and the Department of Mathematics at KTH for hosting me. Lastbut not least, I would like to thank friends and family for always being there –especially Malin, my girlfriend.

Back, 2015. vii

Nomenclature

Most of the recurring abbreviations and symbols are described here.

Symbols

× Vector product〈·, ·〉 Inner productCn Space of n times continuously differentiable functionsCn Complex n-space∂i Partial differentiation with respect to ui

δjk Kronecker delta∆ Flat Laplacian in Rn

det Determinant(gij) Matrix with elements gijΩ Open subset of R2

Rn Euclidean n-spacetr Trace, i.e. the sum of the diagonal entries of a matrix‖X‖ Norm of X induced by the inner productX Vector in an (or map between) inner product space(s)Xi i:th component of X as a vectorXui Partial derivative of X with respect to ui

Abbreviations

iff If and only ifODE Ordinary differential equationON OrthonormalPDE Partial differential equation

Back, 2015. ix

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Topics covered . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Differential geometry of surfaces 32.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The Gauss and Weingarten maps . . . . . . . . . . . . . . . . . . 42.3 Fundamental forms . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The Gauss and Weingarten equations . . . . . . . . . . . . . . . 82.6 Curves on the surface . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Infinitesimal deformations . . . . . . . . . . . . . . . . . . . . . . 102.8 Line congruences and focal surfaces . . . . . . . . . . . . . . . . . 11

3 Minimal surfaces 133.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Isothermal representation . . . . . . . . . . . . . . . . . . . . . . 133.3 Asymptotic line representation . . . . . . . . . . . . . . . . . . . 153.4 Adjoint surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 A Backlund transformation for minimal surfaces 194.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Geometric construction . . . . . . . . . . . . . . . . . . . . . . . 204.3 Algebraic proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Example: catenoid . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A A Matlab script 39A.1 catTrans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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xii Contents

Chapter 1

Introduction

In this first chapter, we give some background and formulate the objective ofthe thesis, and describe what topics are covered.

1.1 Background

Minimal surfaces – surfaces that locally minimize their area – have been studiedin different areas of mathematics for more than 250 years. Although much isknown about them, still new ways of constructing them are being discovered.

In 1908, the American mathematician Luther Pfahler Eisenhart published apaper [5] in which he described geometrically how to construct a transformationfor minimal surfaces in three-dimensional Euclidean space, transforming a givenminimal surface into a family of new minimal surfaces. The transformation,which is a so-called Backlund transformation, also carries with it mappingsbetween solutions of the elliptic Liouville equation

θuu + θvv = e−2θ.

Both the result and the mathematical concepts used in the paper seem to havebeen forgotten, mostly. The purpose of this thesis is to revise some of thoseconcepts and rederive the transformation geometrically as done by Eisenhart in1908, but in a more contemporary mathematical language (for an alternativeformulation see e.g. [4]). We will also give an algebraic proof for the transforma-tion which we have not found elsewhere, and an example using it, transformingthe catenoid into a family of new minimal surfaces.

1.2 Topics covered

There are three chapters (apart from this introduction) and one appendix. Themain topics dealt with are:

Chapter 2: Introduction to general surfaces in R3.

Chapter 3: Definition of what a minimal surface is and derivations of someimportant results related to them.

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2 Chapter 1. Introduction

Chapter 4: Derivation of a Backlund transformation for minimal surfaces interms of geometrical concepts and an algebraic proof for it. We also exploitthe transformation in an example, transforming the catenoid into a familyof new minimal surfaces.

Appendix A: A Matlab script for generating minimal surfaces via the afore-mentioned transformation.

Chapter 2

Differential geometry ofsurfaces

In this chapter, we will study surfaces and the differential geometry in whichthey are described in three-dimensional Euclidean space.

2.1 Preliminaries

Let us start by defining what we mean by a surface.

Definition 2.1.1 (Parametric surface). A parametric surface is taken to be animmersion X : Ω→ R3 where Ω is an open set of R2, i.e. X is a differentiablevector valued function whose derivative is everywhere injective. A point p inΩ is written (u, v), where u and v are called the parameters. If not otherwisestated, X is always assumed to be of class C3 (Ω).

Thus, a parametric surface is a kind of representation X of the surface in R3,and we will exclusively refer to this representation when speaking of surfaces.We write the components of X as

X(u, v) =

X1(u, v)X2(u, v)X3(u, v)

,

and the partial derivatives as

Xui :=∂X

∂ui:=

∂X1

∂ui∂X2

∂ui∂X3

∂ui

, i = 1, 2,

with(u1, u2

)=: (u, v). Demanding that X be an immersion is then equivalent

to having Xu(p) and Xv(p) being linearly independent at all points p ∈ Ω [8],and they therefore span the tangent plane TpX at all points. Moreover, thevector product Xu ×Xv does not vanish, so the normal vector field

N =Xu ×Xv

‖Xu ×Xv‖

Back, 2015. 3

4 Chapter 2. Differential geometry of surfaces

is well defined, ‖ · ‖ denoting the usual norm on R3. Since ‖N‖ = 1, we canview this as a map from Ω to the unit sphere

S2 =

(x, y, z) ∈ R3 : x2 + y2 + z2 = 1.

This map bears the name after the mathematician who first employed it, namelyGauss.

2.2 The Gauss and Weingarten maps

We start as we did in the last section, by a definition.

Definition 2.2.1 (Gauss map). For a surface X : Ω→ R3, the Gauss map

N : Ω→ S2 ⊂ R3

is defined as

N :=Xu ×Xv

‖Xu ×Xv‖,

and the set N (Ω) is called the spherical image of the surface X.

The new notion lies in that we no longer think of the unit normal vectorat a point p ∈ Ω as being attached to the image point X(p), but have insteadmoved it in terms of a translation to the origin of space as seen in Figure 2.1.

Figure 2.1: The normal vector field moved to the origin of space, the imagepoints of the Gauss map being at the tip of the arrowheads.

Definition 2.2.2 (Self-adjoint map). Let (T, 〈·, ·〉) be a finite dimensional realor complex inner product space. A linear map A : T → T is called self-adjointif 〈AV,W 〉 = 〈V,AW 〉 for all vectors V,W ∈ T .

Self-adjoint maps can, as we shall in the next theorem, be used as tools forconstructing bases of inner product spaces.

Theorem 2.2.1 (Existence of ON-basis). If A : T → T is a self-adjoint mapon a real or complex two-dimensional inner product space T , then there existsan orthonormal basis E1, E2 of T consisting of eigenvectors of A. Moreover,

2.2. The Gauss and Weingarten maps 5

the matrix of A in the eigenbasis is diagonal and consists of the corresponding,necessarily real, eigenvalues λ1 and λ2 ≥ λ1, which are given by

λ1 = min

〈AV, V 〉〈V, V 〉

: V ∈ T, V 6= 0

,

λ2 = max

〈AV, V 〉〈V, V 〉

: V ∈ T, V 6= 0

.

Proof. See [2, p. 216].

We recall from the previous section that at every point p ∈ Ω, the tangentvectors Xu(p) and Xv(p) to X : Ω → R3 provide a basis of the tangent planeTpX. Hence, any vector V ∈ TpX can be written as

V = V 1Xu(p) + V 2Xv(p) =∑i

V iXui(p) =: V iXui(p),

where we in the last step have deployed the Einstein summation convention,implying summation over repeated indices. We will continue to use this conven-tion throughout the rest of the thesis, so whenever repeated indices occur in thesame terms (typically as a mix of upper and lower indices as above), summationis implied over those indices.

Definition 2.2.3 (Weingarten map). For a surface X : Ω → R3 with unitnormal vector field

N :=Xu ×Xv

‖Xu ×Xv‖

we define the Weingarten map S : TpX → TpX at a point p ∈ Ω for arbitraryvectors V ∈ TpX written as

V = V iXui(p) via S(p)V := −V iNui(p).

Using the inner product 〈·, ·〉 inherited by the ambient space R3, 〈N,N〉 = 1by definition. Differentiating this then yields 〈N,Nui〉 = 0, so either Nui(p) liein TpX, or in a plane parallel to it which can be identified with TpX. Moreover,since the Weingarten map clearly is linear, using the property 〈N,Xuj 〉 = 0, wecan deduce that the Weingarten map is self-adjoint in TpX. First, by differen-tiation we have

〈Nui , Xuj 〉+ 〈N,Xuiuj 〉 = 0 ⇐⇒ 〈Nui , Xuj 〉 = −〈N,Xuiuj 〉 = −〈N,Xujui〉= 〈Nuj , Xui〉 = 〈Xui , Nuj 〉,

so for arbitrary V,W ∈ TpX written in the basis Xu, Xv as V = V iXui andW = W jXuj

〈SV,W 〉 = −V iW j〈Nui , Xuj 〉 = −V iW j〈Xui , Nuj 〉 = −〈V iXui ,WjNuj 〉

= 〈V, SW 〉.

Here, we have dropped the p for the sake of brevity.


2.3 Fundamental forms

We can define three symmetric bilinear forms (i.e. forms that are symmetricand linear in both arguments) for arbitrary V and W in TpX as

I(V,W ) := 〈V,W 〉, II(V,W ) := 〈SV,W 〉, III(V,W ) := 〈SV, SW 〉.

