Thesis. Bäcklund transformations for minimal surfaces. LiTH-MAT-EX--2015/04--SE

Transcription

1 Thesis Bäcklund transformations for minimal surfaces Per Bäck LiTH-MAT-EX--015/04--SE

2

3 Bäcklund transformations for minimal surfaces Department of Mathematics, Linköping University Per Bäck LiTH-MAT-EX--015/04--SE Master s thesis: 30 hp Level: A Supervisor: Jens Hoppe, Department of Mathematics, Royal Institute of Technology Examiner: Joakim Arnlind, Department of Mathematics, Linköping University Stockholm: September 015

4 iv

5 Abstract In this thesis, we study a Bäcklund transformation for minimal surfaces surfaces 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: Bäcklund transformations, Liouville equation, minimal surfaces, Ribaucour transformations, Thybaut transformations. Bäck, 015. v

6 vi

7 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 adventures. 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 subject, 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. Bäck, 015. vii

8 viii

9 Nomenclature Most of the recurring abbreviations and symbols are described here. Symbols Vector product, Inner product C n Space of n times continuously differentiable functions C n Complex n-space i Partial differentiation with respect to u i δ jk Kronecker delta Flat Laplacian in R n det Determinant g ij Matrix with elements g ij Ω Open subset of R R n Euclidean n-space tr Trace, i.e. the sum of the diagonal entries of a matrix X Norm of X induced by the inner product X Vector in an or map between inner product spaces X i i:th component of X as a vector X u i Partial derivative of X with respect to u i Abbreviations iff If and only if ODE Ordinary differential equation ON Orthonormal PDE Partial differential equation Bäck, 015. ix

10 x

11 Contents 1 Introduction Background Topics covered Differential geometry of surfaces 3.1 Preliminaries The Gauss and Weingarten maps Fundamental forms Curvature The Gauss and Weingarten equations Curves on the surface Infinitesimal deformations Line congruences and focal surfaces Minimal surfaces Preliminaries Isothermal representation Asymptotic line representation Adjoint surface A Bäcklund transformation for minimal surfaces Preliminaries Geometric construction Algebraic proof Example: catenoid A A Matlab script 39 A.1 cattrans Bäck, 015. xi

12 xii Contents

13 Chapter 1 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 50 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 Bäcklund transformation, also carries with it mappings between solutions of the elliptic Liouville equation θ uu + θ vv = e θ. 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 transformation which we have not found elsewhere, and an example using it, transforming the catenoid into a family of new minimal surfaces. 1. Topics covered There are three chapters apart from this introduction and one appendix. The main topics dealt with are: Chapter : Introduction to general surfaces in R 3. Chapter 3: Definition of what a minimal surface is and derivations of some important results related to them. Bäck,

14 Chapter 1. Introduction Chapter 4: Derivation of a Bäcklund 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 aforementioned transformation.

15 Chapter 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..1 Preliminaries Let us start by defining what we mean by a surface. Definition.1.1 Parametric surface. A parametric surface is taken to be an immersion X : Ω R 3 where Ω is an open set of R, 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 C 3 Ω. Thus, a parametric surface is a kind of representation X of the surface in R 3, and we will exclusively refer to this representation when speaking of surfaces. We write the components of X as X 1 u, v Xu, v = X u, v, X 3 u, v and the partial derivatives as X u i := X u i := X 1 u i X u i X 3 u i, i = 1,, with u 1, u =: u, v. Demanding that X be an immersion is then equivalent to having X u p and X v p being linearly independent at all points p Ω [8], and they therefore span the tangent plane T p X at all points. Moreover, the vector product X u X v does not vanish, so the normal vector field N = X u X v X u X v Bäck,

16 4 Chapter. Differential geometry of surfaces is well defined, denoting the usual norm on R 3. Since N = 1, we can view this as a map from Ω to the unit sphere S = { x, y, z R 3 : x + y + z = 1 }. This map bears the name after the mathematician who first employed it, namely Gauss.. The Gauss and Weingarten maps We start as we did in the last section, by a definition. Definition..1 Gauss map. For a surface X : Ω R 3, the Gauss map is defined as N : Ω S R 3 N := X u X v X u X v, 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 Xp, but have instead moved it in terms of a translation to the origin of space as seen in Figure.1. Figure.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.. Self-adjoint map. Let T,, be a finite dimensional real or complex inner product space. A linear map A: T T is called self-adjoint if 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 for constructing bases of inner product spaces. Theorem..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 {E 1, E } of T consisting of eigenvectors of A. Moreover,

17 .. The Gauss and Weingarten maps 5 the matrix of A in the eigenbasis is diagonal and consists of the corresponding, necessarily real, eigenvalues λ 1 and λ λ 1, which are given by { } AV, V λ 1 = min V, V : V T, V 0, { } AV, V λ = max V, V : V T, V 0. Proof. See [, p. 16]. We recall from the previous section that at every point p Ω, the tangent vectors X u p and X v p to X : Ω R 3 provide a basis of the tangent plane T p X. Hence, any vector V T p X can be written as V = V 1 X u p + V X v p = i V i X u ip =: V i X u ip, where we in the last step have deployed the Einstein summation convention, implying summation over repeated indices. We will continue to use this convention 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..3 Weingarten map. For a surface X : Ω R 3 with unit normal vector field N := X u X v X u X v we define the Weingarten map S : T p X T p X at a point p Ω for arbitrary vectors V T p X written as V = V i X u ip via SpV := V i N u ip. Using the inner product, inherited by the ambient space R 3, N, N = 1 by definition. Differentiating this then yields N, N u i = 0, so either N u ip lie in T p X, or in a plane parallel to it which can be identified with T p X. Moreover, since the Weingarten map clearly is linear, using the property N, X u j = 0, we can deduce that the Weingarten map is self-adjoint in T p X. First, by differentiation we have N u i, X u j + N, X ui u j = 0 N u i, X u j = N, X u i u j = N, X u j u i = N u j, X u i = X u i, N u j, so for arbitrary V, W T p X written in the basis {X u, X v } as V = V i X u i W = W j X u j and SV, W = V i W j N u i, X u j = V i W j X u i, N u j = V i X u i, W j N u j = V, SW. Here, we have dropped the p for the sake of brevity.

