Open access peer-reviewed chapter

The Generalized Weierstrass System in Three-Dimensional Euclidean Space

By Paul Bracken

Submitted: July 13th 2018Reviewed: November 21st 2018Published: January 25th 2019

DOI: 10.5772/intechopen.82631

Downloaded: 206

Abstract

In this chapter, some recent advances in the area of generalized Weierstrass representations will be given. This is an approach to the theory of surfaces in Euclidean three space. Weierstrass representations permit the explicit construction of surfaces in the designated space. The discussion proceeds in a novel and introductory manner. The inducing formulas for the coordinates of a surface are derived and important conservation laws are formulated. These lead to the inducing mechanism of a surface in terms of solutions to a system of two-dimensional Dirac equations. A set of fundamental forms as well as expressions for the mean and Gaussian curvatures are derived. The Cartan moving frame picture is also formulated to put everything in a broader perspective. A connection with the nonlinear sigma model is presented, which has important applications in physics. Some relationships are established between integrable systems and geometry by way of conclusion.

Keywords

  • metric
  • tensor
  • manifold
  • Weierstrass representation
  • curvature
  • evolution equation
  • Mathematics Subject Classification: 35Q51
  • 53A10

1. Introduction

The theory of immersions and deformations of surfaces has been an important area of study as far as classical differential geometry is concerned. An inducing mechanism for describing minimal surfaces imbedded in three-dimensional Euclidean space was first put forward by Enneper and Weierstrass in the nineteenth century [1]. Their basic ideas have been extended and generalized by Konopelchenko and colleagues [2, 3, 4]. The connection between certain classes of constant mean curvature surfaces and the trajectories of an infinite-dimensional Hamiltonian system was put forward first by Konopelchenko and Taimanov [2], and has proved to be very useful in investigating types of questions related to this and other types of spaces and in higher dimensions [5, 6].

Surfaces and their dynamics play a very crucial and important role in a great number of phenomena which arise in the physical sciences in general. A longer introduction and more examples can be found in [7, 8]. They appear in the study of surface waves, shock waves, deformations of membranes, as well as in many problems in hydrodynamics connected with the motion of boundaries between regions of differing densities and viscosities. At the present time, they are appearing in string theory models [9, 10, 11] and in the study of integrable systems in general [12, 13]. A special case is that of surfaces which have zero mean curvature. These surfaces are usually referred to as minimal surfaces. The work of Weierstrass and Enneper originally concerned itself with the construction of minimal surfaces in three-dimensional Euclidean space [14, 15].

It is the intention here to present an introduction to the work of Konopelchenko and referred to presently as the generalized Weierstrass representation. The work presents both mathematical and physical developments in the area which should be relevant to both physicists and mathematicians. The development starts by studying a coupled system of two-dimensional Dirac equations in terms of two complex functions that involves a mass term that depends on two coordinates of the space. This equation can then be decomposed into a system of two simpler equations and their respective complex conjugates. By looking at such things as conservation laws, inducing formulas which specify the coordinates of a surface in Euclidean three space can be deduced, as well as the first and second fundamental forms pertaining to the surface. A remarkable result of this development is that the mass which appears in the Dirac system becomes related to the mean curvature of the surface. One might say this indicates that mass is a consequence of geometry in this type of model. To fit these developments in the larger picture of modern differential geometry, the Cartan moving frame for the system is formulated out of which emerges another remarkable result. Namely, the two-dimensional Dirac equation is a way of writing an affine connection on the surface. Finally, by investigating the Gauss map, it is shown that there is a mathematical way of proceeding from the Dirac system and the nonlinear sigma model in two dimensions [16, 17]. The whole construction leads to a very deep link between nonlinear evolution equations and geometry as a whole [18, 19]. The paper finishes with some interesting examples and outlook for further work.

