New constant mean curvature cylinders
M. Kilian, I. McIntosh & N. Schmitt
August 16, 1999
Abstract
We use the DPW construction [5] to present three new
classes of immersed CMC cylinders, each of which includes surfaces
with umbilics. The first class consists of cylinders with one end
asymptotic to a Delaunay surface. The second class presents surfaces with
a closed planar geodesic. In the third class each surface has a
closed curve of points with a common tangent plane.
Introduction
In the article [5] Dorfmeister, Pedit and Wu presented a
method by which all immersed CMC surfaces can, in principle, be constructed.
Their construction is based on the observation that the Gauss map
of every CMC surface is harmonic and every harmonic map from a surface
D to S^{2} is the projection of a horizontal holomorphic map from
its universal cover
into a certain loop group. Thus the
data for the DPW method is a holomorphic 1form with values in a
certain loop algebra: this is called a holomorphic potential. One of
the difficulties in using this method to construct new surfaces is
that if the potential actually lives on D it need not follow that it
produces an immersion of D. We usually only obtain an immersion of
:
this is the closing (or holonomy) problem. Therefore part
of the purpose here is to present some examples of solving the closing
problem in the simplest case, where
.
The simplest known examples of CMC cylinders are the Delaunay
surfaces, which are characterized by being cylinders of revolution
(this includes the standard cylinder). From [7] one knows
that if a CMC cylinder is complete and properly embedded then it must
be a Delaunay surface. Also, every properly embedded
annular end must be a Delaunay end
(i.e. asymptotic to a Delaunay surface) even if the surface
is not embedded. For
example, the `bubbletons' studied by Sterling and Wente in [11]
are immersed cylinders with no umbilics and both ends asymptotic to
the standard cylinder. Their results lead them to pose the question:
are there any CMC cylinders with umbilics? The examples we will
present include cylinders which have one Delaunay end and any number
of umbilics.
In fact we present three new classes of CMC cylinders. The first class
includes surfaces which are best thought of as a Smyth surface
[10,5] with the head replaced by a Delaunay end. Given the
results of [12] on Smyth surfaces we conjecture that these new
examples are complete and proper immersions. Indeed these surfaces
come in oneparameter families each of which includes a Smyth surface
(with the umbilic removed) as a degenerate limit, in the same way
that the Delaunay surfaces are a oneparameter family containing the
sphere (with two points removed) as a degenerate limit. The next class
consists of CMC cylinders which contain a closed planar geodesic.
The third class presents cylinders each of
which admits a closed curve of points with common tangent plane.
Although it is very
easy to read off the Hopf differential from the potential, it is
usually unclear how the geometry of the surface is encoded in the
potential. For example, there is as yet no understanding of the
conditions on a potential which ensure that the surface is either
proper, complete or embedded. The main obstacle in understanding the
passage from the potential to the surface is a loop group
factorization (the Iwasawa decomposition). This motivated us to build
a numerical package which would compute this factorization and produce
images of the surface: the approach is described below.
The result is a computer
laboratory called dpwlab written by the third author.
Other attempts have been made to implement the DPW
method numerically (e.g. [8]) but these find the Iwasawa decomposition by
first turning it into a RiemannHilbert problem (i.e. Birkhoff factorization).
The dpwlab directly computes the Iwasawa decomposition according to the
theory described in [9].
Further Information.
For further information about CMC surfaces, the DPW construction,
and the dpwlab software, visit the Center for Geometry,
Analysis, Numerics and Graphics website (www.gang.umass.edu)
or write to the third author (nick@gang.umass.edu).
Acknowledgments. We are grateful to Franz Pedit and Josef
Dorfmeister for helpful discussions, and to Rob Kusner for his encouragement.
This work was partially supported by NSF grants DMS9626804, DMS9704949 and
DMS9705479. The first author
was also partially supported by
SFB 288 at Technische Universität Berlin.
The DPW Construction
Before stating the DPW recipe, we introduce the ingredients. For
,
denote the analytic maps of the unit circle S^{1}with values in G by
and define the twisted
loops by
Furthermore, define
The principal tool in the DPW method is the loop group Iwasawa
decomposition i.e. any
factorizes uniquely
into a product
where
and
.
Now let us recall the DPW construction. Let D be a Riemann surface
and
its universal cover. Denote by
the
holomorphic 1forms on D. Also define
The following steps (cf. [5]) give an S^{1} family,
called the associated family,
of immersions (possibly with branch points)
with constant mean curvature H.
 1.
 Let
and solve the initial value problem