18 6 Chapter. Differential geometry of surfaces.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 T p X as IV, W := V, W, IIV, W := SV, W, IIIV, W := SV, SW. These can in turn be used for defining three quadratic forms called the first, second and third fundamental form: IV := V, V, IIV := SV, V, IIIV := SV, SV. The first fundamental form is sometimes also called the metric..4 Curvature The quotient κ n V = IIV IV is called the normal curvature, and the minimum and maximum of this are defined as the principal curvatures κ 1 and κ of the surface, { } IIV κ 1 := min IV : V T px, V 0, { } IIV κ := max IV : V T px, V 0. Hence, by Theorem..1, κ 1 and κ are by definition the eigenvalues of the Weingarten map, and the directions of the corresponding eigenvectors E 1 and E are therefore referred to as the principal directions. By the same theorem, {E 1, E } constitutes an ON-basis of T p X, and in this basis the matrix representing S is κ1 0 S =. 0 κ Changing S to this basis can always be done by a similarity transformation S E 1 SE where the columns of E are the orthonormal eigenvectors E 1 and E 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.4.1 Gauss and mean curvature. The functions K := det S = κ 1 κ, H := tr S = κ 1 + κ are called the Gauss curvature and the mean curvature respectively. For computational reasons however, it is often convenient to work in the basis {X u, X v }. As before, we write arbitrary V T p X as V = V i X u i in this basis, and the first and second fundamental form are therefore IV = V, V = V i V j X u i, X u j, IIV = SV, V = V i V j X u i, N u j.

19 .4. Curvature 7 On account of this, g ij := X u i, X u j, h ij := X u i, N u j, are, for each i, j {1, } called the coefficients of the first and second fundamental form. They are also denoted by the letters E := X u, X u, F := X u, X v = X v, X u, G := X v, X v, L := X u, N u = X uu, N, N := X v, N v = X vv, N, M := X u, N v = X v, N u = X uv, N = X vu, N, so in matrix form g11 g g ij = 1 E F h11 h =, h g 1 g F G ij = 1 L M =. h 1 h M N If we denote the inverse of g ij by g ij, so that g ij g 11 g = 1 1 G F g 1 g = EG F, F E the matrix of the Weingarten map in the basis {X u, X v } is equal to [7] S = S i j = g ik g h 11 h kj = 11 + g 1 h 1 g 11 h 1 + g 1 h g 1 h 11 + g h 1 g 1 h 1 + g. h Hence, we can calculate the Gauss and mean curvature as K = det S = det g ik deth ij LN M h kj = = detg ij EG F, H = tr S = gij h ij = LG + N E MF EG F. Example.4.1 -sphere. We shall compute the Gauss and mean curvature of the -sphere of radius r, X = X 1, X, X 3 R 3 : X = r. We parametrize it by Xu, v = X1 u, v sin u cos v X u, v = r sin u sin v, X 3 u cos u cos u cos v sin u sin v X u u, v = r cos u sin v, X v u, v = r sin u cos v, sin u 0 Nu, v = X sin u cos v u X v u, v = sin u sin v, X u X v cos u cos u cos v sin u sin v N u u, v = cos u sin v, N v u, v = sin u cos v, sin u 0

20 8 Chapter. Differential geometry of surfaces for u 0, π and v 0, π. 1 0 g ij = X u i, X u j = g ij = r 0 sin, u 1 0 h ij = X u i, N u j = h ij = r 0 sin = 1 u r g ij. Hence, S = g ik 1 h kj = r where I is the identity matrix, so g ik g kj = 1 r I, 1 K = det S = det r I = 1 r, H = tr S = tr I r = 1 r..5 The Gauss and Weingarten equations When we defined the Weingarten map in Definition..3, we saw that N u ip T p X. We also recall from Section.1 that demanding that X be an immersion is equivalent to having the tangent vectors X u p and X v p being linearly independent at all points p, and that they therefore span all tangent planes T p X. It is therefore natural to seek the expression for N u ip in terms of X u p and X v p, as we shall now do. We start by setting N u i = a k i X u k, for coefficients ai k. By taking the inner product with X uj, we get Nu i, X u j = a k i Xu k, X u j hij = a k i g kj, and by multiplying by the inverse and summing over j as well, h ij g jl = ai k g kj g jl = ai k δk l = ai l, where δk l is the Kronecker delta. Substituting this in our first expression, we have found the Weingarten equations N u i = h ij g jk X u k. Apart from X u p and X v p, at each point we also have access to Np which spans the orthogonal complement T p X. Hence, at each point we have at our disposal a basis of R 3. We shall try to express the second order derivatives of X in this basis, thus putting X ui u j = Γk ijx u k + b ij N.1 for coefficients Γ k ij and b ij. The coefficients Γ k ij of the second kind, while Γ ijk := g il Γ l jk are called Christoffel symbols

21 .6. Curves on the surface 9 are called Christoffel symbols of the first kind. By taking the inner product with X u l in.1 we get Xui u j, X u = Γ k l ij g kl = Γ lij, and hence we have the symmetry relations Γ lij = Γ lji, Γ l jk = Γ l kj. Introducing the shorthand notation by differentiation k := u k, k g ij = k Xu i, X u j = Xu i u k, X u j + Xu i, X u j uk = Γjik + Γ ijk, so using the symmetry relation of the Christoffel symbols of the first kind, k g ij + j g ik + i g kj = Γ jik Γ ijk + Γ kij + Γ ikj + Γ jki + Γ kji = Γ kij = g kl Γ l ij. Multiplying by g mk and dividing by two, we get 1 gmk k g ij + j g ik + i g kj = g mk g kl Γ l ij = δ m lγ l ij = Γ m ij. At last, taking the inner product with N in.1, we see that so that which are called the Gauss equations. b ij = X u i u j, N = h ij, X u i u j = Γk ijx u k + h ij N,.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.6.1 Parametric lines. The curves on a surface X along the direction of X u and X v 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.6. Curvature lines. The curves on a surface in the principal directions are called the curvature lines.