These can in turn be used for defining three quadratic forms called the first,second and third fundamental form:

I(V ) := 〈V, V 〉, II(V ) := 〈SV, V 〉, III(V ) := 〈SV, SV 〉.

The first fundamental form is sometimes also called the metric.

2.4 Curvature

The quotient

κn(V ) =II(V )

I(V )

is called the normal curvature, and the minimum and maximum of this aredefined as the principal curvatures κ1 and κ2 of the surface,

κ1 := min

II(V )

I(V ): V ∈ TpX,V 6= 0

,

κ2 := max

II(V )

I(V ): V ∈ TpX,V 6= 0

.

Hence, by Theorem 2.2.1, κ1 and κ2 are by definition the eigenvalues of theWeingarten map, and the directions of the corresponding eigenvectors E1 andE2 are therefore referred to as the principal directions. By the same theorem,E1, E2 constitutes an ON-basis of TpX, and in this basis the matrix repre-senting S is

S =

(κ1 00 κ2

).

Changing S to this basis can always be done by a similarity transformationS 7→ E−1SE where the columns of E are the orthonormal eigenvectors E1

and E2 of S, and E−1 its inverse. Both the determinant and the trace aresimilarity-invariant, and in differential geometry, those of the Weingarten mapplay a particularly important role.

Definition 2.4.1 (Gauss and mean curvature). The functions

K := detS = κ1κ2,

H :=trS

2=κ1 + κ2

2

are called the Gauss curvature and the mean curvature respectively.

For computational reasons however, it is often convenient to work in thebasis Xu, Xv. As before, we write arbitrary V ∈ TpX as V = V iXui in thisbasis, and the first and second fundamental form are therefore

I(V ) = 〈V, V 〉 = V iV j〈Xui , Xuj 〉,II(V ) = 〈SV, V 〉 = −V iV j〈Xui , Nuj 〉.

2.4. Curvature 7

On account of this,

gij := 〈Xui , Xuj 〉,hij := −〈Xui , Nuj 〉,

are, for each i, j ∈ 1, 2 called the coefficients of the first and second funda-mental form. They are also denoted by the letters

E := 〈Xu, Xu〉, F := 〈Xu, Xv〉 = 〈Xv, Xu〉, G := 〈Xv, Xv〉,L := −〈Xu, Nu〉 = 〈Xuu, N〉, N := −〈Xv, Nv〉 = 〈Xvv, N〉,M := −〈Xu, Nv〉 = −〈Xv, Nu〉 = 〈Xuv, N〉 = 〈Xvu, N〉,

so in matrix form

(gij) =

(g11 g12g21 g22

)=

(E FF G

), (hij) =

(h11 h12h21 h22

)=

(L MM N

).

If we denote the inverse of (gij) by (gij), so that

(gij) =

(g11 g12

g21 g22

)=

1

EG − F2

(G −F−F E

),

the matrix of the Weingarten map in the basis Xu, Xv is equal to [7]

S =(Sij)

=(gikhkj

)=

(g11h11 + g12h21 g11h12 + g12h22g21h11 + g22h21 g21h12 + g22h22

).

Hence, we can calculate the Gauss and mean curvature as

K = detS = det(gikhkj

)=

det(hij)

det(gij)=LN −M2

EG − F2,

H =trS

2=gijhij

2=LG +NE − 2MF

2 (EG − F2).

Example 2.4.1 (2-sphere). We shall compute the Gauss and mean curvatureof the 2-sphere of radius r,

X =(X1, X2, X3

)∈ R3 : ‖X‖ = r.

We parametrize it by

X(u, v) =

X1(u, v)X2(u, v)X3(u)

= r

sinu cos vsinu sin v

cosu

,

Xu(u, v) = r

cosu cos vcosu sin v− sinu

, Xv(u, v) = r

− sinu sin vsinu cos v

0

,

N(u, v) =Xu ×Xv

‖Xu ×Xv‖(u, v) =

sinu cos vsinu sin v

cosu

,

Nu(u, v) =

cosu cos vcosu sin v− sinu

, Nv(u, v) =

− sinu sin vsinu cos v

0

,


for u ∈ (0, π) and v ∈ (0, 2π).

gij = 〈Xui , Xuj 〉 =⇒ (gij) = r2(

1 00 sin2 u

),

hij = −〈Xui , Nuj 〉 =⇒ (hij) = r

(1 00 sin2 u

)=

1

r(gij) .

Hence,

S =(gikhkj

)=

1

r

(gikgkj

)=

1

rI,

where I is the identity matrix, so

K = detS = det

(1

rI

)=

1

r2,

H =trS

2=

tr I

2r=

1

r.

2.5 The Gauss and Weingarten equations

When we defined the Weingarten map in Definition 2.2.3, we saw that Nui(p) ∈TpX. We also recall from Section 2.1 that demanding that X be an immersionis equivalent to having the tangent vectors Xu(p) and Xv(p) being linearlyindependent at all points p, and that they therefore span all tangent planesTpX. It is therefore natural to seek the expression for Nui(p) in terms of Xu(p)and Xv(p), as we shall now do. We start by setting

Nui = a ki Xuk ,

for coefficients a ki . By taking the inner product with Xuj , we get⟨Nui , Xuj

⟩= a ki

⟨Xuk , Xuj

⟩⇐⇒ −hij = a ki gkj ,

and by multiplying by the inverse and summing over j as well,

−hijgjl = a ki gkjgjl = a ki δ

lk = a li ,

where δ lk is the Kronecker delta. Substituting this in our first expression, we

have found the Weingarten equations

Nui = −hijgjkXuk .

Apart from Xu(p) and Xv(p), at each point we also have access to N(p) which

spans the orthogonal complement (TpX)⊥

. Hence, at each point we have at ourdisposal a basis of R3. We shall try to express the second order derivatives ofX in this basis, thus putting

Xuiuj = ΓkijXuk + bijN (2.1)

for coefficients Γkij and bij . The coefficients Γkij are called Christoffel symbolsof the second kind, while

Γijk := gilΓljk

2.6. Curves on the surface 9

are called Christoffel symbols of the first kind. By taking the inner product withXul in (2.1) we get ⟨

Xuiuj , Xul⟩

= Γkijgkl = Γlij ,

and hence we have the symmetry relations

Γlij = Γlji, Γljk = Γlkj .

Introducing the shorthand notation

∂k :=∂

∂uk,

by differentiation

∂kgij = ∂k⟨Xui , Xuj

⟩=⟨Xuiuk , Xuj

⟩+⟨Xui , Xujuk

⟩= Γjik + Γijk,

so using the symmetry relation of the Christoffel symbols of the first kind,

−∂kgij + ∂jgik + ∂igkj = −Γjik − Γijk + Γkij + Γikj + Γjki + Γkji

= 2Γkij = 2gklΓlij .

Multiplying by gmk and dividing by two, we get

1

2gmk (−∂kgij + ∂jgik + ∂igkj) = gmkgklΓ

lij = δmlΓ

lij = Γmij .

At last, taking the inner product with N in (2.1), we see that

bij =⟨Xuiuj , N

⟩= hij ,

so that

Xuiuj = ΓkijXuk + hijN,

which are called the Gauss equations.

2.6 Curves on the surface

In this section we will present some definitions and propositions concerningdifferent curves – although named lines – that may exist on a surface.

Definition 2.6.1 (Parametric lines). The curves on a surface X along thedirection of Xu and Xv respectively are called the parametric lines.

The parametric lines are therefore the curves we get on a surface X byholding u and v constant one at a time. We also say that curves on two differentsurfaces parametrized by the same u and v correspond if they correspond tothe same curves in the uv-plane. For instance, this is always the case for theparametric lines on two different surfaces that are parametrized by the same uand v.

Definition 2.6.2 (Curvature lines). The curves on a surface in the principaldirections are called the curvature lines.


We recall from Section 2.4 that the principal directions were the directionsalong the eigenvectors of the Weingarten map, and that the corresponding eigen-values were known as the principal curvatures, hence the name curvature lines.

Definition 2.6.3 (Asymptotic lines). The directions of V ∈ TpX for which thenormal curvature κn(V ) = 0 are called the asymptotic directions, and curveson the surface which are tangent to these directions at every point are calledasymptotic lines.

In the following two propositions, we shall see what the necessary and suffi-cient conditions are for the parametric lines to be the curvature or asymptoticlines.

Proposition 2.6.1 (Asymptotic parametric lines). The asymptotic lines areparametric if and only if the fundamental coefficients L = N = 0.

Proof. See [6].

Proposition 2.6.2 (Parametric curvature lines). The lines of curvature areparametric if and only if the fundamental coefficients F =M = 0.

Proof. See [6].

2.7 Infinitesimal deformations

Starting with a parametrized surface X(u, v), we can obtain a new surfaceX ′′(u, v) by deforming the former in the direction of a vector X ′(u, v) by setting

X ′′ := X + εX ′, ε ∈ R.

The tangent lines to X ′ are called the generatrices of the deformation, and X ′

itself does also correspond to a parametrized surface. Since

X ′′ui = Xui + εX ′ui , ε ∈ R, (2.2)

by taking the inner product with X ′′uj we obtain

g′′ij = gij + ε (〈Xui , X′uj 〉+ 〈Xuj , X

′ui〉) + ε2g′ij ,

where gij , g′ij and g′′ij are the corresponding coefficients of the first fundamental

forms. If

〈Xui , X′uj 〉+ 〈Xuj , X

′ui〉 = 0 (2.3)

and ε be taken so small that ε2 may be neglected, X and X ′′ are seen to be iso-metric. X ′′ is then said to be obtained from X by an infinitesimal deformation.The problem of making an infinitesimal deformation to X is then equivalent todetermining X ′ by means of (2.3), which in turn is equivalent to

〈Xu, X′u〉 = 0, 〈Xv, X

′v〉 = 0, 〈Xu, X

′v〉 = −〈Xv, X

′u〉 =: w

√EG − F2.

