In 1853 J. H. Jellet showed that if is a compact star-shaped surface in with constant mean curvature, then it is the standard sphere. Subsequently, A. D. Alexandrov proved that a compact embedded surface in with constant mean curvature must be a sphere. Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in must be a standard embedded sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in . In 1984 Henry C. Wente constructed the Wente torus, an immersion into of a torus with constant mean curvature.
Up until this point it had seemed that CMC surfaces were rare; new techniques produced a plethora of examples. In particular gluing methods appear to allow combining CMC surfaces fairly arbitrarily. Delaunay surfaces can also be combined with immersed "bubbles", retaining their CMC properties.
Equal neck sizes
Unequal neck sizes
With nodoid end
Triunduloids with different neck sizes. As neck sizes are varied the asymptotic directions change.
Meeks showed that there are no embedded CMC surfaces with just one end in . Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids. Each end carries a "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. In particular, for coplanar k-unduloids of genus 0 satisfy for odd k, and for even k. At most k − 2 ends can be cylindrical.
Generation methods
Representation formula
Like for minimal surfaces, there exist a close link to harmonic functions. An oriented surface in has constant mean curvature if and only if its Gauss map is a harmonic map. Kenmotsu’s representation formula is the counterpart to the Weierstrass–Enneper parameterization of minimal surfaces:
Let be an open simply connected subset of and be an arbitrary non-zero real constant. Suppose is a harmonic function into the Riemann sphere. If then defined by
with
for is a regular surface having as Gauss map and mean curvature .
For and this produces the sphere. and gives a cylinder where .
Conjugate cousin method
Lawson showed in 1970 that each CMC surface in has an isometric "cousin" minimal surface in . This allows constructions starting from geodesic polygons 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.
CMC Tori
Hitchin, Pinkall, Sterling and Bobenko showed that all constant mean curvature immersions of a 2-torus into the space forms and can be described in purely algebro-geometric data. 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 into and , and spectral data of the form where is a hyperelliptic curve called the spectral curve, is a meromorphic function on , and are points on , is an antiholomorphic involution and is a line bundle on obeying certain conditions.
Discrete numerical methods
Discrete differential geometry can be used to produce approximations to CMC surfaces (or discrete counterparts), typically by minimizing a suitable energy functional.
Besides macroscopic bubble surfaces CMC surfaces are relevant for the shape of the gas–liquid interface on a superhydrophobic surface.
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. CMC structures have been observed in ABC triblock copolymers.
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.
