# 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 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 [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 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. [8]) 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 [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 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 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 1-forms on D. Also define

The following steps (cf. [5]) give an S1 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 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 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 Sym-Bobenko 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 1-forms on the CMC surface. A simple calculation shows that . Further, if we write

for and , then it can be shown that f1 has metric 4r4|a1|2 and Hopf differential . 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 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 w0 as a fixed point and where preserves the subalgebra . Since the base point w0 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 Sym-Bobenko 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) 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 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 zm, for and any constant k, give the Smyth surfaces [10]. 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 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 Sym-Bobenko 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, w0=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 (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

where . 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 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 non-negative 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 F1,F2 be the Gram-Schmidt orthonormalization of the pair , then 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 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 L2-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 Sym-Bobenko formula and taking the trace-free 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 well-defined 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 Pk can be recursively determined provided the operator is invertible. The only difficulty occurs for k=1, since the non-zero 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 two-legged 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)=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

 (10)

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.

 Figure 3 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 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 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 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)=z2r(z)+z-2s(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 two-legged 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, w0=0) will take values in , whence along and by Lemma 3.1 the holonomy is well-defined. 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 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 . 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 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 , then is a 1-form on by the first holonomy condition. It is straightforward to show for any 1-form that whenever for with . In particular we consider the case where a is a primitive n-th 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 n-fold 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 cross-sections 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 left-hand potential by

where z=ew. This left-hand 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 nodoid-like sheath. Experiments suggest that all surfaces in this class are bounded by the outer nodoid-like 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 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 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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