2. Two-dimensional Dirac equation and construction of surfaces

The process of inducing surfaces in three-dimensional space can be generalized by establishing a system of Dirac equations in terms of a mass parameter and two complex valued functions called ψ1and ψ2. In Euclidean space in two dimensions, the Dirac equation can be written in terms of the set of Pauli matrices σμas follows:

Ψ=iσ1x+σ2yΨ+mΨ=0.E1

In (1), the mass term mhas been generalized to be a real function of xand y, which are the Cartesian coordinates of the space. Let us introduce two complex operators defined to be

=12xiy,¯=12x+iy.E2

In terms of a complex variable z=x+iy, we also define =/zand ¯=/z¯.A, and spinor wavefunction Ψis specified in terms of two components ψ1and ψ2as

Ψ=ψ1ψ2.E3

Using (2), the Dirac equation can be developed in terms of the two components of Ψand their complex conjugates to give the following coupled first-order system of equations:

¯ψ1=i2mψ2,ψ¯1=i2mψ¯2,ψ2=i2mψ1,¯ψ¯2=i2mψ¯1.E4

The Dirac equation in the form (4) leads to a variety of differential constraints. The first of which is given by

ψ1ψ¯1+ψ¯1ψ2=ψ1i2mψ¯2+ψ¯2i2mψ1=0,

as well as its complex conjugate equation. There is also the expression for a new real variable p

ψ¯1ψ2ψ2ψ¯1=i2mψ12+ψ22=i2mp,

and its complex conjugate. This also serves to define the real function P

p=ψ12+ψ22.E5

A system of conservation laws can also be formulated

ψ¯1ψ¯1ψ¯2¯ψ¯2=0,ψ¯1ψ2+ψ1¯ψ¯2=0,E6

as well as their complex conjugate equations. The complex quantity Sis defined as follows:

ψ¯2ψ1ψ1ψ¯2=i2pS,ψ¯1¯ψ2ψ2¯ψ1=i2pS¯.E7

Let Φbe the two-by-two matrix spinor given by

Φ=ψ1ψ¯2ψ2ψ¯1,E8

defining the real variable uso that pin (5) is given by p=eu, and it follows that

p=eu=det Φ=ψ12+ψ22.E9

Clearly, we have Φ Φ=pIand there follows another differential constraint:

pu=ψ¯1ψ1+ψ2ψ¯2.E10

Differentiating (8) exteriorly, we obtain that

dΦ=ψ1ψ¯2ψ2ψ1dz+¯ψ1¯ψ¯2¯ψ2¯ψ¯1dz¯.E11

Consequently, we find that

dΦΦ1=1pψ1ψ¯2ψ2ψ1ψ¯1ψ¯2ψ2ψ1dz+1p¯ψ1¯ψ¯2¯ψ2¯ψ¯1ψ¯1ψ¯2ψ2ψ1dz¯=122uiSim0dz+0imiS¯2¯udz¯.E12

Taking the derivative ¯of pSin (7) and substituting system (4), we obtain that

¯pS=mψ¯1ψ1+ψ¯2mψ2mψ2ψ¯2+ψ1mψ¯1=mψ¯1ψ1mψ2ψ¯2+pm.

It follows that

p1pS=mp1p+m=pp1m.

Let us summarize this as

p1¯pS=pp1m.E13

Proceeding in a similar fashion, we calculate the following two derivatives:

p1ψ1=1p2ψ12ψ1ψ1ψ2ψ¯2+ψ12+ψ22ψ1=ψ2p2ψ1ψ¯2+ψ¯2ψ1=i2Sp1ψ2,E14

and as well, we have

¯p1ψ2=1p2ψ1ψ2¯ψ¯1ψ22¯ψ2+ψ12¯ψ2+ψ22¯ψ2;=ψ1pψ2¯ψ¯1+ψ¯1¯ψ2=i2S¯p1ψ1.E15

It should be pointed out that the systems (14) and (15) are summarized here

p1ψ1=i2Sp1ψ2,¯p1ψ2=i2S¯p1ψ1.E16

By comparing with (4), it look very much like a Dirac system in their own right if Sis thought of as a mass variable. Another quantity, a current, was found in [20] and has the form

J=pS.

