Families of conformal tori of revolution in the 3–sphere
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 .
In a recent paper  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  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 , and generic Darboux transforms of a Hamiltonian stationary torus are Hamiltonian stationary .
However, there exist examples, e.g. , 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  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  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  to compute the Darboux transforms of . We can write as
where the so–called Lagrangian angle is given by
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
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
Combining the previous equations, we see that the mean curvature of a conformal immersion of a Riemann surface into is given by
In particular, since computes in the case of Hamiltonian stationary Lagrangians  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 . 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
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
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
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  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 . 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
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, .
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  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 ().
Let be a Hamiltonian stationary torus in with Lagrangian angle and . Then every non–constant, polychromatic Darboux transform of with is given by
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
which shows that allows polychromatic Darboux transforms. In particular, we see that
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
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.
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 .
We consider the case that . In this case (5.1)
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
a lengthy, but straightforward, computation shows that
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.
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 , 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)
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)
Furthermore, for , we have with
and we get for both and
A straightforward computation shows that
so that (5.5) simplifies to
For one easily obtains with
so that (5.6) becomes
Similarly, for we obtain with
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 . ∎
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  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
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 .
We now compute the mean curvature of . To that end, we observe that with
and the left normal of is given (3.2) by
real valued. As before, we compute with (5.5)
and, by evaluating at and , we see that constant is equivalent to . We summarize
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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