(1) 
where
and
.
Then
is defined
on ;
 2.
 Apply the Iwasawa decomposition to
pointwise on
to obtain
;
 3.
 The SymBobenko formula yields
We call the 1form
the holomorphic potential and
the extended holomorphic frame. The unitary factor
is called the extended unitary frame.
Our principal interest in this paper is to construct examples where
and provide sufficient conditions to ensure that the
resultant map
is also defined on
for .
Properties
We list here a number of properties of the construction which
will be relevant for our surfaces.
Group actions.
Notice that the surface depends on the data
.
It is clear from the construction that the
infinite dimensional group
of holomorphic maps
with
acts by
gauge transformation on the fibers of the map
,
since the map
leaves the surface unchanged. In fact one can
always gauge away the diagonal terms of
.
Another group action is
the left action of
on the initial condition,
for
,
which is called the dressing action cf.
[3,6]. It is not hard to see from the SymBobenko formula that
the dressing action of the subgroup
can only
result in Euclidean motions of the surface, therefore it is more usual to
think of the dressing action as being by
.
Metric and Hopf differential (cf. [5]).
We may write
where
are 1forms on the CMC surface. A simple
calculation shows that
.
Further, if we write
for
and
,
then it can
be shown that f_{1} has metric
4r^{4}a_{1}^{2} and Hopf
differential
.
It follows that f_{1} has
branch points at the zeroes of a_{1} and umbilics at the zeroes of
a_{2}. When f_{1} is unbranched the metric can be written as
e^{u}dw^{2} for a local conformal coordinate w on
and we
have
Symmetries.
We cannot usually expect the symmetries of the potential to be
passed on to the CMC immersion because they might
not survive the combination of integration and factorization.
However, there are two situations which occur in our
examples where symmetries will appear in the CMC surface.
A. Suppose
is an
automorphism with w_{0} as a fixed point and
where
preserves the subalgebra
.
Since the base point w_{0} is fixed, the solution to (1) will satisfy
.
Now, since
(or rather,
its lift to the group) preserves
,
we have
by uniqueness of the Iwasawa decomposition.
Thus, by the SymBobenko formula,
.
We will produce examples of this below, where
is a real involution on
and
is either the identity
or
.
These produce reflection symmetries of the immersion.
B. Suppose
induces a finite order automorphism on
D, of order n, and we are given:
a)
;
b)
belongs to
;
c) f_{1} is an immersion of D itself.
Then we may conclude that f_{1} has an nfold rotational symmetry by the
following argument. By (b)
,
so by uniqueness of the Iwasawa
decomposition
.
Therefore
where R is a Euclidean motion. But R
must have order n, therefore it is a
rotation. In the examples below we will have
and
will induce a
rotation on
.
Known examples
The following two classes of known examples will turn out to play an
important role in understanding the new surfaces we will examine
later.
Example 1.
We recall from e.g. [5] that potentials of the form

(2) 
where
p(z)=k z^{m}, for
and any constant k, give
the Smyth surfaces [10]. These surfaces are characterized as CMC
planes which possess an intrinsic isometric S^{1}action (with a fixed
point). If we think of these as singly punctured (topological)
spheres, they have one end with m+2 `legs' and an (m+2)fold
rotational symmetry. We must beware of some degenerate cases: taking
yields the round sphere while
gives the
standard cylinder. The asymptotics of these ends have been quite
thoroughly studied. In [12] it was shown that these surfaces
are proper immersions and that for
,
,
there are polar coordinate rays
which are planar geodesics. The curvature of
these, in the limit as
,
tends to 0 for n even and 1for n odd. The legs develop around those lines with n even, along
which the distance from the origin grows fastest. The angle between
the legs depends upon the coefficient k. Further, it was shown in
[1] that the surface is bounded by a cone.

Figure 1
Sector of a Smyth surface bounded by a nodoidal planar geodesics.