Here, we have in accordance with Eisenhart [6, p. 374] defined the characteristicfunction w(u, v) of the infinitesimal deformation. As usual, E ,F and G are the

2.8. Line congruences and focal surfaces 11

coefficients of the first fundamental form of X. From this, one can also deduce[6, pp. 375–376] the following expressions:

∂v

(Lwv −Mwu

K√EG − F2

)+ ∂u

(Nwu −Mwv

K√EG − F2

)=

2FM− EN − GL√EG − F2

w (2.4)

X ′u =

L (wNv −Nwv)−M (wNu −Nwu)

K√EG − F2

X ′v =M (wNv −Nwv)−N (wNu −Nwu)

K√EG − F2

(2.5)

The former equation is called the characteristic equation, and once we haveobtained the characteristic function w from it, we can also get X ′ by means ofintegrating the latter equations. Here, K denotes the Gaussian curvature andL,M and N the coefficients of the second fundamental form of X.

2.8 Line congruences and focal surfaces

Before studying infinitesimal deformations, we saw that one can define manydifferent types of useful curves on a surface. In this section, we shall studyconnections between such curves and infinitesimal deformations, and also seehow one can define surfaces in terms of lines and lines in terms of surfaces.

Definition 2.8.1 (Line congruence). A two parameter family of straight linesin space is called a line congruence.

Let Ω ⊂ R2 with (u, v) ∈ Ω, X : Ω → R3 and R : Ω → S2 where S2 is theunit sphere in R3, i.e. S2 =

(x, y, z) ∈ R3 : x2 + y2 + z2 = 1

. Then, we can

define a line congruence as

C(u, v, t) = X(u, v) + tR(u, v), t ∈ R,

so that for each pair of fixed parameters (u, v) we get a member of the linecongruence, that is, a straight line. The surface X is called a reference surfaceto C, and it is seen to not be unique; with X+sR for some s ∈ R as a referencesurface instead, we would describe the very same C.

Example 2.8.1 (Normal congruence). The normal lines to a surface constitutea line congruence and is called a normal congruence. If we take the normal fieldN : Ω→ S2 to the surface X : Ω→ R3, we can describe it as

CN (u, v, t) = X(u, v) + tN(u, v), t ∈ R.

The lines belonging to the line congruence were obtained by keeping u andv fixed while varying t. If we instead keep t fixed and vary u and v, we willdescribe a surface. Two such surfaces are described in the next definition.

Definition 2.8.2 (Focal surfaces). For a line congruence

C(u, v, t) = X(u, v) + tR(u, v),

the two surfaces

Fi(u, v) := C(u, v, t = ti) = X(u, v) + tiR(u, v), i = 1, 2,

with common tangent lines belonging to the line congruence are, when theyexist, called focal surfaces.


Hence, when two focal surfaces exist, a line congruence define a map F1 7→ F2

and its inverse F2 7→ F1 between them.

Definition 2.8.3 (W-congruence). A line congruence for which the asymptoticlines on the focal surfaces correspond is called a Weingarten congruence, or justW-congruence.

A W-congruence therefore defines a map mapping asymptotic lines on onefocal surface to asymptotic lines on the other focal surface, and the other wayaround. A way to construct such a map is provided by the next theorem.

Proposition 2.8.1 (Construction of W-congruences). The tangent lines to asurface which are perpendicular to the generatrices of an infinitesimal deforma-tion of the surface constitute a W-congruence. The original, undeformed surfaceis one of the focal surfaces to the W-congruence, and the normal lines to theother focal surface are parallel to these generatrices.

Proof. See [6, p. 420].

This result is rather technical, but will hopefully become clearer within thelast chapter where we make use of it.

Chapter 3

Minimal surfaces

The theory of minimal surfaces originates with Lagrange, who in 1760 formu-lated the problem of what surfaces have the smallest area given a boundary. Bymeans of variational methods, he succeeded with a non-parametric description,and sixteen years later Meusnier proved that they are surfaces with vanishingmean curvature. Later on, different but equivalent definitions have been madein a variety of different areas of mathematics, demonstrating the diversity ofthe subject. In this thesis, we will stick to the definition of being surfaces withvanishing mean curvature, and in this chapter we will study such surfaces andconcepts related to them.

3.1 Preliminaries

We start directly by restating the definition just made in the introduction.

Definition 3.1.1 (Minimal surface). A surface is called a minimal surface ifand only if its mean curvature H ≡ 0.

3.2 Isothermal representation

Verifying that a surface is minimal by calculating its mean curvature usingthe formula in Section 2.4 can sometimes be quite tedious. If one howeverparametrize the surface by so-called isothermal coordinates, the calculationscan be made much simpler.

Definition 3.2.1 (Isothermal coordinates). A surface X is said to be paramet-rized by isothermal coordinates, or in short, to be isothermal, if⟨

Xu, Xu

⟩=⟨Xv, Xv

⟩,⟨Xu, Xv

⟩= 0.

It is a remarkable fact that every surface (as defined in Definition 2.1.1) canbe parametrized by isothermal coordinates (see e.g. Chern [3] for a proof), sowe can always assume that

gij =⟨Xui , Xuj

⟩= λ2δij ,

for some function λ(u, v), denoting by δij the Kronecker delta.

Back, 2015. 13

14 Chapter 3. Minimal surfaces

Theorem 3.2.1 (Isothermal minimal surfaces). A surface X : Ω → R3 para-metrized by isothermal coordinates is minimal if and only if it is harmonic inΩ, i.e. if its Laplacian ∆ be vanishing,

∆X := Xuu +Xvv ≡ 0.

Proof. Since X is isothermal,⟨Xu, Xu

⟩=⟨Xv, Xv

⟩,⟨Xu, Xv

⟩= 0.

By differentiating the first expression with respect to u and the second withrespect to v,⟨

Xuu, Xu

⟩=⟨Xvu, Xv

⟩= −

⟨Xu, Xvv

⟩⇐⇒

⟨Xuu +Xvv, Xu

⟩= 0,

and then differentiating the first expression with respect to v and the secondwith respect to u,⟨

Xvv, Xv

⟩=⟨Xuv, Xu

⟩= −

⟨Xv, Xuu

⟩⇐⇒

⟨Xuu +Xvv, Xv

⟩= 0.

This shows that Xuu + Xvv is orthogonal to both Xu and Xv and thereforeparallel to N . Since the coefficients of the first fundamental form E = G,F = 0,

H =LG +NE − 2MF

2 (EG − F2)=L+N

2E=

⟨Xuu, N

⟩+⟨Xvv, N

⟩2E

≡ 0

⇐⇒⟨Xuu +Xvv, N

⟩≡ 0 ⇐⇒ Xuu +Xvv ≡ 0.

Example 3.2.1 (Catenoid). The catenoid can be constructed by rotating thecatenary X2 = a cosh

(X3/a

), a ∈ R>0 about the X3-axis as seen in Figure 3.1.

It is the only minimal surface that can be constructed this way, i.e. it is theonly minimal surface that is a surface of revolution. By choosing X3 = au, itcan be parametrized by

X(u, v) =

X1(u, v)X2(u, v)X3(u)

= a

coshu cos vcoshu sin v

u

, u ∈ R, v ∈ (0, 2π).

Hence it follows that

Xu = a

sinhu cos vsinhu sin v

1

, Xv = a

− coshu sin vcoshu cos v

0

,

Xuu = a


0

, Xvv = a

− coshu cos v− coshu sin v

0

,

〈Xu, Xu〉 = 〈Xv, Xv〉 = a2 cosh2 u, 〈Xu, Xv〉 = 0, ∆X = Xuu +Xvv = 0.

From the relations of the inner products above, we see that this parametrizationis isothermal, and since ∆X = 0, the catenoid is indeed a minimal surface.

3.3. Asymptotic line representation 15

X1

X2

X3

X2 = a coshX3

a

a

Figure 3.1: The catenoid can be created by rotating the catenary X2 =a cosh

(X3/a

)about the X3-axis.

3.3 Asymptotic line representation

Lemma 3.3.1 (Euler formula). Let W ∈ TpX be arbitrary, E1, E2 an or-thogonal basis of TpX consisting of eigenvectors of the Weingarten map anddenote by α the angle from E1 to W . Then the normal curvature of W is

κn(W ) = κ1 cos2 α+ κ2 sin2 α,

which is called the Euler formula.

Proof. For arbitrary W ∈ TpX, we can always scale the eigenbasis E1, E2 sothat ‖E1‖ = ‖E2‖ = 1/‖W‖. Then we have the well-known relation 〈W,E1〉 =‖W‖‖E1‖ cosα = cosα and similarly 〈W,E2〉 = sinα. Since 〈E1, E2〉 = 0,W = W iEi = E1 cosα+ E2 sinα. The normal curvature of W is then

κn(W ) =II(W )

I(W )=〈SW,W 〉〈W,W 〉

=〈S (E1 cosα+ E2 sinα) , E1 cosα+ E2 sinα〉〈E1 cosα+ E2 sinα,E1 cosα+ E2 sinα〉

= κ1 cos2 α+ κ2 sin2 α.

