Bäcklund transformations for minimal surfaces

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Thesis

B¨

acklund transformations for minimal surfaces

Per B¨

ack

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B¨

acklund transformations for minimal surfaces

Department of Mathematics, Link¨oping University

Per B¨ack

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, Link¨oping University

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Abstract

In this thesis, we study a B¨acklund transformation for minimal surfaces – sur-faces with vanishing mean curvature – transforming a given minimal surface into a possible infinity of new ones.

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

Keywords: B¨acklund transformations, Liouville equation, minimal surfaces, Ribaucour transformations, Thybaut transformations.

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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 as an examiner without blinking, and for great help and discussions regarding the thesis. My greatest gratitude also goes to my opponent Daria Burdakova for comments on the manuscript, Eric Wolter for input on the Matlab script and all 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 the theorems, and the Department of Mathematics at KTH for hosting me. Last but not least, I would like to thank friends and family for always being there – especially Malin, my girlfriend.

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Nomenclature

Most of the recurring abbreviations and symbols are described here.

Symbols

× Vector product h·, ·i Inner product

Cn _{Space of n times continuously differentiable functions} Cn _{Complex n-space}

∂i Partial differentiation with respect to ui δjk Kronecker delta

∆ Flat Laplacian in Rn det Determinant

(gij) Matrix with elements gij Ω Open subset of R2 Rn _{Euclidean n-space}

tr Trace, i.e. the sum of the diagonal entries of a matrix kXk Norm of X induced by the inner product

X Vector in an (or map between) inner product space(s) Xi i:th component of X as a vector

Xui Partial derivative of X with respect to ui

Abbreviations

iff If and only if

ODE Ordinary differential equation

ON Orthonormal

PDE Partial differential equation

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Introduction

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

1.1 Background

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

In 1908, the American mathematician Luther Pfahler Eisenhart published a paper [5] in which he described geometrically how to construct a transformation for minimal surfaces in three-dimensional Euclidean space, transforming a given minimal surface into a family of new minimal surfaces. The transformation, which is a so-called B¨acklund transformation, also carries with it mappings between solutions of the elliptic Liouville equation

θuu+ θvv = e−2θ.

Both the result and the mathematical concepts used in the paper seem to have been forgotten, mostly. The purpose of this thesis is to revise some of those concepts and rederive the transformation geometrically as done by Eisenhart in 1908, but in a more contemporary mathematical language (for an alternative formulation 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, transforming the catenoid into a family of new minimal surfaces.

1.2 Topics covered

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

Chapter 2: Introduction to general surfaces in R3.

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

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2 Chapter 1. Introduction

Chapter 4: Derivation of a B¨acklund transformation for minimal surfaces in terms of geometrical concepts and an algebraic proof for it. We also exploit the transformation in an example, transforming the catenoid into a family of new minimal surfaces.

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

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Chapter 2

Differential geometry of

surfaces

In this chapter, we will study surfaces and the differential geometry in which they 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 an immersion X : Ω → R3 where Ω is an open set of R2, i.e. X is a differentiable vector valued function whose derivative is everywhere injective. A point p in Ω is written (u, v), where u and v are called the parameters. If not otherwise stated, 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_{, u}2_{=: (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, the vector product Xu× Xv does not vanish, so the normal vector field

N = Xu× Xv kXu× Xvk

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4 Chapter 2. Differential geometry of surfaces

is well defined, k · k denoting the usual norm on R3_{. Since kN k = 1, we can} view this as a map from Ω to the unit sphere

S2=(x, y, z) ∈ R3_{: x}2_{+ y}2_{+ z}2_{= 1 .}

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

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 kXu× Xvk

,

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 vector at a point p ∈ Ω as being attached to the image point X(p), but have instead moved 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 image points of the Gauss map being at the tip of the arrowheads.

Definition 2.2.2 (Self-adjoint map). Let (T, h·, ·i) be a finite dimensional real or complex inner product space. A linear map A : T → T is called self-adjoint if hAV, W i = hV, AW i for all vectors V, W ∈ T .

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

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

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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 λ1and λ2≥ λ1, which are given by

λ1= min hAV, V i hV, V i : V ∈ T, V 6= 0 , λ2= max hAV, V i hV, V i : V ∈ T, V 6= 0 . Proof. See [2, p. 216].

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

V = V1Xu(p) + V2Xv(p) = X

i

ViXui(p) =: ViX_ui(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 the same terms (typically as a mix of upper and lower indices as above), summation is implied over those indices.

Definition 2.2.3 (Weingarten map). For a surface X : Ω → R3 _{with unit} normal vector field

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

V = ViXui(p) via S(p)V := −ViN_ui(p).

Using the inner product h·, ·i inherited by the ambient space R3_{, hN, N i = 1} by definition. Differentiating this then yields hN, Nuii = 0, so either N_ui(p) lie

in 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 hN, Xuji = 0, we

can deduce that the Weingarten map is self-adjoint in TpX. First, by differen-tiation we have

hNui, X_uji + hN, X_ui_uji = 0 ⇐⇒ hN_ui, X_uji = −hN, X_ui_uji = −hN, X_uj_uii

= hNuj, X_uii = hX_ui, N_uji,

so for arbitrary V, W ∈ TpX written in the basis {Xu, Xv} as V = ViXui and

W = WjXuj

hSV, W i = −Vi_Wj_hN

ui, X_uji = −ViWjhX_ui, N_uji = −hViX_ui, WjN_uji

= hV, SW i.

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2.3 Fundamental forms

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

I(V, W ) := hV, W i, II(V, W ) := hSV, W i, III(V, W ) := hSV, SW i.

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

I(V ) := hV, V i, II(V ) := hSV, V i, III(V ) := hSV, SV i.

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 are defined as the principal curvatures κ1and 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 the Weingarten map, and the directions of the corresponding eigenvectors E1 and E2 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 0 0 κ2

.

