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Surfaces: from generalised Weierstrass representation to Cartan moving frame (v4)

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Abstract

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 differential equations.
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 dierential 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 dierential 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 dierential
operators. We shall work with complex numbers for compactness, and denote the complex
conjugate by an overbar. We rst dene the Wirtinger dierential 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 denitions:
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 dening relation has been used:
In addition, the following dierential 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 dening 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 satised 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
Dierentiating, 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 dierential 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 dierent 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 dierence 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 dierent 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 innitesimal displacement in the plane is now associated an
innitesimal change of the origin and of the basis vectors, but this time expressed with respect to
the moving frame itself. By using dierential -forms (also called Pfaan forms,) this is written
down as
which is called the rst system of structure equations by Cartan. The vectors satisfy the
orthonormality conditions
Dierentiating them, and using the structure equations (2.1), we get relations among the
dierential 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 dierentiating 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 satised. 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 dierential 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 dierential form (2.11) is a connection, and actually an ane 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 ane 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 dened on the surface, thus the term stands for the ane
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 identication we get
Now using these coecients together with the structure equations (2.1), we are able to
express the data of the surface in term of the Pfaan 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 dened
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 denition 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 dierential 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 dierential 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 dierential
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 dened 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 dierentiating
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
dierence of two expressions that are the complex conjugate of each other. If it is satised, that
means that
and therefore from (3.11) it is easily seen that one of the integrability conditions (2.G) is
fullled. Then a tedious but straightforward calculation shows that the other one (2.CM) is as
well fullled. Therefore it is a sucient 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 satised. But that is just the equation satised 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 Homan 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 dierent 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 modied 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 denition (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 dierent
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 satises
(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 dierential 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)
simplies 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 satised.
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 veries
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 diering 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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