It is possible to construct a vector representation of Φas well. Amatrix such as Φrepresents a rotation matrix multiplied by a scaling in R3as follows V=viσiV=ΦVΦ. So the matrix Φcan be represented by means of a multiple of an orthogona1 3×3real matrix. The matrix elements can be found by using the inner product in V, namely V1V2=1/2Tr V1V2, then

ςij=12TrσiΦσjΦ.E17

ςijdefines a 3×3matrix which can be written down by using the usual representation of the Pauli matrices. In particular, the matrix formed out of the following combinations will be very useful:

ς+=12ς1iς2=12ψ12ψ¯22iψ12+ψ¯222ψ1ψ¯2,ς=12ς1+iς2=12ψ¯12ψ22iψ¯12+ψ222ψ¯1ψ2,ς3=ψ1ψ2ψ¯1ψ¯2iψ1ψ2ψ¯1ψ¯2ψ12ψ22.E18

In terms of matrices, ςand ςare represented as:

ς=12ψ12ψ¯22i2ψ12+ψ¯222ψ1ψ¯212ψ¯12ψ22i2ψ¯12+ψ222ψ¯1ψ2ψ1ψ2ψ¯1ψ2iψ1ψ2ψ¯1ψ¯2ψ12ψ22,E19
ς=12ψ¯12ψ¯2212ψ12+ψ¯22(ψ¯1ψ¯2+ψ1ψ2)i2ψ¯12ψ22i2ψ12+ψ¯22i(ψ1ψ2ψ¯1ψ¯2)2ψ¯1ψ22ψ1ψ¯2ψ12ψ22

Given this explicit representation, it is now possible to evaluate

p2ς=p2ςςdz+¯ςςdz¯.E20

To obtain an expression for (20), both matrices (19) can be expressed in Maple. Apply the operator mapto ς, right multiply by ςthen substitute system (4) of known derivatives to obtain the matrix

2ψ2ψ¯2+ψ¯1ψ1p02ψ1ψ¯2ψ¯2ψ1p00i2mp2i2mp22ψ¯2ψ¯1ψ1ψ¯2pψ2ψ¯2+ψ¯1ψ1pE21

Similarly, applying map¯to ςthen right multiplying by ςyields

00i2mp202ψ1¯ψ1+ψ¯2¯ψ2p2ψ2¯ψ¯1ψ¯1¯ψ2pi2p2S¯i2mp2ψ1¯ψ¯1+ψ2¯ψ2pE22

By (20) and the differential constraints, the vector representation of the Maurer-Cartan form can be expressed as:

p2dςς=2u0i2S00i2mi2mi2Sudz+00i2m02¯ui2S¯i2S¯i2m¯udz.E23

According to the properties of the inner product, we can write Ei=ςijσj=ΦσiΦand calculate that

dςiςj=dEiEj=12TrdΦσiΦΦσjΦ=12TrdΦσiσjΦ+dΦσiσjΦ.E24

If a conserved current can be constructed whose components are divergence free, then a differential one-form exists with values in R3that will induce a surface upon quadrature. Such a current will be given from the global symmetries of the Lagrangian by means of Noether’s theorem. Making the transformations ψ1ψ¯2and ψ2ψ¯1in system (4), it is seen to remain invariant. This can be thought of as a charge conjugation. The same solutions are obtained if we put Φinstead of Ψin the Dirac equation (2). So Φmultiplied on the right by any constant nonsingular matrix is a solution of the equation if Φis. This implies the full symmetry group is GL2C. The transformation above is a member of this group, so can be thought of as a continuous transformation. In terms of matrix Φ, the Lagrangian of the Dirac equation can be written as

L=12TrΦΦ=iψ¯1ψ2iψ2ψ¯1+iψ¯2¯ψ1iψ1¯ψ¯2+mψ12+ψ22.E25

The currents that correspond to the generators of SU2are found to be proportional to the components of ς+and ς; hence, the required conservation law is

