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SURFACES: FROM GENERALISED WEIERSTRASS

REPRESENTATION TO CARTAN MOVING FRAME

Claude Pierre MASSÉ

Laboratoire de Physique Fondamentale, FRANCE

© Claude Pierre MASSÉ 2020

19 April 2020

Konopelchenko’s generalised Weierstraß representation of a surface immersed in the

three dimensional Euclidean space is derived by the way of the conserved current, and

then expressed in the framework of the moving frame. The data of classical surface

theory are given each time. The problem of determining the surfaces from a prescribed

Gauß map is investigated, and this is then applied to inverse scattering transform for

solving some nonlinear dierential equations.

Latest version: http://phy.clmasse.com/surfaces-weierstrass-cartan.html

Introduction

The classical Enneper-Weierstraß representation allows to represent a minimal surface

immersed in by a meromorphic function and a holomorphic -form. Recently,

Konopelchenko [1] used a similar representation with two functions given by a solution of a

two-dimensional Dirac equation. In this article, that formalism is presented in a transparent and

systematic way, so that the Cartan’s moving frame can be introduced naturally. In section §1,

we rst derive the generalised Weierstraß representation by the way of the conserved current,

taking advantage of the vector representation of the spinor. Up to there, everything is expressed

in an arbitrary, xed reference frame. Then in section §2, the representation is recast along the

lines of the Cartan’s moving frame, so that the xed frame becomes unnecessary. The

mathematical expressions of the traditional data of a surface are obtained each time. Singling out

one vector of the moving frame and relating it to the Gauß map, in section §3 is investigated

whether there exist surfaces having this Gauß map, and if yes, how they are determined by it.

This is done through a lift to the moving frame under the condition of remaining in the given

formalism. The geometrical representation of integrable nonlinear dierential equations follows

directly, and in section §4 it is shown how this can be applied to solving the nonlinear Liouville

equation and its generalisations through the inverse scattering transform.

§1. The generalised Weierstraß representation

In the Euclidean two-dimensional space, the Dirac equation can be written using the Pauli

matrices:

as follow (the choice of the Dirac matrices is not a whim…):

Here, the mass parameter (more usually called the potential) is generalised to a real function

of and the Cartesian coordinates, with an obvious notation for the partial dierential

operators. We shall work with complex numbers for compactness, and denote the complex

conjugate by an overbar. We rst dene the Wirtinger dierential operators

or using the complex variable:

Writing the wave function as:

the Dirac equation (1.2) is now decomposed into a system of two simple and handy equations:

We rst focus on the spinor. It is uniquely extended as follow:

It is a matrix multiplied by the factor with the denitions:

so that Let us remark that if we make the substitution

which is in fact the charge conjugaison, the Dirac equation (1.7) still holds. That is, we get the

same solutions if we put instead of in equation (1.2). Then, multiplied on the right by

any constant non singular matrix is a solution of the equation if is so. Thus, the Dirac

equation has the symmetry group The substitution above (1.10) belongs to this group,

which makes charge conjugaison a continuous transformation (i.e. continuously connected to

the identity.)

We then consider the associated (right-invariant) Maurer-Cartan -form:

Its entries can be calculated from the Dirac equation, with the result

where the following dening relation has been used:

In addition, the following dierential constraints have been found:

They prove very useful in the sequel. For completeness, let us list two further constraints:

They are in the form of a conservation laws, a more elaborate version of which will be derived

below.

Taking the derivative of the dening relation (1.13) and using the Dirac equation (1.7), we

directly get

that was rstly proven bei Hopf [2] (chapter VI, section 1.2) We later use the quantity

which is called the Hopf function. The value of could be called the mirror mass, and by

solving (1.13) and (1.14c) for and we have the mirror Dirac equation

It is actually the one satised by

Now we use the well known vector representation. A matrix such as is associated to a

rotation times a scaling in as follow:

The matrix can thus be represented by a multiple of an orthogonal real matrix, whose

elements are easily found by using the inner product in the space of :

then

and after some straightforward calculations:

It is an matrix multiplied by a factor so that

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

with which we calculate

Dierentiating, this gives

from which the vector representation of the Maurer-Cartan form is deduced:

Remark: There is a two-ways fast lane for switching between both representations. Given

that the Lie algebras of and are locally the same, if the generators of

associated with are the corresponding generators of associated with are

Similarly, the generator of the scaling is for he spinor representation, and for the vector

representation, where is the unit matrix of the appropriate size. The Maurer-Cartan form

belongs to the respective Lie algebra. One passes from one to the other just by replacing the

basis of the Lie algebra representation. Denoting this abstract basis by running from

to the form reads

where stands either for or The same works as well for and using the exponential

map of the Lie algebra, from which the Maurer-Cartan form can be derived. It is still

simpler using the step generators and but I shall not spoil

the fun of the reader, who may want to prove the claim too.

