## 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

In (1), the mass term

In terms of a complex variable

Using (2), the Dirac equation can be developed in terms of the two components of

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

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

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

A system of conservation laws can also be formulated

as well as their complex conjugate equations. The complex quantity

Let

defining the real variable

Clearly, we have

Differentiating (8) exteriorly, we obtain that

Consequently, we find that

Taking the derivative

It follows that

Let us summarize this as

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

and as well, we have

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

By comparing with (4), it look very much like a Dirac system in their own right if

It is possible to construct a vector representation of

In terms of matrices,

Given this explicit representation, it is now possible to evaluate

To obtain an expression for (20), both matrices (19) can be expressed in Maple. Apply the operator *map*

Similarly, applying *map*

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

According to the properties of the inner product, we can write

If a conserved current can be constructed whose components are divergence free, then a differential one-form exists with values in

The currents that correspond to the generators of

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

may be added to obtain

Now to describe the surface, define the

which is real since

By Poincare’s lemma, the form is exact since every loop in

Combining the first two equations in (31) and integrating from

In the end, we have set

## 3. Fundamental forms and Cartan moving frame

The necessary information to write down the traditional data for a surface has been obtained. Since

or in a matrix representation,

The inverse of (33) is given by

It is therefore a conformal immersions with isothermal coordinates

and in matrix form,

Collecting (34) and (35), we have

The usual definitions give the mean curvature

Equation (36) relates the mean curvature

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

it also holds that since

Thus, the modulus of

A fixed referential frame in

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

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

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

This collection of results is summarized all together below

As

Assuming structure equations (40) are integrable, differentiating and substituting

hence

and next

hence

Let us summarize these as the pair

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

Let us identify the forms

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

In (47),

Since

This implies that

In terms of

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

where

The first structure equation in (33) is then

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

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

In terms of these matrices, the linear system is

The differential form

Two make further progress,

On account of Cartan’s lemma, both

where

Using

This relation implies that

and moreover, it follows that

This implies that

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

and the second fundamental form can be written as

The element of surface is given by

and the corresponding surface element on the Gauss map is

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

Finally, the mean curvature is given as

## 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

so the north pole corresponds to

then dividing the numerator and denominator by

This quantity is a function of only

Using the differential constraints, the derivatives of

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

Thus, the quantities

Since the spinor product

Combining these last two results, we obtain

If we define the spinor

Let us show that

To obtain an expression for

Dividing this by its complex conjugate gives

Inserting

Differentiating the components of

Returning to the expression for

There is no simple integral of the second term in general. It may be stated that

Suppose it is asked under what condition a given complex function

and the result for the other mixed derivative is

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

If it is satisfied, it has the implication that

Using the previous expressions (73) for the derivatives of

it follows that

Due to cancelations, some shorthand expressions might be quoted

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

It is clear that provided we have

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

This keeps

Substituting (91) into the inducing formulae (30), the Weierstrass representation for

Finally, using (36) and recalling that

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

Taking the conjugate of the first expression in (72) then solving for

Using (70) and the relation

Differentiating

To obtain this, the first two derivatives in (73) have been used to write

Thus, for the parameters that are proportional to a power of

and the current is conserved, of

## 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

where

If we choose

is obtained in terms of the only remaining variable

Due to the spinor representation of the Maurer-Cartan form, from which

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

In (101),

It is straightforward to calculate that

Therefore, we get

and proceeding in a similar fashion, one finds

The linear system for the case in which

where

an AKNS type system is obtained

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.