New constant mean curvature cylinders
M. Kilian, I. McIntosh & N. Schmitt
August 16, 1999
We use the DPW construction  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.
In the article  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 S2 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 1-form 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  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 
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
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  on Smyth surfaces we conjecture that these new
examples are complete and proper immersions. Indeed these surfaces
come in one-parameter 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 one-parameter 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. ) but these find the Iwasawa decomposition by
first turning it into a Riemann-Hilbert problem (i.e. Birkhoff factorization).
The dpwlab directly computes the Iwasawa decomposition according to the
theory described in .
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 (email@example.com).
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 DMS-9626804, DMS-9704949 and
DMS-9705479. 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 S1with values in G by
and define the twisted
The principal tool in the DPW method is the loop group Iwasawa
decomposition i.e. any
into a product
Now let us recall the DPW construction. Let D be a Riemann surface
its universal cover. Denote by
holomorphic 1-forms on D. Also define
The following steps (cf. ) give an S1 family,
called the associated family,
of immersions (possibly with branch points)
with constant mean curvature H.
and solve the initial value problem
- Apply the Iwasawa decomposition to
- The Sym-Bobenko formula yields
We call the 1-form
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
is also defined on
We list here a number of properties of the construction which
will be relevant for our surfaces.
Notice that the surface depends on the data
It is clear from the construction that the
infinite dimensional group
of holomorphic maps
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,
which is called the dressing action cf.
[3,6]. It is not hard to see from the Sym-Bobenko formula that
the dressing action of the subgroup
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. ).
We may write
are 1-forms on the CMC surface. A simple
calculation shows that
Further, if we write
then it can
be shown that f1 has metric
4r4|a1|2 and Hopf
It follows that f1 has
branch points at the zeroes of a1 and umbilics at the zeroes of
a2. When f1 is unbranched the metric can be written as
eu|dw|2 for a local conformal coordinate w on
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.
automorphism with w0 as a fixed point and
preserves the subalgebra
Since the base point w0 is fixed, the solution to (1) will satisfy
its lift to the group) preserves
by uniqueness of the Iwasawa decomposition.
Thus, by the Sym-Bobenko formula,
We will produce examples of this below, where
is a real involution on
is either the identity
These produce reflection symmetries of the immersion.
induces a finite order automorphism on
D, of order n, and we are given:
c) f1 is an immersion of D itself.
Then we may conclude that f1 has an n-fold rotational symmetry by the
following argument. By (b)
so by uniqueness of the Iwasawa
where R is a Euclidean motion. But R
must have order n, therefore it is a
rotation. In the examples below we will have
will induce a
The following two classes of known examples will turn out to play an
important role in understanding the new surfaces we will examine
We recall from e.g.  that potentials of the form
p(z)=k zm, for
and any constant k, give
the Smyth surfaces . These surfaces are characterized as CMC
planes which possess an intrinsic isometric S1-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
standard cylinder. The asymptotics of these ends have been quite
thoroughly studied. In  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
 that the surface is bounded by a cone.
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. ). 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.
All the Delaunay surfaces can be
obtained with the family of potentials
Here z is a coordinate on
and we use as the universal cover
that the map has period
is evaluated at 1in the Sym-Bobenko formula). The Delaunay surface with this potential
has neck radius
can be normalized by conjugation by a diagonal element of
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
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
respectively the holomorphic and unitary extended frames for the
particular, notice that
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
constant along the plane and with g=I, w0=0 in (1).
Recall (from e.g. ) that each such surface is partially
characterized by its spectral curve, which is a Riemann surface with
equation of the form
There is a (g-2)-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
For example, the spectral
curve for the Wente torus has genus 2 and this potential will compute
that surface once the cj 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 Runge-Kutta 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
for some appropriate value of K (typically between 20
and 100). To explicitly construct the Iwasawa factors of
one proceeds as follows (cf. ). Let
denote the Hilbert space
be the subspace of maps whose
Fourier series possess only non-negative powers of .
Notice that this is the
are the columns of ,
has codimension two. Now compute the orthogonal projections
these two span the space
Finally, let F1,F2 be the Gram-Schmidt
orthonormalization of the pair
F=(F1,F2). It is worth recalling from [9, p126] that on
V the L2-inner product and the
-inner product coincide,
hence F is unitary on S1.
The most time-expensive 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 Gram-Schmidt
orthonormalization of the basis
they are computed
more efficiently and stably with the following linear method.
If V is a finite-dimensional inner product space, U a subspace
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
therefore all our calculations take place on finite dimensional
A further speedup is achieved when the twisted structure of the loop
group is exploited.
Two elements of
with opposite polarity are
In this case, the linear system (6) decouples into two
simpler Hermitian systems.