In the usual way, from the Dirac equation we get a conserved current by multiplying it on

the left by writing its hermitian conjugate

and adding them. The result is

In the vector representation (1.21) and with complex variables, this equation becomes the

conservation law

From (1.22) we get the explicit components of the three conserved currents:

According to Noether’s theorem, they are associated to symmetries of the equation of motion,

which are indeed the right multiplication of by a matrix. The currents (1.32)

correspond to the generators of The third one is just the Dirac electromagnetic

current, as expected since the transformation of the electromagnetic gauge is associated to

From this conservation law, we deduce that the -valued dierential form

is closed. In addition it is real because Moreover, it is exact in virtue of the Poincaré’s

lemma since every loop in can be shrunk to a point. Therefore, integrating it along a path

in the plane from a xed point we arrive at the wanted formula of the surface:

for the point of the surface with coordinates in That is the same as in [1]

(equation 2) up to some cosmetic rearrangements. The surface is then normal to This

method is not suited for generalisations, it is just a coincidence working only for surfaces in

We now have all what is necessary to get the traditional data of a surface. Since and

the rst fundamental form is

(it is therefore a conformal immersion with isothermal coordinates and ) The second

fundamental form can be directly calculated too, using :

Finally the usual formulæ gives the mean curvature and the Gauß curvature :

Konopelchenko [1] (equ. 4) gives a dierent expression:

which is known as the Gauß-Riemann curvature. But Ferapontov and Grundland [3] (equation

2.6) proved that they are equivalent, in accordance with Gauß’ Theorema Egregium. Then the

modulus of is a function of and :

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

by

it holds

that is, the modulus of is a measure of the local deformation from a spherical surface, like is

a measure of the local deformation from a minimal surface. Indeed

§2. The Cartan moving frame

So far, we implicitly used a xed referential frame in Indeed, by varying this frame,

with the same solution of the Dirac equation we get a whole set of surfaces that are deduced

from each other by a rigid motion. Expressing the dierent quantities in this xed frame was

deemed awkward by Cartan, so he developed the powerful method of the moving frame [4] from

the Darboux frame. To an innitesimal displacement in the plane is now associated an

innitesimal change of the origin and of the basis vectors, but this time expressed with respect to

the moving frame itself. By using dierential -forms (also called Pfaan forms,) this is written

down as

which is called the rst system of structure equations by Cartan. The vectors satisfy the

orthonormality conditions

Dierentiating them, and using the structure equations (2.1), we get relations among the

dierential forms, e.g.:

By a similar method, we collect the following relations:

As and with the complex conjugates of the same equations we moreover nd

Then assuming the structure equations (2.1) are integrable, by dierentiating them and

substituting (2.1b) wherever possible, we get compatibility equations, called the second

system of structure equations:

The second equality (2.6b) is always true as long as the frames are given, and the rst one (2.6b)

is the equivalent, expressed in the moving frame language, of the requirement that the form

be exact.

From the expression of (1.33) we immediately get

and

The expression of the remaining -forms are easily obtained by writing the structure

equation (2.1b) as a matrix:

where

Multiplying (2.9) by on the right, there results

which is already known, it is the Maurer-Cartan form (1.26), the preceding relations (2.4), (2.5)

are incidentally satised. Further writing

the compatibility equations (2.6) become

If we decompose as follow:

with

the structure equation (2.1b) is then

which is known in the theory of surfaces as the Gauß-Weingarten equations. The second

compatibility equation (2.6b) reads

also known as the Gauß-Codazzi-Mainardi equations. We already met them, here they are

gathered:

Many integrable nonlinear partial dierential equations are expressed within this formalism. The

last equation (2.17) is then called the zero curvature condition, and is the nonlinear equation

represented as a compatibility condition of a linear system. The double is called the Lax

pair.

In the spinor representation, the corresponding matrices are readily obtained from the

Maurer-Cartan form (1.12), they are

and the equations (2.16), (2.17) become

Remark: The dierential form (2.11) is a connection, and actually an ane connection of

that is at on the surface, which is the meaning of the second system of structure

equations (2.6). We have thus the stunning result that the (two-dimensional) Dirac

equation is but a way of writing an ane connection. Its restriction to the surface is

and its intrinsic scalar curvature is precisely the Riemann-Gauß curvature of the surface.