22 10 Chapter. Differential geometry of surfaces We recall from Section.4 that the principal directions were the directions along the eigenvectors of the Weingarten map, and that the corresponding eigenvalues were known as the principal curvatures, hence the name curvature lines. Definition.6.3 Asymptotic lines. The directions of V T p X for which the normal curvature κ n 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 sufficient conditions are for the parametric lines to be the curvature or asymptotic lines. Proposition.6.1 Asymptotic parametric lines. The asymptotic lines are parametric if and only if the fundamental coefficients L = N = 0. Proof. See [6]. Proposition.6. Parametric curvature lines. The lines of curvature are parametric if and only if the fundamental coefficients F = M = 0. Proof. See [6]..7 Infinitesimal deformations Starting with a parametrized surface Xu, v, we can obtain a new surface X u, v by deforming the former in the direction of a vector X u, v by setting X := X + ɛx, ɛ R. The tangent lines to X are called the generatrices of the deformation, and X itself does also correspond to a parametrized surface. Since by taking the inner product with X u j where g ij, g ij forms. If X u = X i u i + ɛx ui, ɛ R,. we obtain g ij = g ij + ɛ X u i, X u j + X u j, X u i + ɛ g ij, and g ij are the corresponding coefficients of the first fundamental X u i, X u + X j u j, X ui = 0.3 and ɛ be taken so small that ɛ may be neglected, X and X are seen to be isometric. 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 to determining X by means of.3, which in turn is equivalent to X u, X u = 0, X v, X v = 0, X u, X v = X v, X u =: w EG F. Here, we have in accordance with Eisenhart [6, p. 374] defined the characteristic function wu, v of the infinitesimal deformation. As usual, E, F and G are the

23 .8. Line congruences and focal surfaces 11 coefficients of the first fundamental form of X. From this, one can also deduce [6, pp ] the following expressions: Lwv Mw u N v K wu Mw v + u EG F K FM EN GL = w.4 EG F EG F X u = L wn v Nw v M wn u Nw u K EG F X v = M wn.5 v Nw v N wn u Nw u K EG F The former equation is called the characteristic equation, and once we have obtained the characteristic function w from it, we can also get X 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..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.8.1 Line congruence. A two parameter family of straight lines in space is called a line congruence. Let Ω R with u, v Ω, X : Ω R 3 and R: Ω S where S is the unit sphere in R 3, i.e. S = { x, y, z R 3 : x + y + z = 1 }. Then, we can define a line congruence as Cu, v, t = Xu, v + tru, 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.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 S to the surface X : Ω R 3, we can describe it as C N u, v, t = Xu, v + tnu, 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.8. Focal surfaces. For a line congruence the two surfaces Cu, v, t = Xu, v + tru, v, F i u, v := Cu, v, t = t i = Xu, v + t i Ru, v, i = 1,, with common tangent lines belonging to the line congruence are, when they exist, called focal surfaces.

24 1 Chapter. Differential geometry of surfaces Hence, when two focal surfaces exist, a line congruence define a map F 1 F and its inverse F F 1 between them. Definition.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.8.1 Construction of W-congruences. The tangent lines to a surface which are perpendicular to the generatrices of an infinitesimal deformation 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. 40]. This result is rather technical, but will hopefully become clearer within the last chapter where we make use of it.

25 Chapter 3 Minimal surfaces The theory of minimal surfaces originates with Lagrange, who in 1760 formulated 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 Minimal surface. A surface is called a minimal surface if and only if its mean curvature H Isothermal representation Verifying that a surface is minimal by calculating its mean curvature using the formula in Section.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..1 Isothermal coordinates. A surface X is said to be parametrized by isothermal coordinates, or in short, to be isothermal, if Xu, X u = Xv, X v, Xu, X v = 0. It is a remarkable fact that every surface as defined in Definition.1.1 can be parametrized by isothermal coordinates see e.g. Chern [3] for a proof, so we can always assume that g ij = X u i, X u j = λ δ ij, for some function λu, v, denoting by δ ij the Kronecker delta. Bäck,

26 14 Chapter 3. Minimal surfaces Theorem 3..1 Isothermal minimal surfaces. A surface X : Ω R 3 parametrized by isothermal coordinates is minimal if and only if it is harmonic in Ω, i.e. if its Laplacian be vanishing, Proof. Since X is isothermal, X := X uu + X vv 0. Xu, X u = Xv, X v, Xu, X v = 0. By differentiating the first expression with respect to u and the second with respect to v, Xuu, X u = Xvu, X v = Xu, X vv Xuu + X vv, X u = 0, and then differentiating the first expression with respect to v and the second with respect to u, Xvv, X v = Xuv, X u = Xv, X uu Xuu + X vv, X v = 0. This shows that X uu + X vv is orthogonal to both X u and X v and therefore parallel to N. Since the coefficients of the first fundamental form E = G, F = 0, LG + N E MF H = EG F = L + N Xuu, N + X vv, N = 0 E E X uu + X vv, N 0 X uu + X vv 0. Example 3..1 Catenoid. The catenoid can be constructed by rotating the catenary X = a cosh X 3 /a, a R >0 about the X 3 -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 X 3 = au, it can be parametrized by X 1 u, v cosh u cos v Xu, v = X u, v = a cosh u sin v, u R, v 0, π. X 3 u u Hence it follows that sinh u cos v cosh u sin v X u = a sinh u sin v, X v = a cosh u cos v, 1 0 cosh u cos v cosh u cos v X uu = a cosh u sin v, X vv = a cosh u sin v, 0 0 X u, X u = X v, X v = a cosh u, X u, X v = 0, X = X uu + X vv = 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.

27 3.3. Asymptotic line representation 15 X 3 X 1 a X = a cosh X3 a X Figure 3.1: The catenoid can be created by rotating the catenary X = a cosh X 3 /a about the X 3 -axis. 3.3 Asymptotic line representation Lemma Euler formula. Let W T p X be arbitrary, {E 1, E } an orthogonal basis of T p X consisting of eigenvectors of the Weingarten map and denote by α the angle from E 1 to W. Then the normal curvature of W is which is called the Euler formula. κ n W = κ 1 cos α + κ sin α, Proof. For arbitrary W T p X, we can always scale the eigenbasis {E 1, E } so that E 1 = E = 1/ W. Then we have the well-known relation W, E 1 = W E 1 cos α = cos α and similarly W, E = sin α. Since E 1, E = 0, W = W i E i = E 1 cos α + E sin α. The normal curvature of W is then κ n W = IIW IW = SW, W W, W = S E 1 cos α + E sin α, E 1 cos α + E sin α E 1 cos α + E sin α, E 1 cos α + E sin α = κ 1 cos α + κ sin α. Theorem 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 κ 1 κ. Let V T p X be any vector. If κ 1 κ = 0 for some points, then κ 1 := min κ n V = max κ n V = 0, so κ n V = 0 for all vectors V T p X at such points, i.e. all directions are asymptotic directions. On the other hand, if κ 1 κ 0, then by the Euler formula κ n V = κ 1 cos α sin α = 0 α = π 4, 3π 4, 5π 4, 7π 4. As can be seen in Figure 3., there then exist four directions of V which are all orthogonal to one another, and therefore also two orthogonal asymptotic lines corresponding to these.