Theorem 3.3.1 (Orthogonal asymptotic lines). A surface is minimal if andonly if there exist two orthogonal asymptotic lines at each of its points.

Proof. A surface is minimal if and only if κ1 ≡ −κ2. Let V ∈ TpX be any vector.If κ1 ≡ −κ2 = 0 for some points, then κ1 := minκn(V ) = maxκn(V ) = 0, soκn(V ) = 0 for all vectors V ∈ TpX at such points, i.e. all directions areasymptotic directions. On the other hand, if κ1 ≡ −κ2 6= 0, then by the Eulerformula

κn(V ) = κ1(cos2 α− sin2 α

)= 0 ⇐⇒ α =

π

4,

3π

4,

5π

4,

7π

4.

As can be seen in Figure 3.2, there then exist four directions of V which are allorthogonal to one another, and therefore also two orthogonal asymptotic linescorresponding to these.


0

π/4

π/2

3π/4

π

5π/4 7π/4

3π/2

Figure 3.2: There are four possible directions of V which all are orthogonal toone another, hence there exist two orthogonal asymptotic lines correspondingto these.

3.4 Adjoint surface

We recall from Theorem 3.2.1 that a surface X : Ω→ R3 is minimal if it satisfies

∆X = 0,⟨Xu, Xu

⟩=⟨Xv, Xv

⟩,⟨Xu, Xv

⟩= 0, ∀u, v ∈ Ω.

For such a surface, we can form an adjoint surface X on Ω as to satisfying

Xu = Xv, Xv = −Xu.

By differentiation,

∆X = Xuu +Xvv = Xvu −Xuv = 0,⟨Xu, Xu

⟩=⟨Xv, Xv

⟩=⟨Xu, Xu

⟩=⟨Xv, Xv

⟩,⟨Xu, Xv

⟩= −

⟨Xu, Xv

⟩= 0,

so the adjoint surface X is also a minimal surface with the same first funda-mental form as X.

Example 3.4.1 (Helicoid). In this example, we shall seek an expression for theadjoint surface to the catenoid. Let us therefore start with the parametrizationX from Example 3.2.1, so that the adjoint X should satisfy

Xu = Xv = a


0

, Xv = −Xu = −a


1

.

By integration, X is determined to within an arbitrary additive constant vector.If we take it to be zero, we arrive at

X = a

− sinhu sin vsinhu cos v−v

which is called the helicoid, depicted in Figure 3.3.

3.4. Adjoint surface 17

X1

X2

X3

Figure 3.3: The helicoid, an adjoint surface to the catenoid.

Chapter 4

A Backlund transformationfor minimal surfaces

In 1883, the Swedish mathematician Albert Victor Backlund established a mapbetween two focal surfaces of constant negative Gaussian curvature of a linecongruence, carrying with it a solution of the sine-Gordon equation

ϕuv = sinϕ

given implicitly by a system of PDEs, relying on an already known solution. Themap given by the line congruence and the system of PDEs and their solutions arenow commonly known as Backlund transformations. In this chapter, we shall seethat minimal surfaces can undergo a similar transformation. The transformationfor minimal surfaces uses a W-congruence, mapping one given minimal surface toa family of new minimal surfaces. Similar to Backlund’s original transformation,it carries with it a solution of the elliptic Liouville equation

θuu + θvv = e−2θ

in terms of a system of PDEs based on an already known solution; hence werecognize it as a Backlund transformation for minimal surfaces.

We will start by constructing this transformation geometrically as describedby Eisenhart in 1908 [5], but using a more contemporary mathematical language.We will also give a direct proof for the transformation using only algebraicmethods, a proof which we have not found elsewhere. At last, we will exploit itin an example transforming the catenoid into a family of new minimal surfaces.

4.1 Preliminaries

We proved in Theorem 3.3.1 that a surface is minimal iff there exist two orthog-onal asymptotic lines at each of its points. By Definition 2.8.3, a W-congruenceprovides a map that maps the asymptotic lines on one of its focal surfaces toasymptotic lines on the other focal surface. By means of Proposition 2.8.1,such a map can be constructed by making an infinitesimal deformation to asurface X. This map then maps the asymptotic lines on X to asymptotic lines

Back, 2015. 19

20 Chapter 4. A Backlund transformation for minimal surfaces

on another surface X, and these two surfaces are the focal surfaces of the W-

congruence. If X be minimal, then X will also be minimal if we demand thatthe asymptotic lines be mapped orthogonally. As we shall see, we can choosethe parameters on the adjoint minimal surface X to X such that the parametriclines on X be its asymptotic lines. The problem is then reduced to finding a map

W that maps the parametric lines from X to X orthogonally, which is simpler

since the parametric lines are those that have Xu, Xv and Xu, Xv as tangentvectors on each surface respectively. As a last step, we transform ”back” from

the surface X to its adjoint minimal surface X. Remarkably, it is parametrizedin the same way as we demanded the surface X to be, and thus it can again betransformed using the very same transformation. Hence, a possible infinity ofminimal surfaces can be found from just one known.

4.2 Geometric construction

Let X : Ω→ R3 be a minimal surface with normal

N :=Xu ×Xv

‖Xu ×Xv‖,

where in accordance with Bianchi [1, p. 335], the parameters (u, v) ∈ Ω ⊂ R2

are chosen such that

(gij) =

(1 00 1

)e2θ, (hij) =

(−1 00 1

), (4.1)

for some function θ(u, v). Since F = M = 0, we recall from Proposition 2.6.2that the parametric lines are the curvature lines. Continuing, the Gaussiancurvature is

K =det(hij)

det(gij)= −e−4θ,

and since gij = e−2θδij , the Christoffel symbols are found to be

Γ111 = θu, Γ1

12 = θv, Γ122 = −θu,

Γ211 = −θv, Γ2

12 = θu, Γ222 = θv.

By the Gauss equations,Xuu = Γk11Xk + h11N = θuXu − θvXv −N,Xvv = Γk22Xk + h22N = −θuXu + θvXv +N,

Xuv = Γk12Xk + h12N = θvXu + θuXv,

(4.2)

and by the Weingarten equations,Nu = −h1jgjkXuk = e−2θXu,

Nv = −h2jgjkXuk = −e−2θXv.(4.3)

4.2. Geometric construction 21

By assumption X ∈ C3, so the condition for cross-differentiation for third orderderivatives has to hold. By differentiation,

Xvuu = θuvXu + θuXuv − θvvXv − θvXvv −Nv= (θuv + 2θuθv)Xu +

(θ2u − θ2v − θvv + e−2θ

)Xv − θvN,

Xuuv = θuvXu + θvXuu + θuuXv + θuXuv

= (θuv + 2θuθv)Xu +(θ2u − θ2v + θuu

)Xv − θvN,

so the conditionXvuu = Xuuv

is equivalent to the elliptic Liouville equation

∆θ = e−2θ. (4.4)

It is seen that no further equations emerge from applying the same condition tothe Weingarten equations. Continuing, the adjoint minimal surface X to X isexpressed via

Xu = Xv, Xv = −Xu,

and since

N :=Xu ×Xv

‖Xu ×Xv‖= − Xv ×Xu∥∥Xv ×Xu

∥∥ =Xu ×Xv∥∥Xu ×Xv

∥∥ =: N, (4.5)

the second fundamental coefficients of X are

L = −〈Xu, Nu〉 = −〈Xv, Nu〉 =M = 0,

M = −〈Xu, Nv〉 = −〈Xv, Nv〉 = N = 1,

N = −〈Xv, Nv〉 = 〈Xu, Nv〉 = −M = 0.

Hence, by Proposition 2.6.1, the parametric lines on X are its asymptotic lines,and they are orthogonal since gij = gij = e2θδij . We now wish to make an

infinitesimal deformation of X, and recall that the surface X ′ proportional tothe direction of the deformation is completely determined by the characteristicfunction w(u, v) which is a solution of the characteristic equation (2.4). SinceK = K = −e−4θ, it takes the form

∂v(wue

2θ)

+ ∂u(wve

2θ)

= 0 ⇐⇒ wuv + θuwv + θvwu = 0. (4.6)

If we introduce the function ψ(u, v) defined by

ψu := wue2θ, ψv := −wve2θ, (4.7)

(4.6) is just the condition that ψuv = ψvu. When w is known from (4.6),

X ′u(2.5)=(wNu −Nwu

)e2θ

(4.5)= (wNu −Nwu) e2θ

(4.3)=(we−2θXu −Nwu

)e2θ, (4.8)

X ′v(2.5)=(Nwv − wNv

)e2θ

(4.5)= (Nwv − wNv) e2θ

(4.3)=(we−2θXv +Nwv

)e2θ. (4.9)


Xu

Xv

N

Y

X ′

α

TpX

Figure 4.1: A possible configuration for which Y ∈ TpX is orthogonal to X ′.