More general surfaces can be obtained by allowing p(z) to be any
polynomial (cf. [5]). The resultant surfaces have m+2legs, where ,
each of which looks like a Smyth surface
leg. To the best of our knowledge, there has not been any work which
describes the strength of this resemblance. Of course, these surfaces
need not possess either intrinsic or extrinsic symmetries.
Example 2.
All the Delaunay surfaces can be
obtained with the family of potentials

(3) 
Here z is a coordinate on
and we use as the universal cover
the map
;
.
The conditions
with
ensure
that the map has period
(here,
is evaluated at 1in the SymBobenko formula). The Delaunay surface with this potential
has neck radius
.
The potential
can be normalized by conjugation by a diagonal element of
so that
.
The parameter c, although not strictly necessary, is sometimes useful.
Its geometric effect is to introduce a phase shift along the profile
curve of the Delaunay surface.
Unduloids and nodoids are obtained when ab>0 and ab<0 respectively.
The limiting case ab=0 yields a sphere
with two points removed. In fact the gauge transformation

(4) 
transforms the potential (2) (on
)
with into the potential
(3) with a=1, b=0, c=0. This gauge transformation will be
useful later on.
Below we will use
and
to denote
respectively the holomorphic and unitary extended frames for the
potential
with
.
In
particular, notice that
.
Remark.
The Delaunay potentials fit into the following more
general context. By a result of Burstall & Pedit [2, Thm 4.3],
each CMC surface with doubly periodic Gauss map can be
obtained from a holomorphic potential
on
which is
constant along the plane and with g=I, w_{0}=0 in (1).
Recall (from e.g. [1]) that each such surface is partially
characterized by its spectral curve, which is a Riemann surface with
equation of the form
There is a (g2)parameter family of CMC surfaces with the same
spectral curve: for g=1 there is one surface for each spectral curve
and this surface is a Delaunay surface. It can be shown (we omit the
proof here) that one of the surfaces with the spectral curve
above can be obtained by taking
where
.
For example, the spectral
curve for the Wente torus has genus 2 and this potential will compute
that surface once the c_{j} are known.
Implementation of the DPW Procedure in Software
Of the three steps in the DPW process, the second step requires the
most attention. The integration step 1 is performed using a standard
fourth order RungeKutta method. We always work with potentials
which are Laurent polynomials in ,
hence we are
always dealing with the Iwasawa decomposition of Laurent polynomial
loops. In software, an element of
is represented as a finite
vector, consisting of the coefficients of
to
for some appropriate value of K (typically between 20
and 100). To explicitly construct the Iwasawa factors of
one proceeds as follows (cf. [9]). Let
denote the Hilbert space
and let
be the subspace of maps whose
Fourier series possess only nonnegative powers of .
Define
.
Notice that this is the
span of
,
where
are the columns of ,
and that
has codimension two. Now compute the orthogonal projections

(5) 
and define
:
these two span the space
.
Finally, let F_{1},F_{2} be the GramSchmidt
orthonormalization of the pair
,
then
F=(F_{1},F_{2}). It is worth recalling from [9, p126] that on
V the L^{2}inner product and the
inner product coincide,
hence F is unitary on S^{1}.
The most timeexpensive part of the software version of the DPW
procedure arises from computing the projections (5).
While these can be found directly (by e.g. the GramSchmidt
orthonormalization of the basis
)
they are computed
more efficiently and stably with the following linear method.
If V is a finitedimensional inner product space, U a subspace
with basis
,
and ,
then
where
solve the
linear system