ς+¯ς+=0.E26

Alternatively, the Dirac equation and its Hermitian conjugate which are given by

Φiσ1x+iσ2y+mΦ=0Φixσ1+iyσ2mΦ=0,E27

may be added to obtain

xΦσ1Φ+yΦσ2Φ=0.E28

Now to describe the surface, define the R3-valued differential form

dr=i2ς+ dz+i2ςdz¯E29

which is real since ς=ς¯+. The differential form (29) is closed under substitution of conservation law (26) since

d2r=i2¯ς+ dz¯dzi2ςdzdz¯=i2¯ς++ςdzdz¯=0.E30

By Poincare’s lemma, the form is exact since every loop in Ccan be collapsed to a point. Therefore, the desired expression for a surface will result when the form is integrated along a path Γ1in the zz¯plane from a fixed point z0. The components are

dx1=i2ψ12ψ¯22dzi2ψ¯12ψ22dz¯,dx2=12ψ12+ψ¯22dz+12ψ¯12+ψ22dz¯,dx3=iψ1ψ¯2dzψ¯1ψ2dz¯.E31

Combining the first two equations in (31) and integrating from z0, the coordinates of a surface in R3are obtained by integrating over any path Γ1in the zz¯plane

x1+ix2=iΓψ12dz+ψ22dz¯x1ix2=iΓψ¯22dz+ψ¯12dz¯,x3=iΓψ1ψ¯2dzψ¯1ψ2dz¯.E32

In the end, we have set z0to be zero, and it may be repeated; the integrals are independent of Γdue to the conservation laws. In (31) and (32), ris the point of the surface with coordinates x1x2x3R3and ς3is normal to the surface.

3. Fundamental forms and Cartan moving frame

The necessary information to write down the traditional data for a surface has been obtained. Since ς±2=0and ς+ς=0, the first fundamental form is given by

I=drdr=p2dzdz¯E33

or in a matrix representation,

I=p220110.

The inverse of (33) is given by

I1=2p20110E34

It is therefore a conformal immersions with isothermal coordinates ζ1,ζ2. The second fundamental form of the surface can also be calculated and using ς3dr=0,

II=dp1ς3dr=p1dς3dr=p2Sdzdz+2mdzdz¯+S¯dz¯dz¯E35

and in matrix form,

II=p2SmmS¯.

Collecting (34) and (35), we have

III1=p2SmmS¯02p22p20=1pSmmS¯0110=1pmSS¯m.

The usual definitions give the mean curvature Hand the Gaussian curvature as well

H=12TrIII1=mp,E36
K=detIII1=1p2m2S2=H2S2p2.E37

Equation (36) relates the mean curvature Hto the mass parameter in the Dirac equation. Konopelchenko obtains the expression

K=4p2¯u,E38

which is known as the Gauss-Riemann curvature. It has been shown however that it is equivalent to (37) in accord with Gauss’ Theorem Egregium.

It is interesting to note that since the difference between the principal curvatures is given as

Δκ2=4H2K,E39

it also holds that since H2K=p2S2,

Δκ=2pS.

Thus, the modulus of Sis a measure of the local deformation from a spherical surface as mis a measure of the local deformation from the case of a minimal surface, so κ=p1m±S.

A fixed referential frame in R3has been implicitly used up to now. By varying the frame with some solution of Dirac system (4), a whole set of surfaces is obtained that may be deduced from each other by means of a rigid motion. Cartan developed a powerful method referred to as the moving frame method to avoid this awkward process.

By introducing differential 1-forms also called Pfaffian forms, we define the system

dr=ωjej,dej=ωijej,i,j=+,, 3.E40

This is the first system of structure equations introduced by Cartan. The vectors eisatisfy orthonormality conditions

e+2=0,e+e=p2,e±e3=0,e32=p2.E41

Differentiating relations (41) and using structure equations (40), the following relations among the differential forms are obtained

de+2=0,2e+de+=0,2e+ω+jej=0,
2e+ω+e=0ω+=0.de2=0,2ede=0,2eωjej=0,2eω+e+=0,ω+=0.de+e+e+de=2pdp,ω++e+e+ωe+e=2pdp,ω+ω++=2u,de+e3+e+de3=0,ωj3+ω3=0.dee3+eω3jej=0,ω3+ω3+=0.3e3de3=p2duω33=du.