Changing S to this basis can always be done by a similarity transformation S 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 are similarity-invariant, and in differential geometry, those of the Weingarten map play a particularly important role.

Definition 2.4.1 (Gauss and mean curvature). The functions

K := det S = κ1κ2,

H :=tr S

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 the basis {Xu, Xv}. As before, we write arbitrary V ∈ TpX as V = ViXui in this

basis, and the first and second fundamental form are therefore

I(V ) = hV, V i = ViVjhXui, X_uji,

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2.4. Curvature 7

On account of this,

gij := hXui, X_uji,

hij := −hXui, N_uji,

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 := hXu, Xui, F := hXu, Xvi = hXv, Xui, G := hXv, Xvi, L := −hXu, Nui = hXuu, N i, N := −hXv, Nvi = hXvv, N i, M := −hXu, Nvi = −hXv, Nui = hXuv, N i = hXvu, N i, so in matrix form (gij) = g11 g12 g21 g22 = E F F G , (hij) = h11 h12 h21 h22 = L M M N .

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

(gij) =g 11 _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 = Si_j = gik_h kj = g11_h 11+ g12h21 g11h12+ g12h22 g21_h 11+ g22h21 g21h12+ g22h22 .

Hence, we can calculate the Gauss and mean curvature as

K = det S = det gikhkj = det(hij) det(gij) =LN − M 2 EG − F2 , H = tr S 2 = gij_h ij 2 = LG + N E − 2MF 2 (E G − F2₎ .

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

X = X1, X2, X3 ∈ R3: kXk = r. We parametrize it by X(u, v) =   X1_{(u, v)} X2_{(u, v)} X3_(u)  = r   sin u cos v sin u sin v cos u  , Xu(u, v) = r   cos u cos v cos u sin v − sin u  , Xv(u, v) = r   − sin u sin v sin u cos v 0  , N (u, v) = Xu× Xv kXu× Xvk (u, v) =   sin u cos v sin u sin v cos u  , Nu(u, v) =   cos u cos v cos u sin v − sin u  , Nv(u, v) =   − sin u sin v sin u cos v 0  ,

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8 Chapter 2. Differential geometry of surfaces for u ∈ (0, π) and v ∈ (0, 2π). gij = hXui, X_uji =⇒ (g_ij) = r2 1 0 0 sin2u , hij = −hXui, N_uji =⇒ (hij) = r 1 0 0 sin2u =1 r(gij) . Hence, S = gikhkj = 1 r g ik_g kj = 1 rI, where I is the identity matrix, so

K = det S = det 1 rI = 1 r2, H = tr S 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 immersion is equivalent to having the tangent vectors Xu(p) and Xv(p) being linearly independent at all points p, and that they therefore span all tangent planes TpX. 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_ikX_uk,

for coefficients ak

i . By taking the inner product with Xuj, we get

Nui, X_uj = a_ikX_uk, X_uj ⇐⇒ −hij= aikgkj, and by multiplying by the inverse and summing over j as well,

−hijgjl= aikgkjgjl = aikδ l k = a l i, where δ l

k 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 our disposal a basis of R3_{. We shall try to express the second order derivatives of} X in this basis, thus putting

Xui_uj = Γk_ijX_uk+ bijN (2.1) for coefficients Γk_ij and bij. The coefficients Γkij are called Christoffel symbols of the second kind, while

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2.6. Curves on the surface 9

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

Xui_uj, X_ul = Γk_ijgkl = Γlij, and hence we have the symmetry relations

Γlij = Γlji, Γljk= Γ l

kj. Introducing the shorthand notation

∂k:= ∂ ∂uk, by differentiation

∂kgij = ∂kXui, X_uj = X_ui_uk, X_uj + X_ui, X_uj_uk = 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 2g mk_(−∂ kgij+ ∂jgik+ ∂igkj) = gmkgklΓlij= δ m lΓ l ij = Γ m ij.

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

bij =Xui_uj, N = h_ij,

so that

Xuiuj = Γk_ijX_uk+ hijN, which are called the Gauss equations.

2.6 Curves on the surface

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

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

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

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

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We recall from Section 2.4 that the principal directions were the directions along 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 the normal curvature κn(V ) = 0 are called the asymptotic directions, and curves on the surface which are tangent to these directions at every point are called asymptotic 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 asymptotic lines.

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

Proof. See [6].

Proposition 2.6.2 (Parametric curvature lines). The lines of curvature are parametric 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 surface X00(u, v) by deforming the former in the direction of a vector X0(u, v) by setting

X00:= X + X0, ∈ R.

The tangent lines to X0 are called the generatrices of the deformation, and X0 itself does also correspond to a parametrized surface. Since

X_u00i = Xui+ X_u0i, ∈ R, (2.2)

by taking the inner product with X_u00j we obtain

g00_ij= gij+ (hXui, X_u0ji + hXuj, X_u0ii) + 2g0_ij,

where gij, gij0 and gij00 are the corresponding coefficients of the first fundamental forms. If

hXui, X_u0ji + hXuj, X_u0ii = 0 (2.3)

and be taken so small that 2may be neglected, X and X00are seen to be iso-metric. X00_{is then said to be obtained from X by an infinitesimal deformation.} The problem of making an infinitesimal deformation to X is then equivalent to determining X0 by means of (2.3), which in turn is equivalent to

hXu, Xu0i = 0, hXv, Xv0i = 0, hXu, Xv0i = −hXv, Xu0i =: w p

EG − F2_. Here, we have in accordance with Eisenhart [6, p. 374] defined the characteristic function w(u, v) of the infinitesimal deformation. As usual, E , F and G are the

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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 N wu− Mwv K√EG − F2 =2F M − E N − GL√ EG − F2 w (2.4)        X_u0 =L (wNv− N wv) − M (wNu− N wu) K√EG − F2 X_v0 = M (wNv− N wv) − N (wNu− N wu) K√EG − F2 (2.5)

The former equation is called the characteristic equation, and once we have obtained the characteristic function w from it, we can also get X0 by means of integrating the latter equations. Here, K denotes the Gaussian curvature and L, 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 many different types of useful curves on a surface. In this section, we shall study connections between such curves and infinitesimal deformations, and also see how 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 lines in space is called a line congruence.