Now, denote by Y ∈ TpX = TpX the vector that is orthogonal to X ′ and by αthe angle from Xu to Y as seen in Figure 4.1. We choose to measure the anglefrom Xu and not Xu for later convenience. Then,⟨

Y ,Xu

⟩=∥∥Y ∥∥∥∥Xu

∥∥ cosα,⟨Y ,Xv

⟩=∥∥Y ∥∥ ∥∥Xv

∥∥ sinα,

so written in terms of Xu and Xv,

Y = Y Xu + Y ⊥Xu = Y Xu + Y Xv =

⟨Y ,Xu

⟩∥∥Xu

∥∥2 Xu +

⟨Y ,Xv

⟩∥∥Xv

∥∥2 Xv

=∥∥Y ∥∥( cosα∥∥Xu

∥∥Xu +sinα∥∥Xv

∥∥Xv

)=∥∥Y ∥∥ e−θ (Xu cosα+Xv sinα) .(4.10)

The condition that Y and X ′ be orthogonal is then⟨X ′, Y

⟩= 0 ⇐⇒

⟨X ′, Xu

⟩cosα+

⟨X ′, Xv

⟩sinα = 0, (4.11)

and the inner products in this equation fulfill the relation

∂v⟨X ′, Xu

⟩=⟨X ′v, Xu

⟩+⟨X ′, Xvu

⟩= −

⟨X ′v, Xv

⟩+⟨X ′, Xvu

⟩(2.3)=⟨X ′, Xvu

⟩=⟨X ′, Xuv

⟩= ∂u

⟨X ′, Xv

⟩.

If we define the function φ(u, v) by

φu := m⟨X ′, Xu

⟩, φv := m

⟨X ′, Xv

⟩, m ∈ R\ 0 , (4.12)

then the former equation is the condition φuv = φvu. For later convenience, weshall define

s(u, v) :=⟨X ′, N

⟩,


and calculate the second order derivatives of φ:

φuu = m⟨X ′u, Xu

⟩+m

⟨X ′, Xuu

⟩ (4.2)= m

⟨X ′u, Xu

⟩+m

⟨X ′, θuXu

⟩−m

⟨X ′, θvXv

⟩−ms (4.12)

= m⟨X ′u, Xu

⟩+ θuφu − θvφv −ms

(4.8)= mwe2θ + θuφu − θvφv −ms,

φuv = m⟨X ′u, Xv

⟩+m

⟨X ′, Xuv

⟩= m

⟨X ′u, Xu

⟩+m

⟨X ′, Xuv

⟩(2.3)= m

⟨X ′, Xuv

⟩ (4.2)= m

⟨X ′, θvXu

⟩+m

⟨X ′, θuXv

⟩(4.13)

(4.12)= θvφu + θuφv,

φvv = m⟨X ′v, Xv

⟩+m

⟨X ′, Xvv

⟩ (4.2)= m

⟨X ′v, Xv

⟩−m

⟨X ′, θuXu

⟩+m

⟨X ′, θvXv

⟩+ms

(4.12)= m

⟨X ′v, Xv

⟩− θuφu + θvφv +ms

(4.9)= mwe2θ − θuφu + θvφv +ms,

We now wish to express the points on the tangent lines to X in the direction ofY in terms of these equations. They can be written as

X = X + t1Y∥∥Y ∥∥ (4.10)

= X + t1e−θ (Xu cosα+Xv sinα)

(4.11)= X + t2e

−θ (⟨X ′, Xv

⟩Xu −

⟨X ′, Xu

⟩Xv

)(4.12)

= X +t2e−θ

m(φvXu − φuXv) = X +

te−2θ

m(φvXu − φuXv) , (4.14)

where t = t(u, v) has been defined in this way for later convenience. We alsorecall from Proposition 2.8.1 that these tangent lines form a W-congruence for

which X is one of the focal surfaces. We shall seek the value of t for which Xis the other focal surface, as depicted in Figure 4.2. First, however, we need to

Xu

Xv

N

Y

X ′

TpX

Xu

XvNTpX

Figure 4.2: The tangent planes to the focal surfaces X and X in terms of theadjoint vectors Xu, Xv, N and Xu, Xv, N .


know the tangent vectors to X. By differentiation,

Xu = Xu +tue−2θ

m(φvXu − φuXv)

+te−2θ

m(2θuφuXv − 2θuφvXu + φuvXu + φvXuu − φuuXv − φuXuv)

(4.2)= Xv +

tue−2θ

m(φvXu − φuXv)

+te−2θ

m((−θuφv − θvφu + φuv)Xu + (θuφu − θvφv − φuu)Xv − φvN)

(4.13)=(ste−2θ − wt+ 1

)Xv −

φvte−2θ

mN +

tue−2θ

m(φvXu − φuXv) ,

Xv = Xv +tve−2θ

m(φvXu − φuXv)

+te−2θ

m(2θvφuXv − 2θvφvXu + φvvXu + φvXuv − φuvXv − φuXvv)

(4.2)= −Xu +

tve−2θ

m(φvXu − φuXv)

+te−2θ

m((θuφu − θvφv + φvv)Xu + (θuφv + θvφu − φuv)Xv − φuN)

(4.13)=(ste−2θ + wt− 1

)Xu −

φute−2θ

mN +

tve−2θ

m(φvXu − φuXv)

Since Xu, Xv, N span R3, for the normal N to X, we can put

N = a1Xu + a2Xv + bN,

for functions a1, a2 and b depending on u and v. Then, since N is orthogonalto all vectors parallel to Y , by (4.14)⟨

N , φvXu − φuXv

⟩= 0 ⇐⇒ a1φv = a2φu,

so N is of the formN = a(φuXu + φvXv) + bN, (4.15)

for some new function a depending on u and v. The two orthogonality conditions⟨N , Xu

⟩= 0,

⟨N , Xv

⟩= 0 (4.16)

then become aφv

(st− e2θ (wt− 1)

)− bφvte

−2θ

m= 0,

aφu(st+ e2θ (wt− 1)

)− bφute

−2θ

m= 0.

(4.17)

We consider the generic case when φ is a function of both u and v and thereforeneither of φu and φv vanish. It is seen from the equations above that if a ≡ 0,then either b ≡ 0 or t ≡ 0. Both cases can be excluded since the former leadsto N being the null vector and the latter to X ≡ X. Hence, by dividing theequations by φv and φu respectively and then subtracting the former from the


latter, we arrive at t = 1/w. Let us evaluate (4.14) and the equations for thecorresponding tangent vectors using this value:

X = X +e−2θ

mw(φvXu − φuXv) ,

Xu = −wuφve−2θ

mw2Xu +

e−2θ

w

(s+

wuφumw

)Xv −

φve−2θ

mwN, (4.18)

Xv =e−2θ

w

(s− wvφv

mw

)Xu +

wvφue−2θ

mw2Xv −

φue−2θ

mwN.

We recall from Proposition 2.8.1 that the asymptotic lines on X now correspondto the asymptotic lines on X. The asymptotic lines on X were its parametriclines, i.e. the curves along Xu and Xv, and by definition these curves correspond

to the parametric lines on X; the curves along Xu and Xv respectively that is.

Hence the asymptotic lines on X are its parametric lines, and thus a necessary

and sufficient condition that X be minimal is that these curves be orthogonal(cp. Theorem 3.3.1), i.e. ⟨

Xu, Xv

⟩= 0.

This is equivalent to

wuwv(φ2u + φ2v

)+msw (wvφu − wuφv) + φuφvw

2e−2θ = 0

(4.7)⇐⇒ e−2θ(φ2u + φ2v

)+msw

(φuψu

+φvψv

)− φuφvw

2

ψuψv= 0. (4.19)

We shall examine this equation further by solving for ξ := φ − ψ. It thenbecomes

e−2θ(φ2u + φ2v

)+ 2msw − w2 + w (ms− w)

(ξuψu

+ξvψv

)− ξuξvψuψv

w2 = 0.

(4.20)

Earlier we defineds(u, v) :=

⟨X ′, N

⟩.

Hence

su =⟨X ′u, N

⟩+⟨X ′, Nu

⟩ (4.3)=⟨X ′u, N

⟩+ e−2θ

⟨X ′, Xu

⟩︸︷︷︸(4.12)=: φu/m

(4.8)= −wue2θ +

φue−2θ

m, (4.21)

sv =⟨X ′v, N

⟩+⟨X ′, Nv

⟩ (4.3)=⟨X ′v, N

⟩− e−2θ

⟨X ′, Xv

⟩︸︷︷︸(4.12)=: φv/m

(4.9)= wve

2θ − φve−2θ

m. (4.22)

We return to the investigation of (4.20), which, when setting

f(u, v) := w (ms− w)

(ξuψu

+ξvψv

)− ξuξvψuψv

w2


and differentiating with respect to u and v respectively becomes

0 = 2(φuφuu + φvφuv − θu

(φ2u + φ2v

))e−2θ + 2m (suw + swu)− 2wwu + fu

(4.13)= 2φum

(we2θ − s

)e−2θ + 2m (suw + swu)− 2wwu + fu

(4.21)= 2φu

(mwe2θ −ms+ w

)e−2θ + 2wu

(ms−mwe2θ − w

)+ fu

(4.7)= 2

(ξu + wue

2θ) (mwe2θ −ms+ w

)e−2θ + 2wu

(ms−mwe2θ − w

)+ fu

= 2ξu(mwe2θ −ms+ w

)e−2θ + fu,

0 = 2(φuφuv + φvφvv − θv

(φ2u + φ2v

))e−2θ + 2m (svw + swv)− 2wwv + fv

(4.13)= 2φvm

(we2θ + s

)e−2θ + 2m (svw + swv)− 2wwv + fv

(4.22)= 2φv

(mwe2θ +ms− w

)e−2θ + 2wv

(ms+mwe2θ − w

)+ fv

(4.7)= 2

(ξv − wve2θ

) (mwe2θ +ms− w

)e−2θ + 2wv

(ms+mwe2θ − w

)+ fv

= 2ξv(mwe2θ +ms− w

)e−2θ + fv.