This collection of results is summarized all together below

ω+=ω+=0,ω3+ω+3=ω3++ω3=0,ω+ω++=2du,ω33=du.E42

As e¯+=eand e¯3=e3, it is found that

ω¯+3=ω3,ω¯++=ω.E43

Assuming structure equations (40) are integrable, differentiating and substituting deiwhere ever possible, compatibility equations are obtained which are referred to as the second system of structure equations, that is first we have

dωjejωjdej=0,

hence

dωs=ωjωjs,

and next

dωijejωijdej=0,

hence

dωij=ωijωjs.

Let us summarize these as the pair

dωs=ωjωjs,dωis=ωijωjs.E44

The second equality is always true as long as the frames are given, and the first is the equivalent, expressed in the formalism of a moving frame, of the requirement that the form drbe exact. Writing dras

dr=i2e+dzi2edz¯=ωjej.E45

Let us identify the forms

ω+=i2dz,ω=i2dz¯,ω3=0,ω¯+=ω.E46

The equations for the remaining one-forms can be represented by writing the structure equation in the form

de=Ωe.E47

In (47), Ωis represented by the 3×3matrix of forms

Ω=ω++ω+ω+3ω+ωω3ω3+ω3ω33E48

Since ee=p2I, (47) can be right multiplied by eto obtain

Ω=p2dee.E49

This implies that Ωcan be identified with the Maurer-Cartan form given in (22). Introduce the vector of differential forms Θas

Θ=ω+ω0,dΘ=0.E50

In terms of Θthe compatibility equations take the form

dΘ=Ω,dΩ=ΩΩ.E51

It is clear from the Maurer-Cartan form that it can be decomposed in the following manner

Ω=M1dz+M2dz¯,E52

where M1and M2are defined to be the matrices

M12u0i2S00i2mi2mi2Su,M2=00i2m02¯ui2S¯S¯2im2¯u.E53

The first structure equation in (33) is then

e=M1e,¯e=M2e.E54

This corresponds to the Gauss-Weingarten equation and the second compatibility equation

¯M1M2+M1M2=0,E55

is also known as the Gauss-Codazzi-Mainardi equations. All of these have been seen here before in (37) and (45). It has been shown that many nonlinear partial differential equations can be expressed within this formalism. In a spinor representation, the corresponding representation in the form of matrices can be obtained out of the Maurer-Cartan form

Z1=122uiSim0,Z2=120imiS¯2¯u.E56

In terms of these matrices, the linear system is

∂Φ=Z1Φ,¯Φ=Z2Φ,
¯Z1Z2+Z1Z2=0.E57

The differential form Ωis a connection and actually an affine connection on R3. It is flat on the surface. This is the meaning of the second system of structure equations. This means that the two-dimensional Dirac equation can be regarded as a way of expressing an affine connection.

Two make further progress, Ωcan be used in the following way. As ω3=0, from the compatibility equation for dω3, we have

ω+ω+3+ωω3=0.

On account of Cartan’s lemma, both ω+3and ω3are equal to a linear combination of ω+and ω

ω+3=h++ω++h+ω,ω3=h+ω++hω,E58

where h+=h+, as can be seen by substituting ω+3and ω3into the constraint above. Since

Ω=2u0i2S00i2mi2mi2Sudz+00i2m0¯ui2S¯i2S¯i2m¯udz¯.E59

Using ω+,ωand ω3from (46), we have

ω+3=h++i2dz+h+i2dz¯=i2Sdzi2mdz¯.E60

This relation implies that

h++=S,h+=m,E61

and moreover, it follows that

ω3=h+ω++hω=h+i2dz+hi2dz¯=i2mdz+i2S¯dz¯.