Let Ω ⊂ R2 with (u, v) ∈ Ω, X : Ω → R3 and R : Ω → S2 where S2 is the unit 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 line congruence, that is, a straight line. The surface X is called a reference surface to C, and it is seen to not be unique; with X + sR for some s ∈ R as a reference surface instead, we would describe the very same C.

Example 2.8.1 (Normal congruence). The normal lines to a surface constitute a line congruence and is called a normal congruence. If we take the normal field N : Ω → S2 _{to the surface X : Ω → R}3_{, 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 and v fixed while varying t. If we instead keep t fixed and vary u and v, we will describe 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 they exist, called focal surfaces.

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Hence, when two focal surfaces exist, a line congruence define a map F17→ F2 and its inverse F27→ F1between them.

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

A W-congruence therefore defines a map mapping asymptotic lines on one focal surface to asymptotic lines on the other focal surface, and the other way around. 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 a surface which are perpendicular to the generatrices of an infinitesimal deforma-tion of the surface constitute a W-congruence. The original, undeformed surface is one of the focal surfaces to the W-congruence, and the normal lines to the other focal surface are parallel to these generatrices.

Proof. See [6, p. 420].

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

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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. By means of variational methods, he succeeded with a non-parametric description, and sixteen years later Meusnier proved that they are surfaces with vanishing mean curvature. Later on, different but equivalent definitions have been made in a variety of different areas of mathematics, demonstrating the diversity of the subject. In this thesis, we will stick to the definition of being surfaces with vanishing mean curvature, and in this chapter we will study such surfaces and concepts 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 if and only if its mean curvature H ≡ 0.

3.2 Isothermal representation

Verifying that a surface is minimal by calculating its mean curvature using the formula in Section 2.4 can sometimes be quite tedious. If one however parametrize the surface by so-called isothermal coordinates, the calculations can 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) can be parametrized by isothermal coordinates (see e.g. Chern [3] for a proof), so we can always assume that

gij =Xui, X_uj = λ2δ_ij,

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

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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 in Ω, 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 with respect to v,

Xuu, Xu = Xvu, Xv = −Xu, Xvv

⇐⇒ Xuu+ Xvv, Xu = 0, and then differentiating the first expression with respect to v and the second with 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 therefore parallel to N . Since the coefficients of the first fundamental form E = G, F = 0,

H = LG + N E − 2MF 2 (E G − 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 the catenary X2_{= a cosh X}3_{/a , a ∈ R}

>0about 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 the only minimal surface that is a surface of revolution. By choosing X3 _{= au, it} can be parametrized by X(u, v) =   X1_{(u, v)} X2(u, v) X3(u)  = a   cosh u cos v cosh u sin v u  , u ∈ R, v ∈ (0, 2π).

Hence it follows that

Xu= a   sinh u cos v sinh u sin v 1  , Xv= a   − cosh u sin v cosh u cos v 0  , Xuu= a   cosh u cos v cosh u sin v 0  , Xvv= a   − cosh u cos v − cosh u sin v 0  , hXu, Xui = hXv, Xvi = a2cosh2u, hXu, Xvi = 0, ∆X = Xuu+ Xvv = 0. From the relations of the inner products above, we see that this parametrization is isothermal, and since ∆X = 0, the catenoid is indeed a minimal surface.

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3.3. Asymptotic line representation 15 X1 X2 X3 X2_{= a cosh}X 3 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 and denote by α the angle from E1 to W . Then the normal curvature of W is

κn(W ) = κ1cos2α + κ2sin2α,

which is called the Euler formula.

Proof. For arbitrary W ∈ TpX, we can always scale the eigenbasis {E1, E2} so that kE1k = kE2k = 1/kW k. Then we have the well-known relation hW, E1i = kW kkE1k cos α = cos α and similarly hW, E2i = sin α. Since hE1, E2i = 0, W = Wi_E

i= E1cos α + E2sin α. The normal curvature of W is then

κn(W ) = II(W )

I(W ) =

hSW, W i hW, W i =

hS (E1cos α + E2sin α) , E1cos α + E2sin αi hE1cos α + E2sin α, E1cos α + E2sin αi = κ1cos2α + κ2sin2α.

Theorem 3.3.1 (Orthogonal asymptotic lines). A surface is minimal if and only 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 := min κn(V ) = max κn(V ) = 0, so κn(V ) = 0 for all vectors V ∈ TpX at such points, i.e. all directions are asymptotic directions. On the other hand, if κ1≡ −κ2 6= 0, then by the Euler formula κn(V ) = κ1 cos2α − sin2α = 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 all orthogonal to one another, and therefore also two orthogonal asymptotic lines corresponding to these.

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16 Chapter 3. Minimal surfaces 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 to one another, hence there exist two orthogonal asymptotic lines corresponding to these.

3.4 Adjoint surface

We recall from Theorem 3.2.1 that a surface X : Ω → R3is 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 the adjoint surface to the catenoid. Let us therefore start with the parametrization X from Example 3.2.1, so that the adjoint X should satisfy

Xu= Xv = a   − cosh u sin v cosh u cos v 0  , Xv= −Xu= −a   sinh u cos v sinh u sin v 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   − sinh u sin v sinh u cos v −v  

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3.4. Adjoint surface 17

X1

X2 X3

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Chapter 4

A B¨

acklund transformation

for minimal surfaces

In 1883, the Swedish mathematician Albert Victor B¨acklund established a map between two focal surfaces of constant negative Gaussian curvature of a line congruence, carrying with it a solution of the sine-Gordon equation

ϕuv= sin ϕ

given implicitly by a system of PDEs, relying on an already known solution. The map given by the line congruence and the system of PDEs and their solutions are now commonly known as B¨acklund transformations. In this chapter, we shall see that minimal surfaces can undergo a similar transformation. The transformation for minimal surfaces uses a W-congruence, mapping one given minimal surface to a family of new minimal surfaces. Similar to B¨acklund’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 we recognize it as a B¨acklund transformation for minimal surfaces.