It is seen that a solution of these two equations is ξ = const., so that in thiscase φ and ψ differ only by a constant. In virtue of this and (4.7),

φu = wue2θ, φv = −wve2θ. (I)

Integrating (4.21) and (4.22),

su(4.21)

= −wue2θ +φue−2θ

m= −φu +

wum⇐⇒ s = −φ+

w

m+ g(v)

⇐⇒ sv = −φv +wvm

+ g′(v)(4.22)

= wve2θ − φve

−2θ

m= −φv +

wvm

⇐⇒ g′(v) ≡ 0 ⇐⇒ g = const.,

for some arbitrary g, so s is also determined to within an additive constant. Wetherefore take

s = −φ+w

m,

so that (4.20) reads

e−2θ(φ2u + φ2v

)+ w2 − 2mφw = 0, (II)

and (4.13)

φuu = mwe2θ + θuφu − θvφv +mφ− w,φuv = θvφu + θuφv, (4.23)

φvv = mwe2θ − θuφu + θvφv −mφ+ w.

Moreover, (4.18) now becomes

X = X +e−2θ

mw(φvXu − φuXv) ,

Xu = −φuφve−4θ

mw2Xu +

e−2θ

mw2

(mφw − φ2ve−2θ

)Xv −

φve−2θ

mwN, (4.24)

Xv =e−2θ

mw2

(mφw − φ2ue−2θ

)Xu −

φuφve−4θ

mw2Xv −

φue−2θ

mwN,


We return to the determination of the functions a and b introduced earlier. By(4.17), we find that

b = amse2θ = a (w −mφ) e2θ,

so (4.15) assumes the form

N = a (φuXu + φvXv) + bN = a(φuXu + φvXv + (w −mφ) e2θN

). (4.25)

The condition ⟨N , N

⟩= 1

is then equivalent to

1 = a2e2θ(φ2u + φ2v + (w −mφ)2e2θ

) (II)=(amφe2θ

)2 ⇐⇒ a = ±e−2θ

mφ.

(4.26)

The possible different signs on a correspond to the orientation ofXu, Xv, N

,

being left- or right-handed with respect to the vector product ×. We shouldtake the canonical, right-handed, so that

N =Xu × Xv∥∥Xu × Xv

∥∥with respect to a right-handed basis of R3. It is sufficient to calculate just onecomponent of the vectors in the left- and right-hand side of this equation. Wedo so using the basis Xu, Xv, N, while the inner product 〈·, ·〉 is the usual one,inherited from R3. Then∥∥Xu × Xv

∥∥⟨N ,Xu

⟩ (4.25)=

∥∥Xu × Xv

∥∥aφue2θ =⟨Xu × Xv, Xu

⟩(4.24)

=φue−2θ

m2w3

(−mφw + φ2ve

−2θ − φ2ve−2θ)

= −φ2e−2θ

w2

e−2θ

mφφue

2θ

(4.26)⇐⇒ a = −e−2θ

mφ,∥∥Xu × Xv

∥∥ =φ2e−2θ

w2.

Using (II) in (4.24), we see that∥∥Xu

∥∥ =∥∥Xv

∥∥,so that, by the definition of the vector product

φ2e−2θ

w2=∥∥Xu × Xv

∥∥ =∥∥Xu

∥∥∥∥Xv

∥∥∥∥N∥∥ sinπ

2=∥∥Xu

∥∥2 =∥∥Xv

∥∥2.Hence the transform is also isothermal, and therefore

gij :=⟨Xui , Xuj

⟩=⇒ (gij) =

φ2e−2θ

w2

(1 00 1

).


We return to the consideration of the normal, which, when a now is known is

N(4.25)

= a(φuXu + φvXv + (w −mφ) e2θN

)= −e

−2θ

mφ(φuXu + φvXv) +

(1− w

mφ

)N.

Differentiating with respect to u and v respectively and making use of (4.2),(4.3) and (4.23),

Nu =e−2θ

mφ2(φ2u −mwφe2θ

)Xu +

φuφve−2θ

mφ2Xv +

wφumφ2

N(4.24)

= −w2e2θ

φ2Xv,

Nv =φuφve

−2θ

mφ2Xu +

e−2θ

mφ2(φ2v −mwφe2θ

)Xv +

wφvmφ2

N(4.24)

= −w2e2θ

φ2Xu,

so that

hij := −⟨Xui , Nuj

⟩=⇒ (hij) =

(0 11 0

).

The transform X thus found is by Eisenhart [5] called a Thybaut transform ofX and it can be obtained by solving (I) and (II) for w and φ. By constructionit constitutes one of the focal surfaces of a W-congruence, X being the otherfocal surface.

As a last step, we shall seek the expression for the adjoint surface X to X.We recall that

φu = wue2θ, φv = −wve2θ,

so with the help of (4.24),

Xu = −Xv =e−2θ

mw2

(φ2ue−2θ −mφw

)Xu +

φuφve−4θ

mw2Xv +

φue−2θ

mwN

= −∂u(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw(φuu − 2θuφu −mφ)Xu

+φue−2θ

mwXuu +

e−2θ

mw(φuv − 2θuφv)Xv +

φve−2θ

mwXuv +

φue−2θ

mwN

(4.2)= −∂u

(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw(φuu − θuφu + θvφv −mφ)Xu

+e−2θ

mw(φuv − θuφv − θvφu)Xv

(4.23)= −∂u

(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw

(mwe2θ − w

)Xu

(4.3)= ∂u

(X − φue

−2θ

mwXu −

φve−2θ

mwXv −

1

mN

),


Xv = Xu = −φuφve−4θ

mw2Xu +

e−2θ

mw2

(mφw − φ2ve−2θ

)Xv −

φve−2θ

mwN

= −∂v(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw(φvv − 2θvφv +mφ)Xv

+φve−2θ

mwXvv +

e−2θ

mw(φuv − 2θvφu)Xu +

φue−2θ

mwXuv −

φve−2θ

mwN

(4.2)= −∂v

(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw(φvv − θvφv + θuφu +mφ)Xv

+e−2θ

mw(φuv − θvφu − θuφv)Xu

(4.23)= −∂v

(φue−2θ

mwXu +

φve−2θ

mwXv

)+e−2θ

mw

(mwe2θ + w

)Xv

(4.3)= ∂v

(X − φue

−2θ

mwXu −

φve−2θ

mwXv −

1

mN

).

By integration, X is determined to within an additive constant which we taketo be zero. Hence,

X = X − 1

m

(φue−2θ

wXu +

φve−2θ

wXv +N

). (A)

As for the case with X, X can also be obtained from (I) and (II), and the normal

N to X is the same as for the adjoint transform. As a consequence, we see thatthe relation

X − φ

wN = X − φ

wN

holds. Hence, at corresponding points, the normals meet in a point at an equaldistance |φ/w| from each surface. As can be seen in Figure 4.3, at corresponding

points the surfaces X and X thus lie on a sphere of radius |φ/w| centered at

X − φwN = X − φ

w N . As both the radius and the center depend on u and v, weget a two-parameter family of spheres called the Ribaucour sphere congruence(recall from Definition 2.8.1 that a two-parameter family of lines were called a

line congruence), and consequently X is called a Ribaucour transform of X [9,

p. 175]. As u and v vary, the surfaces X and X therefore envelop these spheresand are accordingly called the sheets of the envelope of these spheres.

Continuing, gij is also the same as for the adjoint transform, so

(gij) = (gij) =φ2e−2θ

w2

(1 00 1

)=φ2e−4θ

w2(gij),

Nu = Nu = −w2e2θ

φ2Xv =

w2e2θ

φ2Xu,

Nv = Nv = −w2e2θ

φ2Xu = −w

2e2θ

φ2Xv.

It then follows that

hij := −⟨Xui , Nuj

⟩=⇒ (hij) =

(−1 00 1

)= (hij),


O

X− φwN

X

− φw N

∣∣∣ φw ∣∣∣

Figure 4.3: At corresponding points, the surfaces X and X lie on a sphere ofradius |φ/w| centered at X − φ

wN = X − φw N .

so the coefficients of the first and second fundamental form F = M = 0, andthus by Proposition 2.6.2 the parametric lines are the curvature lines. This wasalso the case for X, so the Ribaucour sphere congruence maps curvature linesof X to curvature lines of X.

If we define

θ := ln

∣∣∣∣ φw∣∣∣∣− θ,

then

(gij) = e2θ(

1 00 1

)which is of the same form as (gij), and thus θ is yet another solution of theelliptic Liouville equation

∆θ = e−2θ

that can be obtained from solving the simpler first order system of PDEs (I)and (II).