This implies that

h+=m,h=S¯.E62

It is important to note that these coefficients can be used together with the structure equations to express the fundamental forms of the surface in terms of Pfaffian forms. The first fundamental form is given as

I=2p2ω+ω=2p212dzdz¯,E63

and the second fundamental form can be written as

II=pω+ω+3+ωω3=ph++ω+ω++h++h+ω+ω+hωω.E64

The element of surface is given by

dS=ip2ω+ω,E65

and the corresponding surface element on the Gauss map is

=iω3ω+3=ih+h+h++hω+ω.E66

The total curvature would be the ratio of the former to the latter,

K=p2h+h+h++h.E67

Finally, the mean curvature is given as

H=12ph++h+.E68

4. The Gauss map and nonlinear Sigma model

Under the condition that a given moving frame is integrable, the surface is defined up to a translation. Conversely, given the three vectors which constitute the frame, only one is determined uniquely by the surface, and that is the normal vector. For this reason, it is often referred to as the Gauss or spherical map, as it maps the parameter plane to the sphere of radius one in two dimensions. The map in this instance is given as

ϕ=e3p,E69

so the north pole corresponds to ψ2=0, while the south pole to ψ1=0. If the first column of Φis considered as well as the associated fundamental field

ρ=ψ2ψ1,E70

then dividing the numerator and denominator by ψ12in (69), we obtain

ϕ=11+ρ2ρ+ρ¯iρ¯ρ1ρ2.E71

This quantity is a function of only ρ. It may be thought that ρplays the role of stereographic projection of the Gauss map from the south pole. Moreover, for a minimal surface where m=0, it is readily shown that ρ¯is an analytic function of z.

Using the differential constraints, the derivatives of ρare found to be

ρ=ipm2ψ¯12,ρ¯=ipS2ψ12.E72

By using derivatives (72), the following three relations can be worked out

4ρ¯ρ¯1+ρ22=m2,4ρρ¯1+ρ22=mS,4¯ρρ¯1+ρ22=S2.E73

Thus, the quantities mand Scan be written as a function of only ρ. It may be asked, can the component of the Maurer-Cartan form ube written in a similar way? Starting with the differential constraint for u,

u=1pψ¯1ψ1+ψ2ψ¯2=eu/2ψ1ψ¯2eu/2ψ¯1ψ2.E74

Since the spinor product ψ1ψ¯2ψ1ψ¯2=eu, we have

eu/2ψ1ψ¯2eu/2ψ¯1ψ2=12uψ1ψ¯2euψ¯1ψ2+eu/2ψ1ψ¯2eu/2ψ¯1ψ2.

Combining these last two results, we obtain

12u=eu/2ψ1ψ¯2eu/2ψ¯1ψ2.E75

If we define the spinor α=eu/2ψ1ψ¯2which satisfies αα=1, (75) becomes

12u=αα.E76

Let us show that αcan be expressed as a function of ρ. Using the definition of ρ, a parameterization for αexists as

α=11+ρ21ρ¯ψ1ψ¯21/2.E77

To obtain an expression for ψ1, use differential constraint (2) its conjugate and (70) to arrive at

ψ¯12ρ=i2pm.E78

Dividing this by its complex conjugate gives αas a function of ρas

α=11+ρ21ρρ¯ρ¯1/4ei2n+1π/4,nZ.E79

Inserting αinto the expression for uprovides expressions for uand ¯u

ρ¯ρ¯=2logρ¯ρ¯ρ¯ρ¯.E80

Differentiating the components of αwith respect to z, we find that

αα=ρ¯ρ+ρρ¯21+ρ22+141+ρ2logρ¯ρ¯+ρρ¯2ρρ2ρ¯ρ¯+2ρ1+ρ2ρ¯21+ρ22+ρ241+ρ2logρ¯ρ¯=ρ¯ρ+ρρ¯21+ρ2+14logρ¯ρ¯.