(6) 
Since this system is Hermitian it can be solved by Cholesky
decomposition. Notice that if
denotes the Laurent polynomials with zero coefficient of for k>K then
for j>2K,
therefore all our calculations take place on finite dimensional
subspaces of
.
A further speedup is achieved when the twisted structure of the loop
group is exploited.
Two elements of
with opposite polarity are
L^{2}orthogonal.
In this case, the linear system (6) decouples into two
simpler Hermitian systems.
This also means that the columns of
are already
orthogonal i.e.
is unitary. Hence the map can be obtained by using
in place of F in the
SymBobenko formula and taking the tracefree part of the result.
CMC Cylinders
In this section we will present some new classes of CMC cylinders
for which
.
First let us describe
some conditions under which the map
will be periodic on
(see also [4] for similar results). For any holomorphic potential
the extended
holomorphic frame
has holonomy
where we recall we have chosen to identify
with
.
We would like to define a similar notion for the unitary extended
frame
but a priori we do not know that the quantity
is independent of w. However, we can prove the following crucial
lemma.Lemma 3.1
Suppose
.
Then
is independent
of
and equals
.
Proof. Since
we
have
which implies
.
The result now follows by uniqueness
of the Iwasawa decomposition.
Therefore, under the conditions of the lemma, we can sensibly call
the holonomy of
(and in fact this implies
is periodic). Notice that if one knows that
the surface is a cylinder then
necessarily has
welldefined holonomy. These observations allow us to formulate an
elementary characterization of the conditions under which a periodic
potential produces a periodic immersion.
Proposition 3.2
Let
and
be a solution of (1). Suppose
,
then, for a given
,
the
holonomy of
satisfies
iff the associated family member
obtained by the DPW construction is a CMC immersion of a cylinder.
We will usually work with
.
Let us now consider some
classes of potentials which satisfy the conditions of this
proposition.
Cylinders with One End Asymptotic to a Delaunay Surface
An interesting class of surfaces is obtained by perturbing the
Delaunay potential (3) by a potential on
which
extends holomorphically into z=0. The key to this construction is
that
has holonomy
,
which belongs to
.
Proposition 3.3
Let
extend holomorphically to z=0 with
.
Then using the potential
in (1), with
an appropriate initial condition, produces a cylinder with one end
asymptotic to the Delaunay surface with potential
.
Proof. Consider the system (1) as a first order system of ODE with
a regular singular point at z=0. We will show below that a solution
can be written in the form

(7) 
where
extends holomorphically to z=0, with
.
Given this, we have
since
has trivial holonomy about z=0,
so
.
By lemma 3.1 we have both
and
hence
.
Since the
Delaunay surface satisfies proposition 3.2 for
,
so does the perturbed surface.
Now let us verify (7). For
to exist, there must
be a solution to the differential equation

(8) 
When we examine the expansion
we discover we must have
Therefore the coefficients P_{k} can be recursively determined
provided the operator
is invertible. The
only difficulty occurs for k=1, since the nonzero eigenvalues of
are
for
(and the reader
can easily verify that
for
). But for k=1 we have
which is solved by
since
.
Therefore a solution
exists of the type required.
Notice that in this class of examples we have more or less complete
freedom to specify the location of the umbilics.

Figure 2
CMC cylinder with two umbilics, one marked with a dot.
Asymptotically, one end is a Delaunay nodoid with a thin neck
and the other is a twolegged Smyth surface.
Figure 3 shows a larger piece of this surface. 
Example 3.
Let us begin by considering the simplest class of perturbations which
produce unbranched surfaces with umbilics. Here we take

(9) 
where q(z)=kz^{m} for
and k some constant. By the previous
proof, to obtain a cylinder we must use the initial condition
in (1): this means
first computing the solution
to (8). It is
quite remarkable to see that the surfaces obtained appear to be the
result of attaching a Delaunay end to the head of a Smyth surface. We
have observed that the end opposite to the Delaunay end has m legs
which have all the visible characteristics of the Smyth surface legs
described above, and possesses an mfold symmetry. It appears that
there are 2m planar coordinate lines, one for each angle and the legs develop around those for n even. The umbilics, which
lie at the mth roots of b/k, lie on these lines just before the
first selfintersections (as we move away from the Delaunay end).
Indeed, each Smyth surface lies in a one real parameter family of
surfaces with potential
(9). To see this, observe that the gauge transformation
(4) transforms the Smyth surface potential as

(10) 
which is (9) with a=1, b,c =0 and
q(z)=z^{2}p(z). Therefore it makes sense to think of the surfaces we
see as deformations of the Smyth surfaces, where the intrinsic
S^{1}symmetry has been broken by the bifurcation of the multiple
umbilic at z=0 into m umbilics at equal distance from the origin
and at equal angles. Because of this, we conjecture that these
cylinders are complete and proper immersions for which the end for
is bounded by a cone.

