Families of conformal tori of revolution in the 3–sphere

Families of conformal tori of revolution in the 3–sphere

Abstract.

For all positive integers we construct a 1–parameter family of conformal tori of revolution in the 3–sphere with bulges. These tori arise by Darboux transformations of constant mean curvature tori in the 3–sphere but do not have constant mean curvature in .

Author partially supported by DFG SPP 1154 “Global Differential Geometry”

1. Introduction

In a recent paper [2] it is shown that the multiplier spectral curve of a conformal torus is essentially given by the set of closed Darboux transforms of : to each multiplier on the spectral curve there exists a quaternionic holomorphic section with the given multiplier in the associated quaternionic holomorphic line bundle of . The prolongation of the holomorphic section defines a new conformal torus , and it turns out that and satisfy a “weak enveloping” condition. Thus form a generalized Darboux pair: Classically, the Darboux transformation is defined for isothermic surfaces and a map is called a classical Darboux transform [5] of an isothermic if there exists a sphere congruence enveloping both and .

For every conformal torus the set of holomorphic sections with a given multiplier is generically 1–dimensional, and at generic points the Darboux transformation preserves geometric properties: e.g., generic Darboux transforms of a constant mean curvature torus have constant mean curvature [4], and generic Darboux transforms of a Hamiltonian stationary torus are Hamiltonian stationary [8].

However, there exist examples, e.g. [8], of conformal tori which allow non–trivial multiplier on the spectral curve with high dimensional space of holomorphic sections. The existence of these singular multipliers should allow a deformation of the spectral curve: in the case of constant mean curvature tori in the 3–sphere of spectral genus zero [7] one can deform the spectral curve to obtain a family of Delaunay tori by removing this singularity of the spectral curve, and thus by adding geometric genus. By contrast the Darboux transformation preserves the geometric spectral genus [2] in the case when the Darboux transform is immersed but it may change geometric properties (e.g. break the constant mean curvature condition). In particular, the Darboux transformation at singular points is expected to allow to add or remove arithmetic genus of the spectral curve, and a thorough understanding of the singular points of the multiplier spectral curve may play an important role in understanding the reconstruction of conformal tori by their spectral data [6, 10, 1, 9] and the study of minimum energy tori in presence of a variational principle [11, 7].

In this short note, we concentrate on the geometric properties of the Darboux transformation in the case when the conformal torus is a rectangular torus in the 3–sphere: in particular, we construct for each a 1–parameter family of conformal tori of revolution in with bulges which do not have constant mean curvature in . Using a similar argument we also construct for each a 1–parameter family of cylinder of revolution with non–constant mean curvature.

Acknowledgments. The author would like to thank U. Hertrich–Jeromin, F. Pedit and N. Schmitt for fruitful discussions during the preparation of this work.

2. Rectangular tori

In the following, we will apply the Darboux transformation on rectangular tori with lattice , , to construct the families of conformal tori of revolution. We identify Euclidean 4–space with the quaternions and parametrize a rectangular torus with parameters by

In particular, we will use the fact that a rectangular torus is Hamiltonian stationary, and use the methods and settings developed in [8] to compute the Darboux transforms of . We can write as

where the so–called Lagrangian angle is given by

and

with scale . Moreover, The derivative of can be written as

and is a conformal immersion, that is [3, Sec. 2.2]

with left normal

and right normal

For every conformal immersion with right and left normals and the normal bundle of is given [3, Sec 2.2] by

In particular, if is a conformal map into the 3–sphere, then and are unit normals. Thus, the second fundamental form of as a map into computes with

as

From this we see that the mean curvature vector in relates to the mean curvature vector via

where is the mean curvature of in . We denote by

the and –parts of the derivative of with respect to the complex structure , and define by . Then it is shown in [3, Sec. 7.2] that the mean curvature vector of a conformal immersion into is given by

(2.1)

Combining the previous equations, we see that the mean curvature of a conformal immersion of a Riemann surface into is given by