28 16 Chapter 3. Minimal surfaces π/ 3π/4 π/4 π 0 5π/4 7π/4 3π/ Figure 3.: 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..1 that a surface X : Ω R 3 is minimal if it satisfies X = 0, Xu, X u = Xv, X v, Xu, X v = 0, u, v Ω. For such a surface, we can form an adjoint surface X on Ω as to satisfying By differentiation, X u = X v, X v = X u. X = X uu + X vv = X vu X uv = 0, Xu, X u = Xv, X v = Xu, X u = Xv, X v, Xu, X v = Xu, X v = 0, so the adjoint surface X is also a minimal surface with the same first fundamental form as X. Example 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..1, so that the adjoint X should satisfy cosh u sin v X u = X v = a cosh u cos v, 0 sinh u cos v X v = X u = a 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 sinh u sin v X = a sinh u cos v v which is called the helicoid, depicted in Figure 3.3.

29 3.4. Adjoint surface 17 X 3 X 1 X Figure 3.3: The helicoid, an adjoint surface to the catenoid.

30 18 Chapter 3. Minimal surfaces

31 Chapter 4 A Bäcklund transformation for minimal surfaces In 1883, the Swedish mathematician Albert Victor Bäcklund 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 ϕ uv = 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 Bäcklund 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 Bäcklund s original transformation, it carries with it a solution of the elliptic Liouville equation θ uu + θ vv = e θ in terms of a system of PDEs based on an already known solution; hence we recognize it as a Bäcklund 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 that a surface is minimal iff there exist two orthogonal asymptotic lines at each of its points. By Definition.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.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 Bäck,

32 0 Chapter 4. A Bäcklund 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 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 X orthogonally, which is simpler since the parametric lines are those that have X u, X v and X u, X v as tangent vectors on each surface respectively. As a last step, we transform back from the surface X to its adjoint minimal surface X. 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. Geometric construction Let X : Ω R 3 be a minimal surface with normal N := X u X v X u X v, where in accordance with Bianchi [1, p. 335], the parameters u, v Ω R are chosen such that 1 0 g ij = e θ 1 0, h 0 1 ij =, for some function θu, v. Since F = M = 0, we recall from Proposition.6. that the parametric lines are the curvature lines. Continuing, the Gaussian curvature is K = deth ij detg ij = e 4θ, and since g ij = e θ δ ij, the Christoffel symbols are found to be Γ 1 11 = θ u, Γ 1 1 = θ v, Γ 1 = θ u, Γ 11 = θ v, Γ 1 = θ u, Γ = θ v. By the Gauss equations, X uu = Γ k 11 X k + h 11 N = θ u X u θ v X v N, X vv = Γ k X k + h N = θ u X u + θ v X v + N, X uv = Γ k 1 X k + h 1 N = θ v X u + θ u X v, and by the Weingarten equations, { N u = h 1j g jk X u k = e θ X u, N v = h j g jk X u k = e θ X v

33 4.. Geometric construction 1 By assumption X C 3, so the condition for cross-differentiation for third order derivatives has to hold. By differentiation, so the condition X vuu = θ uv X u + θ u X uv θ vv X v θ v X vv N v = θ uv + θ u θ v X u + θ u θ v θ vv + e θ X v θ v N, X uuv = θ uv X u + θ v X uu + θ uu X v + θ u X uv = θ uv + θ u θ v X u + θ u θ v + θ uu Xv θ v N, X vuu = X uuv is equivalent to the elliptic Liouville equation θ = e θ. 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 X u = X v, X v = X u, and since N := X u X v X u X v = X v X u = X u X v =: N, 4.5 Xv X u Xu X v the second fundamental coefficients of X are L = X u, N u = X v, N u = M = 0, M = X u, N v = X v, N v = N = 1, N = X v, N v = X u, N v = M = 0. Hence, by Proposition.6.1, the parametric lines on X are its asymptotic lines, and they are orthogonal since g ij = g ij = e θ δ ij. We now wish to make an infinitesimal deformation of X, and recall that the surface X proportional to the direction of the deformation is completely determined by the characteristic function wu, v which is a solution of the characteristic equation.4. Since K = K = e 4θ, it takes the form v wu e θ + u wv e θ = 0 w uv + θ u w v + θ v w u = If we introduce the function ψu, v defined by ψ u := w u e θ, ψ v := w v e θ, is just the condition that ψ uv = ψ vu. When w is known from 4.6, X u X v.5 = wn u Nw u e θ 4.5 = wn u Nw u e θ 4.3 = we θ X u Nw u e θ, = Nw v wn v e θ 4.5 = Nw v wn v e θ 4.3 = we θ X v + Nw v e θ. 4.9

34 Chapter 4. A Bäcklund transformation for minimal surfaces N X X v T p X α Y X u Figure 4.1: A possible configuration for which Y T p X is orthogonal to X. Now, denote by Y T p X = T p X the vector that is orthogonal to X and by α the angle from X u to Y as seen in Figure 4.1. We choose to measure the angle from X u and not X u for later convenience. Then, Y, Xu = Y Xu cos α, Y, Xv = Y Xv sin α, so written in terms of X u and X v, Y = Y Xu + Y Xu = Y Xu + Y Xv = Y, Xu Xu X u + Y, Xv Xv X v = Y cos α Xu X u + sin α Xv X v = Y e θ X u cos α + X v sin α.4.10 The condition that Y and X be orthogonal is then X, Y = 0 X, X u cos α + X, X v sin α = 0, 4.11 and the inner products in this equation fulfill the relation v X, X u = X v, X u + X, X vu = X v, X v + X, X vu If we define the function φu, v by.3 = X, X vu = X, X uv = u X, X v. φ u := m X, X u, φv := m X, X v, m R\ {0}, 4.1 then the former equation is the condition φ uv = φ vu. For later convenience, we shall define su, v := X, N,