We will start by constructing this transformation geometrically as described by Eisenhart in 1908 [5], but using a more contemporary mathematical language. We will also give a direct proof for the transformation using only algebraic methods, a proof which we have not found elsewhere. At last, we will exploit it in 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-congruence provides a map that maps the asymptotic lines on one of its focal surfaces to asymptotic 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 a surface X. This map then maps the asymptotic lines on X to asymptotic lines

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20 Chapter 4. A B¨acklund transformation for minimal surfaces

on another surface bX, and these two surfaces are the focal surfaces of the W-congruence. If X be minimal, then bX will also be minimal if we demand that the asymptotic lines be mapped orthogonally. As we shall see, we can choose the parameters on the adjoint minimal surface X to X such that the parametric lines on X be its asymptotic lines. The problem is then reduced to finding a map W that maps the parametric lines from X to bX orthogonally, which is simpler since the parametric lines are those that have Xu, Xv and bXu, bXv as tangent vectors on each surface respectively. As a last step, we transform ”back” from the surface bX to its adjoint minimal surface bX. Remarkably, it is parametrized in the same way as we demanded the surface X to be, and thus it can again be transformed using the very same transformation. Hence, a possible infinity of minimal surfaces can be found from just one known.

4.2 Geometric construction

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

,

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

(gij) = 1 0 0 1 e2θ, (hij) = −1 0 0 1 , (4.1)

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

K = det(hij) det(gij)

= −e−4θ,

and since gij _{= e}−2θ_δij_{, the Christoffel symbols are found to be}

Γ111= θu, Γ112= θv, Γ122= −θu, Γ2₁₁= −θv, Γ212= 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−2θX_u,

Nv= −h2jgjkXuk= −e−2θXv.

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4.2. Geometric construction 21

By assumption X ∈ C3_{, so the condition for cross-differentiation for third order} derivatives has to hold. By differentiation,

Xvuu= θuvXu+ θuXuv− θvvXv− θvXvv− Nv = (θuv+ 2θuθv) Xu+ θu2− θ 2 v− θvv+ e−2θ Xv− θvN, Xuuv= θuvXu+ θvXuu+ θuuXv+ θuXuv = (θuv+ 2θuθv) Xu+ θu2− θv2+ θuu Xv− θvN, so the condition Xvuu = 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 to the Weingarten equations. Continuing, the adjoint minimal surface X to X is expressed via Xu= Xv, Xv= −Xu, and since N := Xu× Xv kXu× Xvk = − Xv× Xu Xv× Xu = Xu× Xv Xu× Xv =: N , (4.5)

the second fundamental coefficients of X are

L = −hXu, Nui = −hXv, Nui = M = 0, M = −hXu, Nvi = −hXv, Nvi = N = 1,

N = −hXv, Nvi = hXu, Nvi = −M = 0.

Hence, by Proposition 2.6.1, the parametric lines on X are its asymptotic lines, and they are orthogonal since g_ij = gij = e2θδij. We now wish to make an infinitesimal deformation of X, and recall that the surface X0 _{proportional to} the direction of the deformation is completely determined by the characteristic function w(u, v) which is a solution of the characteristic equation (2.4). Since K = K = −e−4θ, it takes the form

∂v wue2θ + ∂u wve2θ = 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),

X0 u (2.5) = wNu− N wu e2θ (4.5)= (wNu− N wu) e2θ (4.3) = we−2θXu− N wu e2θ, (4.8) X0 v (2.5) = N wv− wNv e2θ (4.5)= (N wv− wNv) e2θ (4.3) = we−2θXv+ N wv e2θ. (4.9)

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22 Chapter 4. A B¨acklund transformation for minimal surfaces Xu Xv N Y X0 α TpX

Figure 4.1: A possible configuration for which Y ∈ TpX is orthogonal to X0.

Now, denote by Y ∈ TpX = TpX the vector that is orthogonal to X0 and by α the angle from Xuto Y as seen in Figure 4.1. We choose to measure the angle from Xu and not Xu for later convenience. Then,

Y , Xu = Y Xu cos α, Y , Xv = Y Xv sin α,

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 cos α Xu Xu+ sin α Xv Xv ! = Y e−θ(Xucos α + Xvsin α) .(4.10)

The condition that Y and X0 _{be orthogonal is then}

X0_{, Y}_{= 0 ⇐⇒ X}0_{, X}

u cos α + X0, Xv sin α = 0, (4.11) and the inner products in this equation fulfill the relation

∂vX0, Xu = X0v, Xu + X0, Xvu = − X0v, Xv + X0, Xvu (2.3)

= X0_{, X}

vu = X0, Xuv = ∂uX0, Xv . If we define the function φ(u, v) by

φu:= mX0, Xu , φv:= mX0, Xv , m ∈ R\ {0} , (4.12) then the former equation is the condition φuv = φvu. For later convenience, we shall define

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and calculate the second order derivatives of φ:

φuu = mX0u, Xu + m X0, Xuu (4.2) = mX0 u, Xu + m X0, θuXu −mX0_{, θ} vXv − ms (4.12) = mX0 u, Xu + θuφu− θvφv− ms (4.8) = mwe2θ+ θuφu− θvφv− ms, φuv = mX0u, Xv + m X0, Xuv = m X0u, Xu + m X0, Xuv (2.3) = mX0_{, X} uv (4.2) = mX0_{, θ} vXu + m X0, θuXv (4.13) (4.12) = θvφu+ θuφv, φvv = mX0v, Xv + m X0, Xvv (4.2) = mX0 v, Xv − m X0, θuXu +mX0_{, θ} vXv + ms (4.12) = mX0 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 of Y in terms of these equations. They can be written as

b X = X + t1 Y Y (4.10) = X + t1e−θ(Xucos α + Xvsin α) (4.11) = X + t2e−θ X0, Xv Xu−X0, 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 also recall 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 bX is the other focal surface, as depicted in Figure 4.2. First, however, we need to

Xu Xv N Y X0 TpX b Xu b Xv b N TpXb

Figure 4.2: The tangent planes to the focal surfaces X and bX in terms of the adjoint vectors Xu, Xv, N and bXu, bXv, bN .