4.3 Algebraic proof

In the last section, we constructed a transformation for minimal surfaces interms of geometrical concepts that highly relied on the theory of congruences.In this section, we will give a direct proof for the transformation describingnew minimal surfaces parametrized by isothermal coordinates, which can bedetermined solely by solving the coupled system of PDEs (I) and (II). Moreover,we will also prove that a new solution of the elliptic Liouville equation can befound from the very same PDEs. We summarize this in the following theorem:

Theorem 4.3.1 (A Backlund transformation for minimal surfaces).Let X : Ω→ R3 be a minimal surface with normal

N :=Xu ×Xv

‖Xu ×Xv‖,

4.3. Algebraic proof 31

and parameters (u, v) ∈ Ω ⊂ R2 chosen such that

(gij) =

(1 00 1

)e2θ, (hij) =

(−1 00 1

),

for some function θ(u, v). Then θ is a solution of the elliptic Liouville equation∆θ = e−2θ. Furthermore, let φ(u, v) and w(u, v) be solutions of the coupledsystem of PDEs

φu = wue2θ, φv = −wve2θ, (I)

e−2θ(φ2u + φ2v

)+ w2 − 2mφw = 0, (II)

for any m ∈ R\0. Then

X = X − 1

m

(φue−2θ

wXu +

φve−2θ

wXv +N

)(A)

is a new minimal surface parametrized by isothermal coordinates, and yet an-

other solution of ∆θ = e−2θ is given by θ := ln∣∣∣ φw ∣∣∣− θ.

Proof. We proved in the beginning of Section 4.2 that ∆θ = e−2θ follows fromthis particular parametrization. Continuing, we also recall that (I) was thesolution of (4.6),

wuv + θuwv + θvwu = 0,

which could be retrieved by applying the condition φuv = φvu to (I). Some newconditions can be derived from these equations by differentiation:

φ(II)=

φ2u + φ2v2mw

e−2θ +w

2m

(I)=w2u + w2

v

2mwe2θ +

w

2m,

φu =

(wuu + θuwu −

w2u + w2

v

2w+w

2e−2θ

)wue

2θ

mw+ (θuwv + wuv)

wve2θ

mw

(4.6)=

(wuu + θuwu − θvwv −

w2u + w2

v

2w+w

2e−2θ

)wue

2θ

mw

(I)= wue

2θ

⇐⇒ wuu + θuwu − θvwv −w2u + w2

v

2w+w

2e−2θ −mw = 0, (4.27)

φv =

(wvv + θvwv −

w2u + w2

v

2w+w

2e−2θ

)wve

2θ

mw+ (θvwv + wuv)

wve2θ

mw

(4.6)=

(wvv + θvwv − θuwu −

w2u + w2

v

2w+w

2e−2θ

)wve

2θ

mw

(I)= −wve2θ

⇐⇒ wvv + θvwv − θuwu −w2u + w2

v

2w+w

2e−2θ +mw = 0. (4.28)

Adding (4.27) to (4.28) and subtracting (4.28) from (4.27) gives

wuu + wvv −w2u + w2

v

w+ we−2θ = 0, (4.29)

wuu − wvv + 2 (θuwu − θvwv −mw) = 0. (4.30)

By differentiating

X(A)= X − 1

m

(φue−2θ

wXu +

φve−2θ

wXv +N

)(I)= X − 1

m

(wuwXu −

wvwXv +N

),


and using the Gauss and Weingarten equations (4.2) and (4.3) for X derived inSection 4.2,

Xu =1

mw

(mw − wuu +

w2u

w

)Xu +

1

mw

(wuv −

wuwvw

)Xv −

1

mNu

− wumw

Xuu +wvmw

Xuv

=1

mw

(mw − θuwu + θvwv︸︷︷︸

(4.30)= (wuu−wvv)/2

−wuu +w2u

w− we−2θ

)Xu

+1

mw

(wuv + θuwv + θvwu︸︷︷︸

(4.6)= 0

−wuwvw

)Xv +

wumw

N

=1

mw

(−wuu + wvv

2+w2u

w− we−2θ︸︷︷︸

(4.29)= (w2

u−w2v)/(2w)−we−2θ/2

)Xu −

wuwvmw2

Xv +wumw

N

=1

2m

(w2u − w2

v

w2− e−2θ

)Xu −

wuwvmw2

Xv +wumw

N,

Xv = − 1

mw

(wuv −

wuwvw

)Xu +

1

mw

(mw + wvv −

w2v

w

)Xv −

1

mNv

wvmw

Xvv −wumw

Xuv

= − 1

mw

(wuv + θuwv + θvwu︸︷︷︸

(4.6)= 0

−wuwvw

)Xu

+1

mw

(mw − θuwu + θvwv︸︷︷︸

(4.30)= (wuu−wvv)/2

+wvv −w2v

w+ we−2θ

)Xv +

wvmw

N

=wuwvmw2

Xu +1

mw

(wuu + wvv

(2w)− w2

v

w+ we−2θ︸︷︷︸

(4.29)= (w2

u−w2v)/2+we

−2θ/2

)Xv +

wvmw

N

=wuwvmw2

Xu +1

2m

(w2u − w2

v

w2+ e−2θ

)Xv +

wvmw

N.

Since 〈Xu, Xu〉 = 〈Xv, Xv〉 = e2θ, 〈N,N〉 = 1 and Xu, Xv, N being orthogo-nal, the inner products are

〈Xu, Xv〉 =wuwv

2 (mw)2

(w2u − w2

v

w2− e−2θ

)e2θ

− wuwv

2 (mw)2

(w2u − w2

v

w2+ e−2θ

)e2θ +

wuwv

(mw)2 = 0,

〈Xv, Xv〉 − 〈Xu, Xu〉 =1

(2m)2

(4w2u − w2

v

w2e−2θ

)e2θ +

w2v − w2

u

(mw)2 = 0,

4.3. Algebraic proof 33

so the parametrization is indeed isothermal. Continuing,

Xuu =

(∂u + θu

2m

(w2u − w2

v

w2− e−2θ

)+we−2θ − θvwv

mw2wu

)Xu

−(θv2m

(w2u − w2

v

w2− e−2θ

)+∂u + θum

wuwvw2

)Xv

+

(∂u

wumw− 1

2m

(w2u − w2

v

w2− e−2θ

))N,

Xvv =

(∂v + θvm

wuwvw2

− θu2m

(w2u − w2

v

w2+ e−2θ

))Xu

+

(∂v + θv

2m

(w2u − w2

v

w2+ e−2θ

)+θuwu − we−2θ

mw2wv

)Xv

+

(∂v

wvmw

+1

2m

(w2u − w2

v

w2+ e−2θ

))N,

∆X = Xuu + Xvv

=1

m

(∂u2

(w2u − w2

v

w2− e−2θ

)− θue−2θ +

wuwe−2θ + ∂v

wuwvw2

)Xu

+1

m

(∂v2

(w2u − w2

v

w2+ e−2θ

)+ θve

−2θ − wvwe−2θ − ∂u

wuwvw2

)Xv

+1

m

(∂uwuw

+ ∂vwvw

+ e−2θ)N

=1

mw

(wuu + wvv −

w2u + w2

v

w+ we−2θ︸︷︷︸

(4.29)= 0

)(wuwXu −

wvwXv +N

)= 0,

so by Theorem 3.2.1, X describes minimal surfaces parametrized by isothermalcoordinates. We shall continue by proving that θ as defined by

θ := ln

∣∣∣∣ φw∣∣∣∣− θ,

is a new solution of∆θ = e−2θ,

once a solution θ is known. We first note that

−∆θ = −e−2θ (4.29)=

wuu + wvvw

− w2u + w2

v

w2= ∂u

(wuw

)+ ∂v

(wvw

)= ∆ ln |w|,

so by the linearity of ∆,

∆θ = ∆

(ln

∣∣∣∣ φw∣∣∣∣ − θ) = ∆ ln |φ| −∆ ln |w| −∆θ = ∆ ln |φ|

=φuu + φvv

φ− φ2u + φ2v

φ2(I)=

(wuu + 2θuwu − wvv − 2θvwv) e2θ

φ− φ2u + φ2v

φ2

(4.30)=

2mwe2θ

φ− φ2u + φ2v

φ2(II)=

w2e2θ

φ2= e−2(ln|

φw |−θ) = e−2θ.


4.4 Example: catenoid

In Example 3.2.1 we saw that the catenoid is a minimal surface. We shall showthat the parametrization used there (with a = 1) is actually of the desired form(4.1) for using our transformation. Hence, for u ∈ R and v ∈ (0, 2π),

X =


u

, Xu =


1

, Xv =


0

,

N =1

coshu

− cos v− sin vsinhu

, Nu =1

cosh2 u


1

, Nv =1

coshu

sin v− cos v

0

.

gij =⟨Xui , Xuj

⟩=⇒ (gij) = cosh2 u

(1 00 1

),

hij = −⟨Xui , Nuj

⟩=⇒ (hij) =

(−1 00 1

),

which is of the same form as in (4.1) iff e2θ = cosh2 u ⇐⇒ θ = ln coshu.We continue by solving (4.6) with respect to w, which gives

wuv + θuwv + θvwu = 0 ⇐⇒ ∂uwv + wv tanhu = 0

⇐⇒ e∫tanhudu∂uwv + e

∫tanhuduwv tanhu = 0

⇐⇒ ∂u

(wve

∫tanhudu

)= 0 ⇐⇒ ∂u

(wve

ln(coshu))

= 0

⇐⇒ wv =d′(v)

coshu⇐⇒ w =

d(v) + e(u)

coshu, wu =

e′ − (e+ d) tanhu

coshu

for some unknown functions d and e. We then use the above relations togetherwith (I) in (II) and differentiate with respect to v, which gives

0(II)= ∂v

(e−2θ

(φ2u + φ2v

)w

+ w − 2mφ

)(I)= ∂v

(e2θ(w2u + w2

v

)w

+ w − 2mφ

)

= coshu ∂v

(e′2 + d′2 − 2(e+ d)e′ tanhu+ (e+ d)2 tanh2 u

e+ d

)+

d′

coshu+ 2md′ coshu = coshu ∂v

(e′2 + d′2

e+ d+ (2m+ 1)d

)It is readily verified that the only nontrivial solution of this is given by

e′2 + d′2 = (2m+ 1)(e2 − d2) + 2c1(e+ d), c1 ∈ R, (4.31)

for nonvanishing e+ d. Differentiating with respect to u and v give

e′′ − (2m+ 1)e− c1 = 0, (4.32)

d′′ + (2m+ 1)d− c1 = 0, (4.33)

which hold for nonvanishing e′ and d′ respectively.

