This CMC cylinder is a Bäcklund transform of a perturbed Delaunay unduloid.
Smooth surfaces of constant mean curvature in Euclidean three space are characterized by the fact that their Gauss map is harmonic. Such harmonic maps can be obtained by a Weierstrass Type Representation , commonly referred to as the DPW method, that naturally incorporates dressing  as a means of perturbing the harmonic map. The classical Bäcklund transformation corresponds to dressing by 'simple factors' and adds bubbles to the surface , .
A family of surfaces similar to this model are discussed in detail in  as pertubations of Delaunay surfaces: The idea is to take a DPW potential of a Delaunay unduloid - a holomorphic loop valued 1-form on the twice punctured Riemann sphere with simple poles at the origin and the point at infinity and perturb it, by adding an entire term, to change the geometry of the end at infinity. The initial condition in the DPW method then needs to be computed numerically to ensure that the resulting surface again closes . Such surfaces look like our model, but without the bubbles. They may have an arbitrary number of umbilic points and have one Delaunay end. The other highly self intersecting end resembles that of a Smyth surface .
The novelty of this model is that by choosing an appropriate simple factor, it is possible to add bubbles onto the surface: This is possible since the perturbed surface still has Delaunay monodromy, which is known explicitly  and thus allows the choice of singularity in the simple factor to attain any value where the monodromy is reducible. For a recent account of these techniques we refer to .
Model produced with: GANGLab 1.0, Nicholas Schmitt
|Keywords||Bäcklund transformation; umbilic point; constant mean curvature; cylinder|
Submitted: Wed Dec 12 23:52:28 GMT 2001.
Revised: Thu Jul 22 13:50:36 CEST 2004.
Accepted: Sat Sep 4 11:32:21 GMT 2004.
Amherst CollegeNicholas Schmitt
Amherst, MA 01002
University of Massachusetts
Amherst, MA 01002