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know the tangent vectors to bX. By differentiation,

b 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 m N + tue−2θ m (φvXu− φuXv) , b 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 m N + tve−2θ m (φvXu− φuXv) Since {Xu, Xv, N } span R3, for the normal bN to bX, we can put

b

N = a1Xu+ a2Xv+ bN,

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

b N , φvXu− φuXv = 0 ⇐⇒ a1φv= a2φu, so bN is of the form b N = a(φuXu+ φvXv) + bN, (4.15)

for some new function a depending on u and v. The two orthogonality conditions

b N , bXu = 0, N , bb X_v = 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 therefore neither 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 leads to bN being the null vector and the latter to bX ≡ X. Hence, by dividing the equations by φv and φu respectively and then subtracting the former from the

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

b X = X +e −2θ mw (φvXu− φuXv) , b Xu= − wuφve−2θ mw2 Xu+ e−2θ w s + wuφu mw Xv− φve−2θ mw N, (4.18) b Xv = e−2θ w s −wvφv mw Xu+ wvφue−2θ mw2 Xv− φue−2θ mw N.

We recall from Proposition 2.8.1 that the asymptotic lines on bX now correspond to the asymptotic lines on X. The asymptotic lines on X were its parametric lines, i.e. the curves along Xuand Xv, and by definition these curves correspond to the parametric lines on bX; the curves along bXuand bXv respectively that is. Hence the asymptotic lines on bX are its parametric lines, and thus a necessary and sufficient condition that bX be minimal is that these curves be orthogonal (cp. Theorem 3.3.1), i.e. b Xu, bXv = 0. This is equivalent to wuwv φ2u+ φ2v + msw (wvφu− wuφv) + φuφvw2e−2θ= 0 (4.7) ⇐⇒ e−2θ φ2u+ φ2v + msw φu ψu +φv ψv −φuφvw 2 ψuψv = 0. (4.19)

We shall examine this equation further by solving for ξ := φ − ψ. It then becomes e−2θ φ2u+ φ 2 v + 2msw − w 2 + w (ms − w) ξu ψu + ξv ψv − ξuξv ψuψv w2= 0. (4.20) Earlier we defined s(u, v) :=X0_{, N.} Hence su = X0u, N + X0, Nu (4.3) = X0 u, N + e−2θX0, Xu | {z } (4.12) =: φu/m (4.8) = −wue2θ+ φue−2θ m , (4.21) sv = X0v, N + X0, Nv (4.3) = X0 v, N − e−2θX0, Xv | {z } (4.12) =: φv/m (4.9) = wve2θ− φve−2θ m . (4.22)

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

f (u, v) := w (ms − w) ξu ψu + ξv ψv − ξuξv ψuψv w2

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and differentiating with respect to u and v respectively becomes

0 = 2 φuφuu+ φvφuv− θu φ2u+ φ 2 v 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+ wue2θ mwe2θ− ms + w e−2θ+ 2wu ms − mwe2θ− w + fu = 2ξu mwe2θ− ms + w e−2θ+ fu, 0 = 2 φuφuv+ φvφvv− θv φ2u+ φ 2 v 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 this case 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+ wu m ⇐⇒ s = −φ + w m + g(v) ⇐⇒ sv= −φv+ wv m + g 0_(v)(4.22)₌ _w ve2θ− φve−2θ m = −φv+ wv m ⇐⇒ g0(v) ≡ 0 ⇐⇒ g = const.,

for some arbitrary g, so s is also determined to within an additive constant. We therefore take s = −φ + w m, so that (4.20) reads e−2θ φ2_u+ φ2_v + 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

b X = X +e −2θ mw (φvXu− φuXv) , b Xu= − φuφve−4θ mw2 Xu+ e−2θ mw2 mφw − φ 2 ve−2θ Xv− φve−2θ mw N, (4.24) b Xv = e−2θ mw2 mφw − φ 2 ue −2θ_X u− φuφve−4θ mw2 Xv− φue−2θ mw N,

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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

b N = a (φuXu+ φvXv) + bN = a φuXu+ φvXv+ (w − mφ) e2θN . (4.25) The condition b N , bN = 1 is then equivalent to 1 = a2e2θ φ2_u+ φ2_v+ (w − mφ)2e2θ(II) = amφe2θ2 ⇐⇒ a = ±e −2θ mφ. (4.26)

The possible different signs on a correspond to the orientation of

n b

Xu, bXv, bN o

,

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

b N = Xbu× bXv bXu× bXv

with respect to a right-handed basis of R3_{. It is sufficient to calculate just one} component of the vectors in the left- and right-hand side of this equation. We do so using the basis {Xu, Xv, N }, while the inner product h·, ·i is the usual one, inherited from R3_{. Then}

bXu× bXv b N , Xu (4.25) = bXu× bXv aφue2θ=Xb_u× bX_v, X_u (4.24) = φue −2θ m2_w3 −mφw + φ 2 ve−2θ− φ 2 ve−2θ = −φ 2_e−2θ w2 e−2θ mφφue 2θ (4.26) ⇐⇒ a = −e −2θ mφ, bXu× bXv = φ2_e−2θ w2 . Using (II) in (4.24), we see that

bXu = bXv ,

so that, by the definition of the vector product

φ2_e−2θ w2 = bXu× bXv = bXu bXv bN sin π 2 = bXu 2 = bXv 2 .