4.4. Example: catenoid 35

Solving these ODE:s yield

e =

a1 cos

(√−2m− 1u

)+ a2 sin

(√−2m− 1u

)− c1

2m+ 1if m < − 1

2 ,c12u2 + a3u+ a4 if m = − 1

2 ,

a5 cosh(√

2m+ 1u)

+ a6 sinh(√

2m+ 1u)− c1

2m+ 1if m > − 1

2 ,

d =

b1 cosh

(√−2m− 1v

)+ b2 sinh

(√−2m− 1v

)+

c12m+ 1

if m < − 12 ,

c12v2 + b3v + b4 if m = − 1

2 ,

b5 cos(√

2m+ 1v)

+ b6 sin(√

2m+ 1v)

+c1

2m+ 1if m > − 1

2 ,

for constants ai, bj . When using these together with (4.31), we get the algebraicrelations

a21 + a22 − b21 + b22 = 0,

a23 + b23 − 2c1 (a4 + b4) = 0,

−a25 + a26 + b25 + b26 = 0.

According to (A), the new minimal surfaces are then given by the transform

X = X − 1

m

e′

e+ dXu −

d′

e+ dXv − coshu

cos vsin v

0

.

Taking some different values on m and ai, bj , c1 satisfying the algebraic relationsabove, we get a set of new minimal surfaces according to this parametrization:

X|m=−1 = − 1

cosu+ cosh v

coshu sin v sinh v − cos v sinu sinhusinu sinhu sin v + coshu cos v sinh v

sinu

+

00u

,

if a1 = b1 = 1, a2 = b2 = 0, c1 ∈ R,

X|m=− 12

=4

u2 + v2

u cos v sinhu+ v coshu sin vu sinhu sin v − v coshu cos v

u

−coshu cos v

coshu sin v−u

,

if a3 = a4 = b3 = b4 = 0, c1 = 1,

X|m=1 = −√

3

cosh√

3u+ cos√

3v

sinh√

3u cos v sinhu− sin√

3v coshu sin v

sin√

3v coshu cos v + sinh√

3u sinhu sin v

sinh√

3u

+

2 coshu cos v2 coshu sin v

u

, if a5 = b5 = 1, a6 = b6 = 0, c1 ∈ R.

The transforms can be seen in Figure 4.4, Figure 4.5 and Figure 4.6, and are tobe compared with the original catenoid in Figure 3.1. To generate the transformsand the pictures of them, we have used a Matlab script which also verifies thatthe transforms are isothermal and that their Laplacians are zero, hence givingminimal surfaces as expected. The source code can be found in Appendix A.


Figure 4.4: Part of the transform X|m=−1 of the catenoid, using initial valuesa1 = b1 = 1, a2 = b2 = 0, c1 ∈ R.

Figure 4.5: Part of the transform X|m=− 12

of the catenoid, using initial valuesa3 = a4 = b3 = b4 = 0, c1 = 1.

Figure 4.6: Part of the transform X|m=1 of the catenoid, using initial valuesa5 = b5 = 1, a6 = b6 = 0, b4 = 0, c1 ∈ R.

Bibliography

[1] L. Bianchi, Lezioni di geometria differenziale. Vol. 2, 2nd ed., Enrico Spoerri,Pisa, 1903.

[2] M.P.D. Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall,Englewood Cliffs, N.J., 1976.

[3] S.S. Chern, An Elementary Proof of the Existence of Isothermal Parameterson a Surface, Proc. Am. Math. Soc. 6 (1955), no. 5, pp. 771–782.

[4] A.V. Corro, W. Ferreira, and K. Tenenblat, Minimal Surfaces Obtained byRibaucour Transformations, Geom. Dedic. 96 (2003), pp. 117–150.

[5] L.P. Eisenhart, Surfaces with Isothermal Representation of Their Lines ofCurvature and Their Transformations, Trans. Am. Soc. 9 (1908), no. 2, pp.149–177.

[6] , A Treatise on the Differential Geometry of Curves and Surfaces,Ginn, Boston, 1937.

[7] J. Hoppe, Unpublished notes.

[8] W. Kuhnel, Differential geometry: curves - surfaces - manifolds, 2nd ed.,American Mathematical Society, Providence, R.I., 2006.

[9] C. Rogers and W.K. Schief, Backlund and Darboux transformations: geom-etry and modern applications in soliton theory, Cambridge University Press,New York, 2002.

Back, 2015. 37

38 Bibliography

Appendix A

A Matlab script

The source code for the Matlab script catTrans can be found here below. Towork properly, it requires the Symbolic Math Toolbox. When saved, the fileshould be given the extension .m.

A.1 catTrans

%% catTrans: Transforming the catenoid for given constants%% m,A1,A2,B1,B2,c1 fulfilling any of the three conditions in putConst.function X=catTrans(m,A1,A2,B1,B2,c1)

syms u v real;[e,d]=putConst(m,A1,A2,B1,B2,c1,u,v);X=paramCat(m,d,e,u,v);minCheck(X,u,v);

end

%% Checking if any of the three conditions are fulfilled.function [e,d]=putConst(m,A1,A2,B1,B2,c1,u,v)

if(m<-1/2 && A1ˆ2 + A2ˆ2 -B1ˆ2 + B2ˆ2==0)e=A1*cos(u*sqrt(-2*m-1))+A2*sin(u*sqrt(-2*m-1))-c1/(2*m+1);d=B1*cosh(v*sqrt(-2*m-1))+B2*sinh(v*sqrt(-2*m-1))+c1/(2*m+1);

elseif(m==-1/2 && A1ˆ2+B2ˆ2-2*c1*(A2+B2)==0)e=c1/2*uˆ2 + A1*u+A2;d=c1/2*vˆ2 + B1*v+B2;

elseif(m>-1/2 && -A1ˆ2 + A2ˆ2 + B1ˆ2 + B2ˆ2==0)e=A1*cosh(u*sqrt(2*m+1))+A2*sinh(u*sqrt(2*m+1))-c1/(2*m+1);d=B1*cos(v*sqrt(2*m+1))+B2*sin(v*sqrt(2*m+1))+c1/(2*m+1);

elseerror('Algebraic relations were not fulfilled.');

endend

%% Parametrizing the transform.function X=paramCat(m,d,e,u,v)

X1=[cosh(u)*cos(v);cosh(u)*sin(v);u];X=simp(X1-1/m*(diff(e,u)*diff(X1,u)/(e+d)...-diff(d,v)*diff(X1,v)/(e+d)-cosh(u)*[cos(v);sin(v);0]));

end

%% Verifying that the transform is minimal, calculating its curvature,%% and then plotting it.

Back, 2015. 39

40 Appendix A. A Matlab script

function minCheck(X,u,v)

% Calculating tangent vectors and their derivatives.Xu=simp(diff(X,u));Xv=simp(diff(X,v));Xuu=simp(diff(Xu,u));Xvv=simp(diff(Xv,v));Xuv=simp(diff(Xu,v));

% Calculating the normal.N=cross(Xu,Xv);N=simp(N/norm(N));

% Calculating the coefficients of the first fundamental form.E=simp(dot(Xu,Xu));F=simp(dot(Xu,Xv));G=simp(dot(Xv,Xv));

% Calculating the coefficients of the second fundamental form.e=simp(dot(N,Xuu));f=simp(dot(N,Xuv));g=simp(dot(N,Xvv));

% Calculating the mean curvature H and the Gauss curvature K.H=simp((E*g+G*e-2*F*f)/(2*(E*G-Fˆ2)));K=simp((e*g-f*f)/(E*G-F*F));

% Verifying isothermal coordinates and zero Laplacian.if(simp(E-G)==0) && (simp(F)==0)

fprintf('Isothermal coordinates. ');if (simp(Xuu+Xvv)==0)

fprintf('Zero Laplacian; minimal surface!\n');end

elsefprintf('Not isothermal coordinates,');if(H==0)

fprintf(' but zero mean curvature; minimal surface!\n');else

fprintf([' and mean curvature H=',char(H),'.\n']);end

end

% Plotting the transform.fprintf(['Gaussian curvature K=',char(K),...

'.\nPlotting the surface.\n']);ezmesh(X(1),X(2),X(3));axis off;title('');colormap([0,0,0]);

end

%% Simplifying the simplify function.function S=simp(X)

S=simplify(X,'Steps',50);end

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