35 4.. Geometric construction 3 and calculate the second order derivatives of φ: φ uu = m X u, X u + m X, X uu 4. = m X u, X u + m X, θ u X u m X, θ v X v ms 4.1 = m X u, X u + θu φ u θ v φ v ms 4.8 = e θ + θ u φ u θ v φ v ms, φ uv = m X u, X v + m X, X uv = m X u, X u + m X, X uv.3 = m X, X uv 4. = m X, θ v X u + m X, θ u X v 4.1 = θ v φ u + θ u φ v, φ vv = m X v, X v + m X, X vv 4. = m X v, X v m X, θ u X u +m X, θ v X v + ms 4.1 = m X v, X v θu φ u + θ v φ v + ms 4.9 = e θ θ u φ u + θ v φ v + ms, 4.13 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 X Y 4.10 = X + t 1 = X + t Y 1 e θ X u cos α + X v sin α 4.11 = X + t e θ X, X v Xu X, X u Xv 4.1 = X + t e θ m φ vx u φ u X v = X + te θ m φ vx u φ u X v, 4.14 where t = tu, v has been defined in this way for later convenience. We also recall from Proposition.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 X is the other focal surface, as depicted in Figure 4.. First, however, we need to N X v T p X X v X u X T p X Y N X u Figure 4.: The tangent planes to the focal surfaces X and X in terms of the adjoint vectors X u, X v, N and X u, X v, N.

36 4 Chapter 4. A Bäcklund transformation for minimal surfaces know the tangent vectors to X. By differentiation, X u X v = X u + t ue θ m φ vx u φ u X v + te θ m θ uφ u X v θ u φ v X u + φ uv X u + φ v X uu φ uu X v φ u X uv 4. = X v + t ue θ m φ vx u φ u X v te θ m θ uφ v θ v φ u + φ uv X u + θ u φ u θ v φ v φ uu X v φ v N = ste θ wt + 1 X v φ vte θ = X v + t ve θ m m φ vx u φ u X v N + t ue θ m φ vx u φ u X v, + te θ m θ vφ u X v θ v φ v X u + φ vv X u + φ v X uv φ uv X v φ u X vv 4. = X u + t ve θ m φ vx u φ u X v te θ m θ uφ u θ v φ v + φ vv X u + θ u φ v + θ v φ u φ uv X v φ u N = ste θ + wt 1 X u φ ute θ m N + t ve θ m φ vx u φ u X v Since {X u, X v, N} span R 3, for the normal N to X, we can put N = a 1 X u + a X v + bn, for functions a 1, a and b depending on u and v. Then, since N is orthogonal to all vectors parallel to Y, by 4.14 N, φv X u φ u X v = 0 a1 φ v = a φ u, so N is of the form N = aφ u X u + φ v X v + bn, 4.15 for some new function a depending on u and v. The two orthogonality conditions then become N, Xu = 0, N, Xv = aφ v st e θ wt 1 bφ vte θ = 0, m aφ u st + e θ wt 1 bφ ute θ = 0. m 4.17 We consider the generic case when φ is a function of both u and v and therefore neither of φ u and φ v 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 N being the null vector and the latter to X X. Hence, by dividing the equations by φ v and φ u respectively and then subtracting the former from the

37 4.. Geometric construction 5 latter, we arrive at t = 1/w. Let us evaluate 4.14 and the equations for the corresponding tangent vectors using this value: X = X + e θ φ vx u φ u X v, X u = w uφ v e θ X u + e θ s + w uφ u X v φ ve θ N, 4.18 w X v = e θ s w vφ v X u + w vφ u e θ w X v φ ue θ N. We recall from Proposition.8.1 that the asymptotic lines on X now correspond to the asymptotic lines on X. The asymptotic lines on X were its parametric lines, i.e. the curves along X u and X v, and by definition these curves correspond to the parametric lines on X; the curves along X u and X v 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, X v = 0. This is equivalent to w u w v φ u + φ v + msw wv φ u w u φ v + φ u φ v w e θ = e θ φ u + φ φu v + msw + φ v φ uφ v w = ψ u ψ v ψ u ψ v We shall examine this equation further by solving for ξ := φ ψ. It then becomes e θ φ u + φ v + msw w ξu + w ms w + ξ v ξ uξ v w = 0. ψ u ψ v ψ u ψ v 4.0 Earlier we defined Hence su, v := X, N. s u = X u, N + X, N u 4.3 = X u, N + e θ X, X u }{{} 4.1 =: φ u/m 4.8 = w u e θ + φ ue θ m, 4.1 s v = X v, N X, N v = X v, N e θ X, X v }{{} 4.1 =: φ v/m 4.9 = w v e θ φ ve θ m. 4. We return to the investigation of 4.0, which, when setting ξu fu, v := w ms w + ξ v ξ uξ v w ψ u ψ v ψ u ψ v

38 6 Chapter 4. A Bäcklund transformation for minimal surfaces and differentiating with respect to u and v respectively becomes 0 = φ u φ uu + φ v φ uv θ u φ u + φ v e θ + m s u w + sw u ww u + f u 4.13 = φ u m we θ s e θ + m s u w + sw u ww u + f u 4.1 = φ u e θ ms + w e θ + w u ms e θ w + f u 4.7 = ξ u + w u e θ e θ ms + w e θ + w u ms e θ w + f u = ξ u e θ ms + w e θ + f u, 0 = φ u φ uv + φ v φ vv θ v φ u + φ v e θ + m s v w + sw v ww v + f v 4.13 = φ v m we θ + s e θ + m s v w + sw v ww v + f v 4. = φ v e θ + ms w e θ + w v ms + e θ w + f v 4.7 = ξ v w v e θ e θ + ms w e θ + w v ms + e θ w + f v = ξ v e θ + ms w e θ + f v. 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, Integrating 4.1 and 4., s u 4.1 = w u e θ + φ ue θ m φ u = w u e θ, φ v = w v e θ. I = φ u + w u m s v = φ v + w v m + g v 4. = w v e θ φ ve θ m g v 0 g = const., s = φ + w m + gv = φ v + w v m for some arbitrary g, so s is also determined to within an additive constant. We therefore take s = φ + w m, so that 4.0 reads and 4.13 Moreover, 4.18 now becomes e θ φ u + φ v + w mφw = 0, II φ uu = e θ + θ u φ u θ v φ v + mφ w, φ uv = θ v φ u + θ u φ v, 4.3 φ vv = e θ θ u φ u + θ v φ v mφ + w. X = X + e θ φ vx u φ u X v, X u = φ uφ v e 4θ X u + e θ mφw φ v e θ X v φ ve θ N, 4.4 X v = e θ mφw φ u e θ X u φ uφ v e 4θ X v φ ue θ N,