Hence the transform is also isothermal, and therefore

bgij := b Xui, bX_uj =⇒ (bg_ij) = φ2e−2θ w2 1 0 0 1 .

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

b 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), b Nu= e−2θ mφ2 φ 2 u− mwφe 2θ_X u+ φuφve−2θ mφ2 Xv+ wφu mφ2N (4.24) = −w 2_e2θ φ2 Xbv, b Nv= φuφve−2θ mφ2 Xu+ e−2θ mφ2 φ 2 v− mwφe 2θ_X v+ wφv mφ2N (4.24) = −w 2_e2θ φ2 Xbu, so that b hij := − b Xui, bN_uj =⇒ (bhij) = 0 1 1 0 .

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

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

φu= wue2θ, φv = −wve2θ,

so with the help of (4.24),

b Xu = − bXv= e−2θ mw2 φ 2 ue−2θ− mφw Xu+ φuφve−4θ mw2 Xv+ φue−2θ mw N = −∂u φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw (φuu− 2θuφu− mφ) Xu +φue −2θ mw Xuu+ e−2θ mw (φuv− 2θuφv) Xv+ φve−2θ mw Xuv+ φue−2θ mw N (4.2) = −∂u φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw (φuu− θuφu+ θvφv− mφ) Xu +e −2θ mw (φuv− θuφv− θvφu) Xv (4.23) = −∂u φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw mwe 2θ_{− w X} u (4.3) = ∂u X −φue −2θ mw Xu− φve−2θ mw Xv− 1 mN ,

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4.2. Geometric construction 29 b Xv = bXu= − φuφve−4θ mw2 Xu+ e−2θ mw2 mφw − φ 2 ve −2θ_X v− φve−2θ mw N = −∂v φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw (φvv− 2θvφv+ mφ) Xv +φve −2θ mw Xvv+ e−2θ mw (φuv− 2θvφu) Xu+ φue−2θ mw Xuv− φve−2θ mw N (4.2) = −∂v φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw (φvv− θvφv+ θuφu+ mφ) Xv +e −2θ mw (φuv− θvφu− θuφv) Xu (4.23) = −∂v φue−2θ mw Xu+ φve−2θ mw Xv +e −2θ mw mwe 2θ_{+ w X} v (4.3) = ∂v X − φue −2θ mw Xu− φve−2θ mw Xv− 1 mN .

By integration, bX is determined to within an additive constant which we take to be zero. Hence, b X = X − 1 m φue−2θ w Xu+ φve−2θ w Xv+ N . (A)

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

N to bX is the same as for the adjoint transform. As a consequence, we see that the relation b X − φ wN = X −b φ wN

holds. Hence, at corresponding points, the normals meet in a point at an equal distance |φ/w| from each surface. As can be seen in Figure 4.3, at corresponding points the surfaces X and bX thus lie on a sphere of radius |φ/w| centered at X −_wφN = bX −_wφN . As both the radius and the center depend on u and v, web get 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 bX is called a Ribaucour transform of X [9, p. 175]. As u and v vary, the surfaces X and bX therefore envelop these spheres and are accordingly called the sheets of the envelope of these spheres.

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

(_bgij) = (bgij) = φ2e−2θ w2 1 0 0 1 =φ 2_e−4θ w2 (gij), b Nu= bNu= − w2_e2θ φ2 Xbv= w2_e2θ φ2 Xbu, b Nv= bNv= − w2_e2θ φ2 Xbu= − w2_e2θ φ2 Xbv. It then follows that

b hij := −Xbui, bN_uj =⇒ (bhij) = −1 0 0 1 = (hij),

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30 Chapter 4. A B¨acklund transformation for minimal surfaces O X −_wφN b X −_wφNb φ w

Figure 4.3: At corresponding points, the surfaces X and bX lie on a sphere of radius |φ/w| centered at X −_wφN = bX − _wφN .b

so the coefficients of the first and second fundamental form bF = cM = 0, and thus by Proposition 2.6.2 the parametric lines are the curvature lines. This was also the case for X, so the Ribaucour sphere congruence maps curvature lines of X to curvature lines of bX. If we define b θ := ln φ w − θ, then (_bgij) = e2bθ 1 0 0 1

which is of the same form as (gij), and thus bθ is yet another solution of the elliptic 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 in terms of geometrical concepts that highly relied on the theory of congruences. In this section, we will give a direct proof for the transformation describing new minimal surfaces parametrized by isothermal coordinates, which can be determined 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 be found from the very same PDEs. We summarize this in the following theorem:

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

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4.3. Algebraic proof 31

and parameters (u, v) ∈ Ω ⊂ R2_{chosen such that}

(gij) = 1 0 0 1 e2θ, (hij) = −1 0 0 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 coupled} system of PDEs

φu= wue2θ, φv= −wve2θ, (I) e−2θ φ2_u+ φ2_v + w2

− 2mφw = 0, (II)

for any m ∈ R\{0}. Then

b X = X − 1 m φue−2θ w Xu+ φve−2θ w Xv+ N (A)

is a new minimal surface parametrized by isothermal coordinates, and yet an-other solution of ∆θ = e−2θ is given by bθ := ln

φ w − θ.