(2.2)

In particular, since computes in the case of Hamiltonian stationary Lagrangians [8] to

the constant mean curvature of a rectangular torus in is given by

3. The Darboux transformation

We will briefly recall the construction of Darboux transforms in the case when is a conformal immersion from a Riemann surface into Euclidean 4–space. For the general case of conformal immersions into the 4–sphere and details of the construction compare [2]. In our situation, the associated quaternionic holomorphic line bundle of the immersion can be identified with the trivial quaternionic bundle equipped with the (quaternionic) holomorphic structure given by

for where is the left normal of . We denote by the set of holomorphic sections of the holomorphic line bundle . The prolongation of a local holomorphic section is given by the local section

(3.1)

of the trivial bundle where is defined by . Then spans locally a quaternionic line bundle , and if is nowhere vanishing, the corresponding map is a branched conformal immersion into , a so–called a Darboux transform of . If we denote by , then the derivative of is given away from the zeros of by

From we see that , in other words, since has right normal . In particular, has left normal

(3.2)

To compute the mean curvature vector of the Darboux transform , it remains (2.1) to compute using the defining equation . To this end, note that the derivative of computes with as

so that the -part of with respect to is given by

(3.3)

where .

To obtain Darboux transforms which are globally defined, we consider holomorphic sections with multiplier, that is holomorphic sections of the trivial bundle over the universal cover of which satisfy

with for all . From and (3.1) we see that the prolongation of has multiplier , that is for , so that defines a branched conformal immersion if is nowhere vanishing.

In the case when is a conformal 2–torus, the existence of global Darboux transforms is guaranteed by the link [2] between Darboux transforms and the multiplier spectral curve of : to every multiplier there exists at least one holomorphic section with multiplier , and each such holomorphic section gives by prolongation a Darboux transform of . In other words, there is at least a Riemann surface worth of Darboux transforms of a conformal torus.

4. Darboux transforms of Hamiltonian stationary Lagrangian tori

In the following we summarize notations and results of [8]. In the case of an Hamiltonian stationary Lagrangian torus with Lagrangian angle , every multiplier of a holomorphic section is of the form

with such that

is not empty. A holomorphic section with multiplier is called monochromatic if it is given by a Fourier monomial, that is if

(4.1)

is given by a single frequency where and

A polychromatic holomorphic section with multiplier is given by a non–trivial linear combination of monochromatic holomorphic sections, .

Definition 4.1.

A branched conformal immersion is called a monochromatic Darboux transform (respectively polychromatic) if it is given by the prolongation of a monochromatic (respectively polychromatic) holomorphic section.

In [8] it is shown that all monochromatic Darboux transforms of a rectangular torus are after reparametrization again a rectangular torus. Moreover, polychromatic holomorphic sections with multiplier , only occur if is real. In particular, the corresponding Darboux transform coincides with a monochromatic Darboux transform. To obtain new tori we therefore have to consider polychromatic Darboux transforms with :

Theorem 4.2 ([8]).

Let be a Hamiltonian stationary torus in with Lagrangian angle and . Then every non–constant, polychromatic Darboux transform of with is given by

(4.2)

where the finite set

parametrizes the admissible frequencies and are chosen so that the map

is nowhere vanishing.

5. Families of conformal tori of revolution in

In this section we discuss the Darboux transforms of a rectangular torus . If the parameters of satisfy , we show that there exist polychromatic Darboux transforms which are conformal tori in the 3–sphere.

Fix and consider all rectangular tori in the 3–sphere with parameters satisfying . The multiplier with

has since

which shows that allows polychromatic Darboux transforms. In particular, we see that

with

(5.1)

and

so that (4.1) all monochromatic holomorphic sections with multiplier are given by

Using Theorem 4.2 we see that the corresponding polychromatic Darboux transforms are given by

where

(5.2)

and have to be chosen so that

is nowhere vanishing where

Using (5.1) and the denominatior simplifies to

In general, the above Darboux transforms will be conformal immersions into the 4–sphere. However, there exist constants so that the polychromatic Darboux transform is a conformal immersion into the 3–sphere.