39 4.. Geometric construction 7 We return to the determination of the functions a and b introduced earlier. By 4.17, we find that b = amse θ = a w mφ e θ, so 4.15 assumes the form N = a φ u X u + φ v X v + bn = a φ u X u + φ v X v + w mφ e θ N. 4.5 The condition N, N = 1 is then equivalent to 1 = a e θ φ u + φ v + w mφ e θ II = amφe θ a = ± e θ mφ. The possible different signs on a correspond to the orientation of { Xu, X v, N}, 4.6 being left- or right-handed with respect to the vector product. We should take the canonical, right-handed, so that N X u X v = Xu X v with respect to a right-handed basis of R 3. 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 {X u, X v, N}, while the inner product, is the usual one, inherited from R 3. Then Xu X v 4.5 N, Xu = Xu X aφu v e θ = Xu X v, X u 4.4 = φ ue θ m w a = e θ mφ, Using II in 4.4, we see that = φ e θ w mφw + φ v e θ φ ve θ e θ mφ φ ue θ Xu X v = φ e θ w. Xu = Xv, so that, by the definition of the vector product φ e θ w = Xu X v = Xu Xv N sin π = Xu = Xv. Hence the transform is also isothermal, and therefore ĝ ij := Xu i, X u j = ĝij = φ e θ w

40 8 Chapter 4. A Bäcklund transformation for minimal surfaces We return to the consideration of the normal, which, when a now is known is N 4.5 = a φ u X u + φ v X v + w mφ e θ N = e θ mφ φ ux u + φ v X v + 1 w mφ N. Differentiating with respect to u and v respectively and making use of 4., 4.3 and 4.3, N u = e θ φ mφ u φe θ X u + φ uφ v e θ mφ N v = φ uφ v e θ mφ X v + wφ u mφ N 4.4 = w e θ φ Xv, X u + e θ mφ φ v φe θ X v + wφ v mφ N 4.4 = w e θ φ Xu, so that h ij := Xu i, N u j = ĥ ij = The transform X 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 X to X. We recall that so with the help of 4.4, X u φ u = w u e θ, φ v = w v e θ, = X v = e θ φ u e θ mφw X u + φ uφ v e 4θ φu e θ = u X u + φ ve θ X v + φ ue θ X uu + e θ φu e θ = u X u + φ ve θ X v 4. X v + φ ue θ N + e θ φ uu θ u φ u mφ X u φ uv θ u φ v X v + φ ve θ + e θ φ uu θ u φ u + θ v φ v mφ X u + e θ φ uv θ u φ v θ v φ u X v 4.3 φu e θ = u X u + φ ve θ X v 4.3 = u X φ ue θ + e θ X u φ ve θ X v 1 m N e θ w X u, X uv + φ ue θ N

41 4.. Geometric construction 9 X v = X u = φ uφ v e 4θ X u + e θ mφw φ v e θ X v φ ve θ N φu e θ = v X u + φ ve θ X v + e θ φ vv θ v φ v + mφ X v + φ ve θ X vv + e θ φu e θ = v X u + φ ve θ X v 4. φ uv θ v φ u X u + φ ue θ + e θ φ vv θ v φ v + θ u φ u + mφ X v + e θ φ uv θ v φ u θ u φ v X u 4.3 φu e θ = v X u + φ ve θ X v 4.3 = v X φ ue θ + e θ X u φ ve θ X v 1 m N e θ + w X v. X uv φ ve θ N By integration, X is determined to within an additive constant which we take to be zero. Hence, X = X 1 φu e θ m w X u + φ ve θ w X v + 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 that the relation X φ w N = X φ w N 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 X thus lie on a sphere of radius φ/w centered at X φ w N = X φ N. w As both the radius and the center depend on u and v, we get a two-parameter family of spheres called the Ribaucour sphere congruence recall from Definition.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 spheres and are accordingly called the sheets of the envelope of these spheres. Continuing, ĝ ij is also the same as for the adjoint transform, so ĝ ij = ĝ ij = φ e θ 1 0 w = φ e 4θ 0 1 w g ij, It then follows that N u = N u = w e θ Xv φ = w e θ φ Xu, N v = N v = w e θ φ ĥ ij := Xu i, N u j = ĥ ij = Xu = w e θ φ Xv. 1 0 = h 0 1 ij,

42 30 Chapter 4. A Bäcklund transformation for minimal surfaces X φ w φ w N φ w N X O Figure 4.3: At corresponding points, the surfaces X and X lie on a sphere of radius φ/w centered at X φ w N = X φ w N. so the coefficients of the first and second fundamental form F = M = 0, and thus by Proposition.6. 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 X. If we define θ := ln φ w θ, then ĝ ij = e θ which is of the same form as g ij, and thus θ is yet another solution of the elliptic Liouville equation θ = e θ 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 A Bäcklund transformation for minimal surfaces. Let X : Ω R 3 be a minimal surface with normal N := X u X v X u X v,

43 4.3. Algebraic proof 31 and parameters u, v Ω R chosen such that 1 0 g ij = e θ 1 0, h 0 1 ij =, 0 1 for some function θu, v. Then θ is a solution of the elliptic Liouville equation θ = e θ. Furthermore, let φu, v and wu, v be solutions of the coupled system of PDEs for any m R\{0}. Then φ u = w u e θ, φ v = w v e θ, I e θ φ u + φ v + w mφw = 0, II X = X 1 m φu e θ w X u + φ ve θ w X v + N is a new minimal surface parametrized by isothermal coordinates, and yet another solution of θ = e θ is given by θ := ln θ. Proof. We proved in the beginning of Section 4. that θ = e θ follows from this particular parametrization. Continuing, we also recall that I was the solution of 4.6, w uv + θ u w v + θ v w u = 0, which could be retrieved by applying the condition φ uv = φ vu to I. Some new conditions can be derived from these equations by differentiation: φ II = φ u + φ v e θ + w I = w u + wv m eθ + w m, φ u = w uu + θ u w u w u + wv + w wu e θ w e θ + θ uw v + w uv w ve θ 4.6 = w uu + θ u w u θ v w v w u + wv + w wu e θ w e θ I = w u e θ w uu + θ u w u θ v w v w u + wv + w w e θ = 0, 4.7 φ v = w vv + θ v w v w u + wv + w wv e θ w e θ + θ vw v + w uv w ve θ 4.6 = w vv + θ v w v θ u w u w u + wv + w wv e θ w e θ I = w v e θ w vv + θ v w v θ u w u w u + wv + w w e θ + = Adding 4.7 to 4.8 and subtracting 4.8 from 4.7 gives By differentiating φ w A w uu + w vv w u + wv + we θ = 0, w 4.9 w uu w vv + θ u w u θ v w v = φu e θ w X u + φ ve θ w I = X 1 wu m w X u w v w X v + N X A = X 1 m X v + N,