Returning to the expression for 1/2u, we can now write

12u=14logρ¯ρ¯+12ρρ¯ρ¯ρ1+ρ2,E81
12¯u=14¯logρ¯ρ¯+12ρ¯¯ρρ¯ρ¯1+ρ2E82

There is no simple integral of the second term in general. It may be stated that 1/2uhas the form of a potential with a fixed gauge, because ρis given as a function of zand z¯, so the directions of the axes e+and ehave been fixed so that a gauge transformation is a rotation of them.

Suppose it is asked under what condition a given complex function ρzz¯is the Gauss map of some surface. A necessary condition can be obtained by working out the compatibility condition for the linear system (81) and (82), that is, first

¯u=12¯logρ¯ρ¯+ρ¯¯ρ+ρ¯¯ρρ¯ρ¯ρ¯ρ¯1+ρ2ρ¯¯ρρ¯ρ¯1+ρ22ρρ¯+ρ¯ρ=12¯logρ¯ρ¯+¯ρρ¯¯ρ¯ρ¯ρρρ¯2+ρ¯¯ρ¯ρ21+ρ2ρ¯ρ¯ρ¯¯ρ1+ρ22,

and the result for the other mixed derivative is

¯u=12¯logρ¯ρ¯+ρ¯¯ρ¯ρ¯ρ+1+ρ2ρ¯ρ¯ρ¯¯ρ+ρ¯2ρ¯ρρ2ρ¯¯ρ¯1+ρ22.

Equating these mixed partial derivatives, the necessary condition takes the form

¯logρ¯ρ¯+21+ρ2ρ¯ρ¯ρ¯¯ρ+ρ¯2ρ¯ρρ2ρ¯¯ρ¯1+ρ22.E83

If it is satisfied, it has the implication that

¯u=¯ρρ¯ρ¯ρ¯1+ρ22.E84

Using the previous expressions (73) for the derivatives of ρ, this can be put into the form of the Gauss equation. Consequently, one of the integrability conditions is fulfilled. Since

¯ρρ¯ρ¯ρ¯1+ρ22=12ρ¯¯ρρ¯ρ¯1+ρ2+¯12ρρ¯ρ¯ρ1+ρ2,E85

it follows that

¯u=12ρ¯¯ρρ¯ρ¯1+ρ2+¯12ρρ¯ρ¯ρ1+ρ2=¯ρρ¯ρ¯ρ¯1+ρ22=14p2K.E86

Due to cancelations, some shorthand expressions might be quoted

¯u=ρ¯¯ρ1+ρ2+¯ρ¯ρ1+ρ2=ρ¯ρ¯1+ρ2+¯ρρ¯1+ρ2=¯ρρ¯ρ¯ρ¯1+ρ22.E87

The integrability condition can be expressed in the form of a zero curvature condition

¯ρρ2ρ¯¯ρ1+ρ2¯¯ρ¯¯ρ¯2ρρ¯1+ρ2=0.E88

It is clear that provided we have

Bρρ¯=¯ρ2ρ¯1+ρ2ρ¯ρ=0,E89

the condition is satisfied automatically. This may be recognized as the equation describing the nonlinear sigma model. As well it is the equation which is satisfied by the Gauss map of a constant mean curvature surface which is harmonic.

It is well known that for a given Gauss map ρsuch that ρ=0, there is a one parameter family of surfaces called the associated family which is obtained through the transformation

ψ1q1/2ψ1,ψ2q¯1/2ψ2.E90

This keeps pand ρinvariant if qis a complex constant of modulus one. If m0, it is not possible since mwould not stay real. In the latter case, the only allowed values are q=einπ. To construct the surface, take αand replace the phase factor by q1/2, so p=1, and we obtain

ψ1=q1/21+ρ2,ψ2=q1/2ρ1+ρ2.E91

Substituting (91) into the inducing formulae (30), the Weierstrass representation for q=1can be observed.