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

wuv+ θuwv+ θvwu= 0,

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

φ(II)= φ 2 u+ φ 2 v 2mw e −2θ₊ w 2m (I) = w 2 u+ w 2 v 2mw e 2θ₊ w 2m, φu = wuu+ θuwu− w2 u+ w2v 2w + w 2e −2θ wue2θ mw + (θuwv+ wuv) wve2θ mw (4.6) = wuu+ θuwu− θvwv− w_u2+ w_v2 2w + w 2e −2θ wue2θ mw (I) = wue2θ ⇐⇒ wuu+ θuwu− θvwv− wu2+ wv2 2w + w 2e −2θ_{− mw = 0,} _(4.27) φv = wvv+ θvwv− w2 u+ wv2 2w + w 2e −2θ wve2θ mw + (θvwv+ wuv) wve2θ mw (4.6) = wvv+ θvwv− θuwu− w2u+ w2v 2w + w 2e −2θ wve2θ mw (I) = −wve2θ ⇐⇒ wvv+ θvwv− θuwu− w2u+ w2v 2w + w 2e −2θ_{+ mw = 0.} _(4.28)

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

wuu+ wvv− w2 u+ wv2 w + we −2θ_{= 0,} _(4.29) wuu− wvv+ 2 (θuwu− θvwv− mw) = 0. (4.30) By differentiating b X (A)= X − 1 m φue−2θ w Xu+ φve−2θ w Xv+ N (I) = X − 1 m w_u wXu− wv wXv+ N ,

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and using the Gauss and Weingarten equations (4.2) and (4.3) for X derived in Section 4.2, b Xu= 1 mw mw − wuu+ w2 u w Xu+ 1 mw wuv− wuwv w Xv− 1 mNu −wu mwXuu+ wv mwXuv = 1 mw mw − θuwu+ θvwv | {z } (4.30) = (wuu−wvv)/2 −wuu+ w2 u w − we −2θ Xu + 1 mw wuv+ θuwv+ θvwu | {z } (4.6) = 0 −wuwv w Xv+ wu mwN = 1 mw −wuu+ wvv 2 + w2 u w − we −2θ | {z } (4.29) = (w2 u−w2v)/(2w)−we−2θ/2 Xu− wuwv mw2Xv+ wu mwN = 1 2m w2 u− w2v w2 − e −2θ Xu− wuwv mw2Xv+ wu mwN, b Xv = − 1 mw wuv− wuwv w Xu+ 1 mw mw + wvv− w2 v w Xv− 1 mNv wv mwXvv− wu mwXuv = − 1 mw wuv+ θuwv+ θvwu | {z } (4.6) = 0 −wuwv w Xu + 1 mw mw − θuwu+ θvwv | {z } (4.30) = (wuu−wvv)/2 +wvv− w2_v w + we −2θ_X v+ wv mwN =wuwv mw2Xu+ 1 mw wuu+ wvv (2w) − w2 v w + we −2θ | {z } (4.29) = (w2 u−wv2)/2+we−2θ/2 Xv+ wv mwN =wuwv mw2Xu+ 1 2m w2 u− w2v w2 + e −2θ_X v+ wv mwN.

Since hXu, Xui = hXv, Xvi = e2θ, hN, N i = 1 and {Xu, Xv, N } being orthogo-nal, the inner products are

h bXu, bXvi = wuwv 2 (mw)2 w2 u− w2v w2 − e −2θ e2θ − wuwv 2 (mw)2 w2 u− wv2 w2 + e −2θ e2θ+ wuwv (mw)2 = 0, h bXv, bXvi − h bXu, bXui = 1 (2m)2 4w 2 u− w2v w2 e −2θ e2θ+w 2 v− w2u (mw)2 = 0,

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4.3. Algebraic proof 33

so the parametrization is indeed isothermal. Continuing,

b Xuu= ∂u+ θu 2m w2 u− wv2 w2 − e −2θ +we −2θ_{− θ} vwv mw2 wu Xu − θv 2m w2 u− w2v w2 − e −2θ +∂u+ θu m wuwv w2 Xv + ∂u wu mw− 1 2m w2 u− w2v w2 − e −2θ N, b Xvv= ∂v+ θv m wuwv w2 − θu 2m w2 u− wv2 w2 + e −2θ Xu + ∂v+ θv 2m w2 u− w2v w2 + e −2θ₊θuwu− we−2θ mw2 wv Xv + ∂v wv mw+ 1 2m w2 u− w 2 v w2 + e −2θ_N, ∆ bX = bXuu+ bXvv = 1 m ∂u 2 w2 u− wv2 w2 − e −2θ − θue−2θ+ wu w e −2θ_{+ ∂} v wuwv w2 Xu +1 m ∂v 2 w2 u− wv2 w2 + e −2θ + θve−2θ− wv w e −2θ_{− ∂} u wuwv w2 Xv +1 m ∂u wu w + ∂v wv w + e −2θ_N = 1 mw wuu+ wvv− w2 u+ wv2 w + we −2θ | {z } (4.29) = 0 _w u wXu− wv wXv+ N = 0,

so by Theorem 3.2.1, bX describes minimal surfaces parametrized by isothermal coordinates. We shall continue by proving that bθ as defined by

b θ := ln φ w − 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+ wvv w − w2 u+ w2v w2 = ∂u w_u w + ∂v w_v w = ∆ ln |w|, so by the linearity of ∆, ∆bθ = ∆ ln φ w − θ = ∆ ln |φ| − ∆ ln |w| − ∆θ = ∆ ln |φ| = φuu+ φvv φ − φ2 u+ φ2v φ2 (I) = (wuu+ 2θuwu− wvv− 2θvwv) e 2θ φ − φ2 u+ φ2v φ2 (4.30) = 2mwe 2θ φ − φ2u+ φ2v φ2 (II) = w 2_e2θ φ2 = e −2(ln_|φ w|−θ) = e−2bθ_.