44 3 Chapter 4. A Bäcklund transformation for minimal surfaces and using the Gauss and Weingarten equations 4. and 4.3 for X derived in Section 4., X u = 1 w uu + w u X u + 1 w 4.30 = w uu w vv/ w uv w uw v w w u X uu + w v X uv = 1 θ u w u + θ v w v w }{{} uu + w u w we θ X u + 1 = 1 = 1 m w uv + θ u w v + θ v w }{{ u } 4.6 = 0 w uw v w w uu + w vv + w u w we θ }{{} 4.9 = wu w v /w we θ / w u wv e θ w X v + w u N X v 1 m N u X u w uw v X v + w u N X u w uw v X v + w u N, X v = 1 w v w uv w uw v w X vv w u X uv = 1 w uv + θ u w v + θ v w u w uw v }{{} w + 1 = w uw v X u + 1 = w uw v X u + 1 m 4.6 = 0 θ u w u + θ v w }{{ v } 4.30 = w uu w vv/ X u w vv w v X v 1 w m N v X u +w vv w v w + we θ X v + w v N wuu + w vv w v w w + we θ }{{} 4.9 = wu w v /+we θ / w u wv + e θ w X v + w v N. X v + w v N Since X u, X u = X v, X v = e θ, N, N = 1 and {X u, X v, N} being orthogonal, the inner products are X u, X v = w uw v w u wv w w u wv w uw v X v, X v X u, X u = 1 m w 4 w u wv w e θ e θ e θ + e θ e θ + w uw v = 0, e θ + w v w u = 0,

45 4.3. Algebraic proof 33 so the parametrization is indeed isothermal. Continuing, u + θ u w X uu = u w v m w e θ + we θ θ v w v w u θv w u w v m w e θ + u + θ u w u w v m w X v w u + u 1 w u w v m w e θ N, v + θ v w u w v X vv = m w θ u w u w v m w + e θ X u v + θ v w + u w v m w + e θ + θ uw u we θ w v + v + 1 w u w v m w + e θ N, X u w v X v X = X uu + X vv = 1 u w u w v m w e θ θ u e θ + w u w u w v w e θ + v w X u + 1 v w u w v m w + e θ + θ v e θ w v w u w v w e θ u w X v + 1 w u u m w + w v v w + e θ N = 1 w uu + w vv w u + w wu v + we θ }{{ w } w X u w v w X v + N = 0, 4.9 = 0 so by Theorem 3..1, X describes minimal surfaces parametrized by isothermal coordinates. We shall continue by proving that θ as defined by θ := ln φ w θ, is a new solution of θ = e θ, once a solution θ is known. We first note that θ 4.9 θ = e = w uu + w vv w u + wv w w = u wu w so by the linearity of, θ = ln φ w θ = ln φ ln w θ = ln φ = φ uu + φ vv φ φ u + φ v φ 4.30 = eθ φ u + φ v φ φ wv + v = ln w, w I = w uu + θ u w u w vv θ v w v e θ φ u + φ v φ φ II = w e θ φ = e ln φ w θ = e θ.

46 34 Chapter 4. A Bäcklund transformation for minimal surfaces 4.4 Example: catenoid In Example 3..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, π, cosh u cos v sinh u cos v cosh u sin v X = cosh u sin v, X u = sinh u sin v, X v = cosh u cos v, N = 1 cosh u u 1 cos v sinh u cos v sin v 1, N u = sinh u cosh sinh u sin v u 1 0, N v = 1 cosh u g ij = X u i, X u j = gij = cosh 1 0 u, 0 1 h ij = 1 0 X u i, N u j = hij =, 0 1 sin v cos v 0 which is of the same form as in 4.1 iff e θ = cosh u θ = ln cosh u. We continue by solving 4.6 with respect to w, which gives w uv + θ u w v + θ v w u = 0 u w v + w v tanh u = 0 e tanh udu u w v + e tanh udu w v tanh u = 0 u w v e tanh udu = 0 u w v e lncosh u = 0 w v = d v cosh u dv + eu w =, w u = e e + d tanh u cosh 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 e θ φ u + φ v = v + w mφ w I e θ wu + wv = v + w mφ w e + d e + de tanh u + e + d tanh u = cosh u v e + d + d e cosh u + + d md cosh u = cosh u v + m + 1d e + d It is readily verified that the only nontrivial solution of this is given by e + d = m + 1e d + c 1 e + d, c 1 R, 4.31 for nonvanishing e + d. Differentiating with respect to u and v give which hold for nonvanishing e and d respectively.. e m + 1e c 1 = 0, 4.3 d + m + 1d c 1 = 0, 4.33

47 4.4. Example: catenoid 35 Solving these ODE:s yield a 1 cos m 1u + a sin m 1u c 1 if m < 1 m + 1, c 1 e = u + a 3 u + a 4 if m = 1, a 5 cosh m + 1u + a 6 sinh m + 1u c 1 if m > 1 m + 1, b 1 cosh m 1v + b sinh m 1v + c 1 if m < 1 m + 1, c 1 d = v + b 3 v + b 4 if m = 1, b 5 cos m + 1v + b 6 sin m + 1v + c 1 if m > 1 m + 1, for constants a i, b j. When using these together with 4.31, we get the algebraic relations a 1 + a b 1 + b = 0, a 3 + b 3 c 1 a 4 + b 4 = 0, a 5 + a 6 + b 5 + b 6 = 0. According to A, the new minimal surfaces are then given by the transform X = X 1 e m e + d X u d e + d X v cosh u cos v sin v. 0 Taking some different values on m and a i, b j, c 1 satisfying the algebraic relations above, we get a set of new minimal surfaces according to this parametrization: cosh u sin v sinh v cos v sin u sinh u 1 X m= 1 = sin u sinh u sin v + cosh u cos v sinh v + 0 0, cos u + cosh v sin u u if a 1 = b 1 = 1, a = b = 0, c 1 R, u cos v sinh u + v cosh u sin v cosh u cos v X m= 1 = 4 u sinh u sin v v cosh u cos v u + v cosh u sin v, u u if a 3 = a 4 = b 3 = b 4 = 0, c 1 = 1, sinh 3u cos v sinh u sin 3v cosh u sin v 3 X m=1 = cosh 3u + cos sin 3v cosh u cos v + sinh 3u sinh u sin v 3v sinh 3u cosh u cos v + cosh u sin v, if a 5 = b 5 = 1, a 6 = b 6 = 0, c 1 R. u 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.