Theorem 5.3.

For each there exists a 1–parameter family of conformal tori of revolution in the 3–sphere with bulges. Only one conformal torus in this family has constant mean curvature in .

Proof.

We consider the case that . In this case (5.1)

shows that

From (5.2)

we see that is independent of the choice of . In particular, we may assume from now on that . Then

is independent of while

is independent of . In particular, the Darboux transform is given by where , with

and

Writing where

(5.3)

a lengthy, but straightforward, computation shows that

(5.4)

In other words, is a conformal map into the 3–sphere. On the other hand, we have

where is a real valued function, and both and only depend on , that is, is a surface of revolution in . Note that for , the Darboux transform is a rectangular torus with constant mean curvature in the 3–sphere. If then is extremal at

in particular, is a torus of revolution in with bulges.

Figure 1. Darboux transforms of rectangular tori with parameter (1.8,1) and (2.1, 1)

If had constant mean curvature in and is not a rectangular torus, we would expect to be a Delaunay torus and thus to be parametrized by elliptic functions. However, the Darboux transformation essentially preserves spectral genus [2], and in our case is parametrized by (rational functions of) trigonometric functions. Indeed, we compute the mean curvature of in explicitly when : first, is defined by , that is (3.3)

where with

Figure 2. Non–embedded Darboux transform of the rectangular torus with parameter (2.6, 1)

Moreover, we compute with real valued

The left normal of is given by (3.2)

so that with and

Next observe that for any we have which shows that

and the mean curvature of in is, using , given by (2.2)

(5.5)

Furthermore, for , we have with

and we get for both and

whereas for

A straightforward computation shows that

so that (5.5) simplifies to

(5.6)
Figure 3. Darboux transforms of rectangular tori with parameter (2.9,1), (3.2, 1) and (3.5,1)

For one easily obtains with

that

so that (5.6) becomes

Similarly, for we obtain with

that

which gives

In particular, if has constant mean curvature in then gives

which is equivalent to

that is, . Finally, we notice that all rectangular tori are, up to reparametrization , , rectangular tori with parameter . Thus, we obtain for each , a 1–parameter family of tori of revolution in with bulges, each torus given by a polychromatic Darboux transform of a rectangular torus with parameter . ∎

Figure 4. Darboux transforms of rectangular tori with parameter (4.3,1), (5.3, 1) and (6.3,1)

6. Polychromatic Darboux transforms of cylinders

We use similar methods to compute polychromatic Darboux transforms of a standard cylinder . To stay close to the notations and computations in the previous sections, our maps will take values in . Note that in this case has mean curvature [3] given by

where is again given by . A standard cylinder

is then a Hamiltonian stationary immersion with harmonic left and right normals

and Lagrangian angle with Moreover, we have with and With the same methods as before (with the obvious adaptions to the situation of a cylinder), we consider for all cylinder with , and obtain again for

holomorphic sections with multiplier. The corresponding frequencies are

and the monochromatic holomorphic sections with multiplier are

Again, we apply Theorem 4.2 with constants , and obtain, after a similar computation as in the case of rectangular tori, the monochromatic Darboux transforms of a cylinder for as

with real valued functions

and

Here, we have with

and thus, both and only depend on . In particular, is a surface of revolution in the 3–space spanned by , and obviously, is a round cylinder whenever for .

Figure 5. polychromatic Darboux transforms of cylinder of radius 2.1 and 2.9

We now compute the mean curvature of . To that end, we observe that with

and the left normal of is given (3.2) by

with

real valued. As before, we compute with (5.5)

and, by evaluating at and , we see that constant is equivalent to . We summarize

Theorem 6.4.

For all , the Darboux transformation gives a 1–parameter family of cylinder of revolution which are not constant mean curvature cylinder in the 3–space.

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