Delaunay Surfaces - Constant Mean Curvature Cylinders in Euclidean 3-Space
Nicholas Schmitt
Media
Description
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.
References
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Charles Delaunay.
Sur la surface de révolution dont la courbure moyenne est constante.
Journal de mathématiques pures et appliquées, 1841.
URL: http://sites.mathdoc.fr/JMPA/feuilleter.php?id=JMPA_1841_1_6. -
Martin Kilian.
On the associated family of Delaunay surfaces.
Proc. Amer. Math. Soc., 2004.
URL: http://sites.mathdoc.fr/JMPA/feuilleter.php?id=JMPA_1841_1_6. -
Michael Melko and Ivan Sterling.
Application of soliton theory to the construction of pseudospherical surfaces in R^3.
Ann. Global Anal. Geom., 1993.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1201412. -
Michael Melko and Ivan Sterling.
Integrable systems, harmonic maps and the classical theory of surfaces.
Aspects Math., E23,, 1994.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1264184. -
Ivan Sterling and Henry Wente.
Existence and classification of constant mean curvature multibubbletons of finite and infinite type.
Indiana Univ. Math. J., 1993.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1266092. -
Bifurcations of the nodoids, volume 24, Mathematisches Forschungsinstitut Oberwolfach, 2007. Progress in surface theory.
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Rafe Mazzeo.
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URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1941630. -
Uwe Abresch.
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J. Reine Angew. Math., 1987.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=876223. -
Rolf Walter.
Explicit examples to the H-problem of Heinz Hopf.
Geom. Dedicata, 1987.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=892400. -
Brian Smyth.
A generalization of a theorem of Delaunay on constant mean curvature surfaces.
IMA Vol. Math. Appl., 51,, 1993.
URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1226924.