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4.4 Example: catenoid

In Example 3.2.1 we saw that the catenoid is a minimal surface. We shall show that 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 =   cosh u cos v cosh u sin v u  , Xu=   sinh u cos v sinh u sin v 1  , Xv =   − cosh u sin v cosh u cos v 0  , N = 1 cosh u   − cos v − sin v sinh u  , Nu= 1 cosh2u   sinh u cos v sinh u sin v 1  , Nv = 1 cosh u   sin v − cos v 0  . gij=Xui, X_uj =⇒ (gij) = cosh2u 1 0 0 1 , hij= −Xui, N_uj =⇒ (h_ij) = −1 0 0 1 ,

which is of the same form as in (4.1) iff e2θ= cosh2u ⇐⇒ θ = ln cosh u. We continue by solving (4.6) with respect to w, which gives

wuv+ θuwv+ θvwu= 0 ⇐⇒ ∂uwv+ wvtanh u = 0 ⇐⇒ eR tanh udu∂uwv+ eR tanh uduwvtanh u = 0 ⇐⇒ ∂u wveR tanh udu = 0 ⇐⇒ ∂u wveln(cosh u) = 0 ⇐⇒ wv= d0(v) cosh u ⇐⇒ w = d(v) + e(u) cosh u , wu= e0− (e + d) tanh u cosh u

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

0(II)= ∂v e−2θ φ2_u+ φ2_v w + w − 2mφ ! (I) = ∂v e2θ _w2 u+ w2v w + w − 2mφ ! = cosh u ∂v

e02_{+ d}02_{− 2(e + d)e}0_{tanh u + (e + d)}2_tanh2 u e + d + d 0 cosh u + 2md 0_{cosh u = cosh u ∂} v e02_{+ d}02 e + d + (2m + 1)d

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

e02+ d02= (2m + 1)(e2− d2) + 2c1(e + d), c1∈ R, (4.31) for nonvanishing e + d. Differentiating with respect to u and v give

e00− (2m + 1)e − c1= 0, (4.32)

d00+ (2m + 1)d − c1= 0, (4.33)

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4.4. Example: catenoid 35

Solving these ODE:s yield

e =          a1cos √ −2m − 1u + a2sin √ −2m − 1u − c1 2m + 1 if m < − 1 2, c1 2u 2_{+ a} 3u + a4 if m = −1₂, a5cosh √ 2m + 1u + a6sinh √ 2m + 1u − c1 2m + 1 if m > − 1 2, d =          b1cosh √ −2m − 1v + b2sinh √ −2m − 1v + c1 2m + 1 if m < − 1 2, c1 2v 2_{+ b} 3v + b4 if m = −1₂, b5cos √ 2m + 1v + b6sin √ 2m + 1v + c1 2m + 1 if m > − 1 2,

for constants ai, bj. When using these together with (4.31), we get the algebraic relations a2₁+ a2₂− b2 1+ b 2 2= 0, a23+ b 2 3− 2c1(a4+ b4) = 0, −a2 5+ a 2 6+ b 2 5+ b 2 6= 0.

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

b X = X − 1 m   e0 e + dXu− d0 e + dXv− cosh u   cos v sin v 0    .

Taking some different values on m and ai, bj, c1satisfying the algebraic relations above, we get a set of new minimal surfaces according to this parametrization:

b X|m=−1= − 1 cos u + cosh v  

cosh u sin v sinh v − cos v sin u sinh u sin u sinh u sin v + cosh u cos v sinh v

sin u  +   0 0 u  , if a1= b1= 1, a2= b2= 0, c1∈ R, b X|_m=−1 2 = 4 u2_{+ v}2  

u cos v sinh u + v cosh u sin v u sinh u sin v − v cosh u cos v

u  −   cosh u cos v cosh u sin v −u  , if a3= a4= b3= b4= 0, c1= 1, b X|m=1= − √ 3 cosh√3u + cos√3v  

sinh√3u cos v sinh u − sin√3v cosh u sin v sin√3v cosh u cos v + sinh√3u sinh u sin v

sinh√3u   +   2 cosh u cos v 2 cosh u 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 to be compared with the original catenoid in Figure 3.1. To generate the transforms and the pictures of them, we have used a Matlab script which also verifies that the transforms are isothermal and that their Laplacians are zero, hence giving minimal surfaces as expected. The source code can be found in Appendix A.

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Figure 4.4: Part of the transform bX|m=−1 of the catenoid, using initial values a1= b1= 1, a2= b2= 0, c1∈ R.

Figure 4.5: Part of the transform bX|_m=−1

2 of the catenoid, using initial values

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

Figure 4.6: Part of the transform bX|m=1 of the catenoid, using initial values a5= b5= 1, a6= b6= 0, b4= 0, c1∈ R.

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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 Parameters on 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 by Ribaucour Transformations, Geom. Dedic. 96 (2003), pp. 117–150.

[5] L.P. Eisenhart, Surfaces with Isothermal Representation of Their Lines of Curvature 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. K¨uhnel, Differential geometry: curves - surfaces - manifolds, 2nd ed., American Mathematical Society, Providence, R.I., 2006.

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

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Appendix A

A Matlab script

The source code for the Matlab script catTrans can be found here below. To work properly, it requires the Symbolic Math Toolbox. When saved, the file should 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);

else

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

end end

%% 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.

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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 else

fprintf('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);

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Bäcklund transformations for minimal surfaces

Thesis

B¨

acklund transformations for minimal surfaces

Per B¨

ack

B¨

acklund transformations for minimal surfaces

Abstract

Acknowledgements

Nomenclature

Symbols

Abbreviations

Contents

Chapter 1

Introduction

1.1

Background

1.2

Topics covered

Chapter 2

Differential geometry of

surfaces

2.1

Preliminaries

2.2

The Gauss and Weingarten maps

2.3

Fundamental forms

2.4

Curvature

2.5

The Gauss and Weingarten equations

2.6

Curves on the surface

2.7

Infinitesimal deformations

2.8

Line congruences and focal surfaces

Chapter 3

Minimal surfaces

3.1

Preliminaries

3.2

Isothermal representation

3.3

Asymptotic line representation

3.4

Adjoint surface

Chapter 4

A B¨

acklund transformation

for minimal surfaces

4.1

Preliminaries

4.2

Geometric construction

4.3

Algebraic proof

4.4

Example: catenoid

Bibliography

Appendix A

A Matlab script

A.1

catTrans