This also means that the columns of
is unitary. Hence the map can be obtained by using
in place of F in the
Sym-Bobenko formula and taking the trace-free part of the result.
In this section we will present some new classes of CMC cylinders
First let us describe
some conditions under which the map
will be periodic on
(see also  for similar results). For any holomorphic potential
where we recall we have chosen to identify
We would like to define a similar notion for the unitary extended
but a priori we do not know that the quantity
is independent of w. However, we can prove the following crucial
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
well-defined holonomy. These observations allow us to formulate an
elementary characterization of the conditions under which a periodic
potential produces a periodic immersion.
be a solution of (1). Suppose
then, for a given
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
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
extends holomorphically into z=0. The key to this construction is
which belongs to
extend holomorphically to z
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
extends holomorphically to z=0, with
Given this, we have
has trivial holonomy about z=0,
By lemma 3.1 we have both
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
When we examine the expansion
we discover we must have
Therefore the coefficients Pk can be recursively determined
provided the operator
is invertible. The
only difficulty occurs for k=1, since the non-zero eigenvalues of
(and the reader
can easily verify that
). But for k=1 we have
which is solved by
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.
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 two-legged Smyth surface.
Figure 3 shows a larger piece of this surface.|
Let us begin by considering the simplest class of perturbations which
produce unbranched surfaces with umbilics. Here we take
where q(z)=kzm 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 m-fold 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 m-th roots of b/k, lie on these lines just before the
first self-intersections (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
which is (9) with a=1, b,c =0 and
q(z)=-z2p(z). Therefore it makes sense to think of the surfaces we
see as deformations of the Smyth surfaces, where the intrinsic
S1-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.
Two-legged Smyth surfaces with Delaunay heads.
The Smyth surface (middle) has a sphere-like head and
is a singular boundary between the unduloidal and nodoidal examples.|
The m-fold rotational symmetry is explained by reference to the earlier discussion
denote translation by .
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
for w0=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
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.
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
have the same Delaunay end. The
rationale here is that this might attach the surfaces for
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
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
The double Mr. Bubble is two two-legged Smyth surfaces
joined by an unduloidal neck.|
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
We will choose
-valued on the unit circle
It follows that the solution
to (1) (with g=I, w0=0) will take
and by Lemma 3.1 the holonomy
well-defined. It is not hard to see that
-valued on the
unit circle if and only if
Since it is always possible to gauge away the
diagonal terms of the potential, we may assume without loss of
is of the form
CMC Cylinders with a Closed Planar Geodesic
In the first class of potentials of this type we will also insist that
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
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
Proof. For ,
the solution of
is given by
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
Using the reality conditions on
this integral vanishes
The simplest example is obtained with the forms
Here the constant
where J0is the Bessel function of order zero. To see this observe that if we
parameterize the unit circle by
z(t)=eit we have
It follows that we have a discrete family of immersed CMC cylinders
indexed by the zeroes of
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
This CMC cylinder has the appearance of an unduloid conjoined with a nodoid.
The figure-eight 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 half-nodoid.|
More examples can be obtained using the following method. The first
holonomy condition is simply
where we consider
as 1-forms on
with an isolated singularity at
z=0. For the second condition, set
is a 1-form on
by the first
holonomy condition. It is straightforward to show for any 1-form
In particular we consider
the case where a is a primitive n-th root of unity. Then
and the reality conditions are satisfied,
are of the form
In this case the potential (11) possesses the
These imply that the surface has an n-fold rotational symmetry
then the surfaces will have n extra planes of reflective symmetry
(for example, see figure ).
The planar geodesic of this CMC cylinder, marked in black, lies
near the beginning of the sequence shown in figure
Figure 7 shows
a sequence of planar geodesic cross-sections for
CMC cylinders with potential (11) for
For c=0 we obtain the round sphere. As c increases (left to right)
the curves acquire more loops.
Planar geodesics which are the cross sections of a CMC cylinder family.
The dots mark the six umbilic points.|
Another class of examples is obtained by asking that the holomorphic
potential satisfy the conditions
Other CMC Cylinders
is the potential for a CMC cylinder with umbilics at the zeroes of
and branch points at the zeroes of
over the unit circle and we deduce
so the first holonomy condition of proposition
3.2 is satisfied. The second holonomy condition
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.
First, this class contains all Delaunay nodoids.
These arise if we take any
The explanation for this lies in the gauge transformation
achieved by gauging the left-hand potential by
where z=ew. This left-hand potential is
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.
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
nodoid-like sheath. Experiments suggest that all surfaces in this
class are bounded by the outer nodoid-like surface.
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 2-legged Smyth end. Divide the region
|z| >1 into its four quadrants. The lines at angles are mapped to unduloidal-like profiles, which decay in amplitude as
the radius increases. The line at angles
are mapped to
nodoidal-like 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
3-space. 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
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
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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