Figure 3
Twolegged Smyth surfaces with Delaunay heads.
The Smyth surface (middle) has a spherelike head and
is a singular boundary between the unduloidal and nodoidal examples. 
The mfold rotational symmetry is explained by reference to the earlier discussion
2.1B. Let
denote translation by .
This induces
on
a rotation through this angle. Then
,
so the same is true for
.
A careful examination of the series
expansion of (8) shows that this implies
.
It follows that
belongs to
for w_{0}=0, therefore we have all
the conditions for this symmetry to exhibit itself on the surface.
More general types of perturbations than (9) do not seem
to alter the end behavior a great deal. Certainly taking q(z) to be
any polynomial has the effect one expects from knowledge of the
generalized Smyth surfaces: the number of legs is
and their
direction depends in some way upon the roots of q(z)b. If we
consider perturbations at higher powers of
we can obtain
surfaces with no umbilics but they still appear to have the same end behaviour.
Example 4.
The form of the potential (9) made us think that to
some extent we may be able to treat holomorphic potentials like
building blocks to patch two types of end behavior together. Therefore
we considered potentials of the form
where
and
have the same Delaunay end. The
rationale here is that this might attach the surfaces for
and
together along a Delaunay
tube centered at the image of z =1 (we can always make
this lie on the Delaunay end by suitable scaling). For
of the type (9) this
amounts to examining potentials of the form (9) where now
q(z)=z^{2}r(z)+z^{2}s(1/z) and
are entire
functions. Although we do not claim that the resultant
surfaces close into cylinders, the experiments show that they are very
close to closing and are quite stable to perturbations of the
coefficients of r and s. As one would hope, each end has the
expected number of legs:
for the end near z=0 and
for the end near infinity. We believe that cylinders of this
type exist with the correct choice of initial condition for
(1).

Figure 4
The double Mr. Bubble is two twolegged Smyth surfaces
joined by an unduloidal neck. 
CMC Cylinders with a Closed Planar Geodesic
Proposition 3.2 gives conditions on the holonomy of
the extended unitary frame
which are in general hard to
verify, since both integration of (1) and the subsequent
Iwasawa decomposition cannot usually be performed explicitly. Here we
will work with a class of potentials for which
(1) can be integrated explicitly at least over the
unit circle.
We will choose
to be
valued on the unit circle
z=1 in
.
It follows that the solution
to (1) (with g=I, w_{0}=0) will take
values in
,
whence
along
and by Lemma 3.1 the holonomy
is
welldefined. It is not hard to see that
is
valued on the
unit circle if and only if
where
.
Since it is always possible to gauge away the
diagonal terms of the potential, we may assume without loss of
generality that
is of the form
with
.
In the first class of potentials of this type we will also insist that
both satisfy
.
In this case, under the conditions of the next proposition, the image
of the unit circle is a planar geodesic: we exhibit some
examples in figure 7.
For the purposes of the next proposition, let z(t) denote the contour
in
.
Proposition 3.4
Let
satisfy:
 1.

,
;
 2.

and
 3.

.
Then

(11) 
is the potential for a CMC cylinder with umbilics at the zeroes of
.
Further, the plane containing the image of the unit circle is a plane of
reflective symmetry.
Proof. For ,
the solution of
along
is given by
.
Therefore
Since
along
the first holonomy condition
from proposition 3.2 becomes
,
which is equivalent to
Similarly, it is straightforward to check that the second holonomy
condition of proposition 3.2 is implied by
.
A computation yields
where
.
Using the reality conditions on
,
this integral vanishes
precisely when
Example 5.
The simplest example is obtained with the forms
Here the constant
must satisfy
where J_{0}is the Bessel function of order zero. To see this observe that if we
parameterize the unit circle by
z(t)=e^{it} we have
It follows that we have a discrete family of immersed CMC cylinders
indexed by the zeroes of
.
Further,
and
.
Therefore each cylinder in this family
has two planar symmetries: one plane containing the image of the unit
circle and the other containing the image of the real axis. From the
graphics we observe that, near the planar geodesic, the image of the
the positive real axis resembles a profile curve of a nodoid while the
image of the negative real axis resembles the profile curve of an
unduloid. Figure 5
displays aspects of the surface for the first positive root of
.

Figure 5
This CMC cylinder has the appearance of an unduloid conjoined with a nodoid.
The figureeight in the transparent image
is the planar geodesic across which the
surface has reflective symmetry. As it evolves toward
an end, one of its loops sweeps out half of the unduloid,
the other traces the opposite halfnodoid. 
More examples can be obtained using the following method. The first
holonomy condition is simply
,
where we consider
as 1forms on
with an isolated singularity at
z=0. For the second condition, set
,
then
is a 1form on
by the first
holonomy condition. It is straightforward to show for any 1form
that
whenever
for
with .
In particular we consider
the case where a is a primitive nth root of unity. Then
,
and the reality conditions are satisfied,
if
are of the form
with
for
.
In this case the potential (11) possesses the
symmetries
and
.
These imply that the surface has an nfold rotational symmetry
(since
is
valued over
).
Further, if
also satisfy
then the surfaces will have n extra planes of reflective symmetry
(for example, see figure ).