The restriction of the structure equations doesn’t necessarily hold, and indeed we have:

But isn’t dened on the surface, thus the term stands for the ane

curvature form, whose explicit expression taken in three dimensions, that is extrinsically, is

the Gauß curvature. That’s still another way of viewing the Theorema Egregium. The

connection has no torsion, i.e. extrinsic part of from the sheer fact that and

thus it is indeed the Levi-Civita connection.

Spelling out the compatibility equations (2.6), we easily derive the already known

equations (2.G) and (2.CM) again. In addition, as from the compatibility equation for

we have

According to the Cartan’s lemma, and are then equal to a linear combination of and

like

and

as can be seen by substituting and in equation (2.23). Finally by identication we get

Now using these coecients together with the structure equations (2.1), we are able to

express the data of the surface in term of the Pfaan forms:

The surface element is

and the corresponding surface element on the Gauß map (see next section) is

The total curvature is the ratio of these two surfaces:

and the mean curvature is

The results are the sames as in the end of the previous section (1.35-38), as expected.

§3. The GAUSS map

Given a moving frame, and under the condition that it is integrable, the surface is dened

up to a translation. Conversely, among the three vectors of this frame, only one is determined

uniquely by the surface, the normal vector. It is called the Gauß map (or spherical map,) it is a

map from the parameter plane to the two-dimensional sphere of radius The question

addressed in this section is then, to which extend is the surface determined by the Gauß map

alone?

From our derivation, it is obvious that the Gauß map is directly given by:

The north pole corresponds to and the south pole to To go further, let us remark

that in the vector representation, we use row vectors while the spinor is a column. If we

consider the rst column of and the associated fundamental eld:

then, dividing the numerator and the denominator by in the expression of (3.1), we

obtain

that is a function of alone. In other words, is the stereographic projection of the Gauß map

from the south pole. Moreover, for a minimal surface ( ) it is readily shown that is

meromorphic, as is known for a long time. Actually, is precisely the meromorphic function of

the Enneper-Weierstraß representation, while the holomorphic -form is That

generalises for any non minimal surface with being any function. On the other hand, if

the denition of (1.13) implies that is, is an antiholomorphic function. In

the following, we shall then use throughout, and call it the Gauß map.

Now we tackle the problem of reconstructing the surface from the Gauß map. In our

setting, it is easily answered through the spinor, and more precisely the unit row spinor

and its orthogonal unit spinor

satisfying

Together they form the unitary matrix Taking into account that a

parametrisation of them is

Next, using the dierential constraints (1.14b) we calculate

and dividing this last equation by it complex conjugate gives

Then we have and as functions of only:

We have achieved the lift of the spherical map to but to get the surface completely, is

still lacking. Yet, from the dierential constraints (1.14b) and (1.13), with (3.4a) and (3.4b) we

have

then with (3.8), cancellations because of (3.4c) lead to

yielding the simpler formulæ

So, and can be written as a function of only. Similarly, using once more the dierential

constraint (1.14c), that is

we get the remaining components of the Maurer-Cartan form

Under the condition that this system be compatible, is also determined by integration up to a

constant multiplicative factor, which corresponds to a mere change of scale of the surface, and it

is obvious that the Gauß map is invariant under such a transformation. However in general,

there is no simple integral of the second term. We see that has the form of a potential with

a xed gauge. Indeed, a gauge transformation is a rotation of the axes and then as is

given as a function of and and these latter variables are dened by this axis system, the

gauge is xed.

Now we are in position to investigate under which condition a given complex function

is the Gauß map of some surface. We get a necessary condition by cross dierentiating

the linear system (3.13), which amounts to the same as requiring that be real. This gives

The necessary condition then reads

It is a single real condition, since the term is purely imaginary, and the fraction is the

dierence of two expressions that are the complex conjugate of each other. If it is satised, that

means that

and therefore from (3.11) it is easily seen that one of the integrability conditions (2.G) is

fullled. Then a tedious but straightforward calculation shows that the other one (2.CM) is as

well fullled. Therefore it is a sucient condition too.

Because of the potential form of in (3.13), an alternative expression can still be given:

where the curvature nature of is manifest. Due to cancellations, in addition we have the short

hand expressions

Similarly, the integrability condition (3.15) can be written as a zero-curvature condition:

It then comes as obvious that if

the condition is automatically satised. But that is just the equation satised by the Gauß map of

a constant mean curvature surface [5], which is harmonic. Thus is proportional to the tension

eld actually:

Up to now, we have derived the results of Homan and Osserman [6] for a surface in

arguably in an more intelligible way. To proceed, notice that the zero-curvature condition

(3.19) means that is the gradient of some real fonction or

Beside constant mean curvature surfaces with harmonic Gauß map, other families of surfaces can

be constructed by prescribing for which solutions of the above equation are known.