Finally, using (36) and recalling that

H=p2¯pS,ψ12ρ¯=i2pSE92

the second equation in (92) is differentiated with respect to ¯to obtain,

imψ1ψ2ρ¯+ψ12¯ρ¯=i2¯pS.E93

Taking the conjugate of the first expression in (72) then solving for ψ12and substituting into (93) we have

2ψ1ψ2pρ¯+¯ρ¯ρ¯=m1p1¯pS.E94

Using (70) and the relation ψ12/p=1/1+ρ2, we obtain the desired result

H1H=¯ρ¯ρ¯2ρρ¯1+ρ2.E95

Differentiating J=pSwith respect to ¯then multiplying by pS1, we obtain

pS1¯pS=m2ψ1ψ2pSρ¯+1S¯ρ¯¯ρ¯ρ¯=mρ¯¯ρ¯2ψ1ψ2pS¯ρ¯+1S¯ρ¯ρ¯
=mSρ¯¯ρ¯¯ρ¯ρ¯2ρ¯ρ¯1+ρ2=¯ρ¯ρ¯2ρ¯ρ¯1+ρ2.

To obtain this, the first two derivatives in (73) have been used to write ρ¯/¯ρ¯=S/m. Summarizing these calculations, the following relations have been proved:

H1H=¯ρ¯ρ¯2ρρ¯1+ρ2=B¯¯ρ¯,J1J=¯ρ¯ρ¯2ρρ¯1+ρ2=B¯ρ¯.E96

Thus, for the parameters that are proportional to a power of p, the logarithmic derivatives can still be computed. For a constant mean curvature surface H=0and so Bρρ¯=0hence

¯J=J¯=0,E97

and the current is conserved, of Jis a holomorphic function.

5. Summary and conclusions

It should be said that this work has deep implications for the study of manifolds and their relationship with integrable systems in general [21, 22, 23, 24]. It would be worth illustrating this more clearly as a way to conclude. As a particular example, consider the case of a spherical surface for which S=0so that

K=H2=m2p,E98

where Kis now a constant and the Gauss equation simplifies to

¯u+m2=0.E99

If we choose K=1, this implies that m=p; hence m=euand (99) is then the nonlinear Liouville equation

¯u+e2u=0

is obtained in terms of the only remaining variable u. This procedure has resulted in a nonlinear equation with a link to surfaces. Since p1m=1, the Codazzi-Mainardi equation is trivially satisfied.

Due to the spinor representation of the Maurer-Cartan form, from which Z1and Z2are deduced, for any nonsingular matrix τ, there is a gauge transformation given by [19]

ΦτΦ,Z1τZ1τ1+ττ1,Z2τZ2τ1+¯ττ1,E100

for which the nonlinear zero curvature equation still holds. For example, suppose we take

τ=λ¯1/200λ1/2eu/200eu/2eu/2.E101

In (101), λcan be thought of as a complex spectral parameter that satisfies λ2=1. Starting with (8), we find that

Φλ=τΦ=λ¯1/2ψ1λ¯1/2ψ¯2λ1/2euψ2λ1/2euψ¯1,det Φλ=1.E102

It is straightforward to calculate that

τZ1τ1=122u00,τ=000λ1/2euu,ττ1=000u,

Therefore, we get

τZ1τ1+ττ1=122u02u,

and proceeding in a similar fashion, one finds

τZ2τ1+¯ττ1=120iλ¯e2u00.

The linear system for the case in which S=0and m=pis given by

Φλ=122u02uΦλ,¯Φλ=120iλ¯e2u00Φλ,E103

where Φλis given by (102). Other choices for the gauge function τwill lead to other systems: for example, taking

τ=121110λ¯1/200λ1/2eu=12λ¯1/2λ1/2euλ¯1/2λ1/2euE104

an AKNS type system is obtained

Φ˜=144u4u+Φ˜λ,¯Φ˜=iλ4e2ue2ue2ue2uΦ˜λ.E105

Therefore, hierarchies may be generated and this linear system which is derived from the Dirac equation and used to create surfaces provides the link between nonlinear evolution equations and geometry.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Paul Bracken (January 25th 2019). The Generalized Weierstrass System in Three-Dimensional Euclidean Space, Manifolds II - Theory and Applications, Paul Bracken, IntechOpen, DOI: 10.5772/intechopen.82631. Available from:

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