48 36 Chapter 4. A Bäcklund transformation for minimal surfaces Figure 4.4: Part of the transform X m= 1 of the catenoid, using initial values a 1 = b 1 = 1, a = b = 0, c 1 R. Figure 4.5: Part of the transform X m= 1 a 3 = a 4 = b 3 = b 4 = 0, c 1 = 1. of the catenoid, using initial values Figure 4.6: Part of the transform X m=1 of the catenoid, using initial values a 5 = b 5 = 1, a 6 = b 6 = 0, b 4 = 0, c 1 R.

49 Bibliography [1] L. Bianchi, Lezioni di geometria differenziale. Vol., nd ed., Enrico Spoerri, Pisa, [] M.P.D. Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, N.J., [3] S.S. Chern, An Elementary Proof of the Existence of Isothermal Parameters on a Surface, Proc. Am. Math. Soc , no. 5, pp [4] A.V. Corro, W. Ferreira, and K. Tenenblat, Minimal Surfaces Obtained by Ribaucour Transformations, Geom. Dedic , pp [5] L.P. Eisenhart, Surfaces with Isothermal Representation of Their Lines of Curvature and Their Transformations, Trans. Am. Soc , no., pp [6], A Treatise on the Differential Geometry of Curves and Surfaces, Ginn, Boston, [7] J. Hoppe, Unpublished notes. [8] W. Kühnel, Differential geometry: curves - surfaces - manifolds, nd ed., American Mathematical Society, Providence, R.I., 006. [9] C. Rogers and W.K. Schief, Bäcklund and Darboux transformations: geometry and modern applications in soliton theory, Cambridge University Press, New York, 00. Bäck,

50 38 Bibliography

51 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,a,b1,b,c1 fulfilling any of the three conditions in putconst. function X=catTransm,A1,A,B1,B,c1 syms u v real; [e,d]=putconstm,a1,a,b1,b,c1,u,v; X=paramCatm,d,e,u,v; mincheckx,u,v; end %% Checking if any of the three conditions are fulfilled. function [e,d]=putconstm,a1,a,b1,b,c1,u,v ifm<-1/ && A1ˆ + Aˆ -B1ˆ + Bˆ==0 e=a1*cosu*sqrt-*m-1+a*sinu*sqrt-*m-1-c1/*m+1; d=b1*coshv*sqrt-*m-1+b*sinhv*sqrt-*m-1+c1/*m+1; elseifm==-1/ && A1ˆ+Bˆ-*c1*A+B==0 e=c1/*uˆ + A1*u+A; d=c1/*vˆ + B1*v+B; elseifm>-1/ && -A1ˆ + Aˆ + B1ˆ + Bˆ==0 e=a1*coshu*sqrt*m+1+a*sinhu*sqrt*m+1-c1/*m+1; d=b1*cosv*sqrt*m+1+b*sinv*sqrt*m+1+c1/*m+1; else error'algebraic relations were not fulfilled.'; end end %% Parametrizing the transform. function X=paramCatm,d,e,u,v X1=[coshu*cosv;coshu*sinv;u]; X=simpX1-1/m*diffe,u*diffX1,u/e+d... -diffd,v*diffx1,v/e+d-coshu*[cosv;sinv;0]; end %% Verifying that the transform is minimal, calculating its curvature, %% and then plotting it. Bäck,

52 40 Appendix A. A Matlab script function mincheckx,u,v % Calculating tangent vectors and their derivatives. Xu=simpdiffX,u; Xv=simpdiffX,v; Xuu=simpdiffXu,u; Xvv=simpdiffXv,v; Xuv=simpdiffXu,v; % Calculating the normal. N=crossXu,Xv; N=simpN/normN; % Calculating the coefficients of the first fundamental form. E=simpdotXu,Xu; F=simpdotXu,Xv; G=simpdotXv,Xv; % Calculating the coefficients of the second fundamental form. e=simpdotn,xuu; f=simpdotn,xuv; g=simpdotn,xvv; % Calculating the mean curvature H and the Gauss curvature K. H=simpE*g+G*e-*F*f/*E*G-Fˆ; K=simpe*g-f*f/E*G-F*F; % Verifying isothermal coordinates and zero Laplacian. ifsimpe-g==0 && simpf==0 fprintf'isothermal coordinates. '; if simpxuu+xvv==0 fprintf'zero Laplacian; minimal surface!\n'; end else fprintf'not isothermal coordinates,'; ifh==0 fprintf' but zero mean curvature; minimal surface!\n'; else fprintf[' and mean curvature H=',charH,'.\n']; end end % Plotting the transform. fprintf['gaussian curvature K=',charK,... '.\nplotting the surface.\n']; ezmeshx1,x,x3; axis off; title''; colormap[0,0,0]; end %% Simplifying the simplify function. function S=simpX S=simplifyX,'Steps',50; end

53 LINKÖPING UNIVERSITY ELECTRONIC PRESS Copyright The publishers will keep this document online on the Internet or its possible replacement for a period of 5 years from the date of publication barring exceptional circumstances. The online availability of the document implies a permanent permission for anyone to read, to download, to print out single copies for your own use and to use it unchanged for any non-commercial research and educational purpose. Subsequent transfers of copyright cannot revoke this permission. All other uses of the document are conditional on the consent of the copyright owner. The publisher has taken technical and administrative measures to assure authenticity, security and accessibility. According to intellectual property law the author has the right to be mentioned when his/her work is accessed as described above and to be protected against infringement. For additional information about the Linköping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: Upphovsrätt Detta dokument hålls tillgängligt på Internet eller dess framtida ersättare under 5 år från publiceringsdatum under förutsättning att inga extraordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det ofärändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida: 015, Per Bäck Bäck,