Let us remark that, having obtained (3.8) and (3.13), we have also found as

a function of up to a constant real factor, therefore the surface is determined up to a scaling

factor by only. But it is not unique as it depends on the chosen fourth root, given by in

(3.8). Its values of same parity give the same surface because the induction formula (1.34) is

quadratic in thus there are two dierent surfaces, which are deduced from each other by a

space inversion with (or by a negative scaling factor if one want.) Yet, is the same

since the constant exponential factor cancels out.

We have tacitly assumed that for then and can’t be determined. We

shall not consider isolated zeros, but the case where in an open set, which corresponds to

a portion of a minimal surface. The Dirac equation (1.7) reduces to and so that

a solution is given by an arbitrary holomorphic function and we have which is

automatically antiholomorphic. Now it is well known that for a given Gauß map such that

there is a one-parameter family of surfaces called the associate family. It is obtained

through the transformation

that keep both and invariant if is a complex constant of modulus while This

is not possible if because wouldn’t remain real, hence in the latter case the only

allowed values of are The surface can now be constructed by putting and in the

induction formula (1.34). Then as already discussed, we recognise the Weierstraß

representation, where the argument of is the Bonnet angle.

For the parameters of the surface that are proportional to a power of one can still

compute the logarithmic derivatives:

Three conclusions can be drawn:

The function in (3.22) depends on alone, viz.

and we fall back to the result of Kenmotsu [5].

For a constant mean curvature surface, and thus as we also have

that is, (see equ. (1.17)) is a holomorphic function.

As

the function

is a rst integral of the harmonic equation (3.20), that is if is a solution.

To conclude this section, let us look back at the road traveled. We got at an integrability

equation of the Gauß map for a constant mean curvature surface, while we started from another

equation, but that is not restricted to particular surfaces, and that contains a prescribed function.

We shall then investigate how it can be modied so that it describes a mean curvature surface

too. One way is to directly replace by in the equation, that becomes nonlinear.

Konopelchenko has shown that it is integrable [1]. But there is another way by keeping the

prescribed function We can use the mirror Dirac equation (1.18) if can be calculated, and

it occurs it can. The equation then remains linear.

The case of identically zero is already known, there are many minimal surfaces. The case

of negative is deduced from the one of the opposite mean curvature, as we have seen the

surface is spatially inverted. So we consider the case of positive. Then must be a strictly

positive function. To begin with, the denition (1.37) yields

Then we make use of the integrability equations in turn. From (2.G) along with the condition

that be constant we rst get

As there is a second constraint on :

The (2.CM) equation reduces to

and using the parametrisation

in it, we nally obtain the result

This equation has a unique solution up to an additive constant, since it is linear and of rst

order, and this determines entirely. As a global change of the argument of gives a dierent

surface, there is a one-parameter family of surfaces, which is the associated family. Remark that

the function is harmonic, which is a mere consequence of the holomophicity of So, is

determined by , the linear Dirac type equation for a mean curvature surface is a system of four

equations, and it has a unique solution if is given at a point, thus all the solutions represent the

same surface up to a rigid motion.

We have then arrived at: provided that the function be strictly positive and satises

(3.33), among the surfaces represented by the solutions of the Dirac equation (1.2) there is an

associate family of constant mean curvature surfaces, and all those deduced from them by a rigid

motion. There is an apparent contradiction with the transformation (3.23), with which we said

that wouldn’t remain real. But here, is xed, and consequently we no longer have this

transformation, so that isn’t invariant.

§4. Application

This application illustrates the use of the equations (2.16), (2.17) or (2.19), (2.20) to express

a nonlinear partial dierential equation and its representation by the compatibility condition of a

linear system. As particular case we shall take a spherical surface, then Equation (1.38)

simplies to

where is now a constant, and equation (2.G), which is deduced from (2.17) or (2.20),

becomes

For the special choice as substituting

it gives

and the only remaining variable is It is the nonlinear Liouville equation as well known. The

other equation (2.CM) is trivially satised.

