Delaunay Surfaces
Constant mean curvature cylinders in euclidean 3-space
The Delaunay surfaces, being surfaces of revolution, are the simplest constant mean curvature cylinders [1]. Delaunay surfaces lie in associate families of twizzlers, which have screw-motion symmetry.
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A Delaunay unduloid, and cutaway view of a Delaunay nodoid.
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A twizzler, in the associate family of a Delaunay surface.
References
- C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures et Appl. Série 1 6 (1841), 309-320 link.
Roulettes of Conics
Profile curves of Delaunay surfaces
The roulette of a conic, traced out by its focus, is the profile curve of a Delaunay surface [1]. 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.
References
- M. Kilian, On the associated family of Delaunay surfaces, Proc. Amer. Math. Soc. 132 (2004), no. 10, 3075—3082 [2063129].
Bubbletons
Constant mean curvature cylinders in euclidean 3-space
Delaunay bubbletons are constructed by dressing a Delaunay surface with a product of simple factor dressings [3,1,2].
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A two-lobed bubble on a straight cylinder.
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Two-lobed bubbles on a Delaunay unduloid and nodoid.
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Double bubbleton: two-lobed and three-lobed bubbles on a Delaunay unduloid.
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Two-lobed bubbles on a Delaunay unduloid and nodoid.
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Two-lobed bubbleton on a Delaunay twizzler.
References
- M. Melko and I. Sterling, Application of soliton theory to the construction of pseudospherical surfaces in R3, Ann. Global Anal. Geom. 11 (1993), no. 1, 65—107 [1201412].
- M. Melko and I. Sterling, Integrable systems, harmonic maps and the classical theory of surfaces, Harmonic maps and integrable systems, Aspects Math., E23, Friedr. Vieweg, Braunschweig (1994), 129—144 [1264184].
- I. Sterling and H. Wente, Existence and classification of constant mean curvature multibubbletons of finite and infinite type, Indiana Univ. Math. J. 42 (1993), no. 4, 1239—1266 [1266092].
Bifurcating Nodoids
Constant mean curvature cylinders in euclidean 3-space
These constant mean curvature cylinders [3,2] have spectral genus two like the Wente tori, though only one period is closed. Their metric is given in terms of Jacobi elliptic functions [1,4].
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A cylinder of spectral genus two, constructed by bifurcating off of the family of Delaunay nodoids.
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A cylinder of spectral genus two, constructed by bifurcating off of the family of Delaunay nodoids.
References
- U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169—192 [876223].
- K. Grosse-Brauckmann, Bifurcations of the nodoids, Oberwolfach Report 24/2007, Progress in surface theory (2007).
- R. Mazzeo and F. Pacard, Bifurcating nodoids, Topology and geometry: commemorating SISTAG, Contemp. Math., Amer. Math. Soc., Providence, RI (2002), 169—186 [1941630].
- R. Walter, Explicit examples to the H-problem of Heinz Hopf, Geom. Dedicata 23 (1987), no. 2, 187—213 [892400].
Smyth Surfaces
Constant mean curvature cylinders in euclidean 3-space
Smyth surfaces are constant mean curvature surfaces with a one-parameter metric symmetry [1]. 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.
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One-parameter family of three-legged Smyth surfaces.
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Smyth surface family.
References
- B. Smyth, A generalization of a theorem of Delaunay on constant mean curvature surfaces, Statistical thermodynamics and differential geometry of microstructured materials (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, New York (1993), 123—130 [1226924].
Perturbed Delaunay Surfaces
Constant mean curvature cylinders in euclidean 3-space
Perturbed Delaunay surfaces with one Smyth end and one Delaunay end.
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A cylinder with a Delaunay and a Smyth end.