Figure 6
The planar geodesic of this CMC cylinder, marked in black, lies
near the beginning of the sequence shown in figure
7. 
Figure 7 shows
a sequence of planar geodesic crosssections for
CMC cylinders with potential (11) for
and
,
where
.
For c=0 we obtain the round sphere. As c increases (left to right)
the curves acquire more loops.

Figure 7
Planar geodesics which are the cross sections of a CMC cylinder family.
The dots mark the six umbilic points. 
Other CMC Cylinders
Another class of examples is obtained by asking that the holomorphic
potential satisfy the conditions
for
and .
Proposition 3.5
Let
with
.
Then

(12) 
is the potential for a CMC cylinder with umbilics at the zeroes of
and branch points at the zeroes of .
Proof. Since
,
the solution
to (1)
with
has .
As above,
over the unit circle and we deduce
,
so the first holonomy condition of proposition
3.2 is satisfied. The second holonomy condition
follows from
as in the proof of proposition 3.4.
The cylinders generated by these potentials have constant frame
over the unit circle. This means that the Gauss map is
constant along the image of the unit circle so that this lies on a
single tangent plane to the surface.
Example 6.
First, this class contains all Delaunay nodoids.
These arise if we take any
and set
.
The explanation for this lies in the gauge transformation
achieved by gauging the lefthand potential by
where z=e^{w}. This lefthand potential is
with
a = b = 1/s and c=1/2
By contrast, if we take
we obtain the surface in
figure 8. This example displays
the characteristic features of the cylinders in this class.

Figure 8
This CMC
cylinder is tangent to a plane along the black curve in solid figure. 
More generally, if
for a polynomial p(z), we have observed that
the resulting surface has
legs emerging within a
nodoidlike sheath. Experiments suggest that all surfaces in this
class are bounded by the outer nodoidlike surface.
Concluding Remarks
It is difficult to convey in static pictures the intuition gained by
being able to rotate, cut away and zoom in on these surfaces. One
feature which struck us was the ubiquity of nodoidal and unduloidal
features in the ends. In fact, the Smyth end itself, which at first
looks impossibly complicated, appears to have the following simple
description. Consider the 2legged Smyth end. Divide the region
z >1 into its four quadrants. The lines at angles are mapped to unduloidallike profiles, which decay in amplitude as
the radius increases. The line at angles
are mapped to
nodoidallike profiles which become more circular as the radius
increases. Between these lines the surface must interpolate between an
unduloid and a nodoid. It does so in a way which strongly resembles
the way a Delaunay unduloid unravels and wraps up into a Delaunay
nodoid as it moves through the associated family.
The surfaces introduced in sections
3.2 and 3.3 have a similar description
as we rotate around
.
But their behaviour as the radius
increases or decreases is quite different. From the figures 5 and 8 we are
lead to ask: are either of these surfaces bounded by a standard cylinder?
For the surface with a
planar geodesic in figure 5, as z increases from z=1 (or as
it decreases) each circle is stretched in two opposite directions in
3space. Since the image of circles of constant z appear
to pass through the central plane of reflection not far from the
planar curve, it is not yet settled whether these examples are properly
immersed.
On the other hand, the surface in figure 8 seems to be made by translating the same shape as
the radius z increases and decreases from z,
although this cannot be literally true since there is only one umbilic
(and branch point): it lies at z=1. This suggests that this map is
proper.
It seems that these surfaces give two new types of end behaviour
which, although they are immersed, do not appear to be significantly
more complicated than the Smyth surfaces.
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 A Bobenko,
Constant mean curvature surfaces and integrable equations,
Russian Math. Surveys 46 (1991), 145.
 2
 F E Burstall & F Pedit,
Harmonic maps via AdlerKostantSymes theory, in
Harmonic maps and integrable systems,
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 F E Burstall & F Pedit,
Dressing orbits of harmonic maps,
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