Delaunay Surfaces - Constant Mean Curvature Cylinders in Euclidean 3-Space

Nicholas Schmitt



The Delaunay surfaces, being surfaces of revolution, are the simplest constant mean curvature cylinders. Delaunay surfaces lie in associate families of twizzlers, which have screw-motion symmetry.[1]
The roulette of a conic, traced out by its focus, is the profile curve of a Delaunay surface.[2] Roulettes of ellipses make unduloids, and those of hyperbola make nodoids. The major and minor axes of the conic determine the neck and bulge sizes of the Delaunay surface. The roulettes in these flipbooks are traced out at constant speed.[3][4][5]

Delaunay bubbletons are constructed by dressing a Delaunay surface with a product of simple factor dressings.

There are constant mean curvature cylinders that [6][7] have spectral genus two like the Wente tori, though only one period is closed. Their metric is given in terms of Jacobi elliptic functions.[8][9]

Smyth surfaces are constant mean curvature surfaces with a one-parameter metric symmetry.[10] Also known as Mr and Mrs Bubble, Smyth surfaces come in many shapes and sizes, with varying numbers of legs. A Delaunay can be surgically attached to the head to make cylinders. This can for instance be seen by the perturbed Delaunay surface with one Smyth end and one Delaunay end.


Dr. Nicholas Schmitt   +

University: TU Berlin