Constant-mean-curvature surface
[1][2] This includes minimal surfaces as a subset, but typically they are treated as special case.
Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere.
In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics.
These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.
[4] Subsequently, A. D. Alexandrov proved that a compact embedded surface in
[6] Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in
This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in
[7] Up until this point it had seemed that CMC surfaces were rare.
Using gluing techniques, in 1987 Nikolaos Kapouleas constructed a plethora of examples of complete immersed CMC surfaces in
[8][9] Subsequently, Kapouleas constructed compact CMC surfaces in
[10][11] In particular gluing methods appear to allow combining CMC surfaces fairly arbitrarily.
[12][13][14] Delaunay surfaces can also be combined with immersed "bubbles", retaining their CMC properties.
Meeks showed that there are no embedded CMC surfaces with just one end in
[16] Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids.
"force" along the asymptotic axis of the unduloid (where n is the circumference of the necks), the sum of which must be balanced for the surface to exist.
Current work involves classification of families of embedded CMC surfaces in terms of their moduli spaces.
[13] Like for minimal surfaces, there exist a close link to harmonic functions.
[19] Kenmotsu’s representation formula[20] is the counterpart to the Weierstrass–Enneper parameterization of minimal surfaces: Let
Lawson showed in 1970 that each CMC surface in
has an isometric "cousin" minimal surface in
, which are spanned by a minimal patch that can be extended into a complete surface by reflection, and then turned into a CMC surface.
Hitchin, Pinkall, Sterling and Bobenko showed that all constant mean curvature immersions of a 2-torus into the space forms
This can be extended to a subset of CMC immersions of the plane which are of finite type.
More precisely there is an explicit bijection between CMC immersions of
[23][24][25] Discrete differential geometry can be used to produce approximations to CMC surfaces (or discrete counterparts), typically by minimizing a suitable energy functional.
[26][27] CMC surfaces are natural for representations of soap bubbles, since they have the curvature corresponding to a nonzero pressure difference.
[28] Like triply periodic minimal surfaces there has been interest in periodic CMC surfaces as models for block copolymers where the different components have a nonzero interfacial energy or tension.
CMC analogs to the periodic minimal surfaces have been constructed, producing unequal partitions of space.
[29][30] CMC structures have been observed in ABC triblock copolymers.
[31] In architecture CMC surfaces are relevant for air-supported structures such as inflatable domes and enclosures, as well as a source of flowing organic shapes.