Because of the spinor representation of the Maurer-Cartan form (1.12) from which and

are deduced (2.18), for any non singular complex matrix function there is the gauge

transformation

for which the nonlinear equation (2.20) still holds. Following Lund and Regge [7], (see also [3])

we choose the matrix

The purpose of each factor, from right to left, is to eliminate the terms in and to introduce

the complex spectral parameter that veries

The result is the linear system

where

In order that it could be used to solve the nonlinear equation by the inverse scattering transform,

it must be reformulated in the coordinates, giving

Using the further factor

this system has the same form as the one used for the sine-Gordon equation in laboratory

coordinates [8].

Our aim is not to work out the full inverse scattering transform. Actually, we already have

the general solution from (4.3), (3.11a), and (3.11c). As we have so that

where is any holomorphic function. The same formula can be obtained by integrating the

linear system (3.13) using the fact that is holomorphic. In addition, can be determined from

(3.8).

As an example, the choice

yields the complete solution

It is an easy exercice to verify that it is actually the case. This stands for all the solutions of the

spectral problem because there is only one spherical surface of curvature

What is shown here is that this linear system is derived from the Dirac equation used to

represent the surface. That’s the explanation of the link between nonlinear evolution equations

and geometry. In fact, it is not the usual representation with a surface of constant negative

gaußian curvature and isometry group but the Lax pair can be derived the same way.

We have seen that if is multiplied on the right by a constant matrix, it is still a

solution of the Dirac equation, and in addition its determinant remains identical to Writing

this matrix as

we get the transformation

and accordingly, dividing numerator and denominator by the so-called nonlinear realisation

of

which is a (proper) isometry of the Gauß sphere, and then of the spherical surface. That is to say,

the isometry group is instead of

Because of the mirror symmetry, it is natural to wonder whether the rôles of and can

be swapped. And that is indeed true, it is a recently discovered duality [9] that is more

transparent in this framework. So, setting for the choice the nonlinear

equation is now

The antiholomorphic functions are replaced by holomorphic ones, and the surface is minimal.

Since by substituting the nonlinear equation is the same as in the previous case, the

general solution is

while with our example becomes

where is a real constant. This family of minimal surfaces is the associate family we have seen in

§3, for it is known as the Enneper’s surface. No further details are given since the

reasoning is strictly parallel. Let us only remark that these two surface families have the same

isometry group, the Gauß map being merely inverted. The improper isometries are generated

by which transform a surface into its mirror image (the mirror is not plane, as it

were.)

It happens we have used two constant mean curvature surfaces. This is no coincidence.

Actually, a full family of nonlinear equations arises from this setting. We have seen that when

the mean curvature is constant, the function (1.17) is holomorphic, so the (2.CM) equation

holds. Then we write the (2.G) equation as follows:

Replacing by and rearranging, we get

Our rst case corresponds to and the second one to and and it is

obvious in this equation that these two cases are deduced from one another through

There is also the elliptic sinh-Poisson (or elliptic sine-Gordon) equation for and many

others more complicated, and indeed less interesting since they explicitly involve

But whatever the equation, it is a reduction of a single one: the harmonic equation (3.20).

The above two special cases correspond to the “ nite action ” solutions satisfying or

For the general case, from a harmonic function and choosing a value of that is

but a scale factor, the solution is given by (3.11a) since (1.37). Conversely, from a

solution of the nonlinear equation (4.22) where and the holomorphic function are given,

we can calculate and The Gauß map is then deduced from the

integration of the Weingarten equations (2.16) or (2.19), starting from any initial value of or

at which xes the Gauß map at and thus selects a single element in the set of

surfaces diering only by a rotation.

For instance, the solutions of the sinh-Poisson equation

are determined by three constraints. First, must be a general harmonic function. Then, from

and (3.11a) we get the solution given by (4.12), and from and (3.11c) we get

another expression of the same solution given by (4.19). Both expressions must agree and this is

a condition on the eligible harmonic functions. But this way doesn’t seem promising, so we

only exhibit the simplest example of a constant mean curvature surface: the cylinder of

revolution. All the normal vectors are in the same plane, so that for a cylinder along the axis

we have

Actually, it can be written as

which is clearly harmonic. The condition that (4.12) and (4.19) should agree imposes

Then the solution is

It is the trivial solution of (4.23). The reader would as easily get the function and see that it is

a plane wave. The other directions of the cylinder are deduced from a rigid rotation with (4.17).

The Lax pair of the sinh-Poisson equation is derived as follows. The conditions

and give

Then from the equation (2.CM) we have

thus is constant since it is real. We take as spectral parameter, the Maurer-Cartan

matrices are

so that by the gauge transformation

we get the linear system

which in the laboratory coordinates becomes

With the further gauge transformation (4.11) it can be directly compared with [8]. This Lax

pair was also derived in the vector representation in [10].

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