Constant-mean-curvature surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. 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. History. 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. In 1853 J. H. Jellet showed that if formula_1 is  a compact star-shaped surface in formula_2 with constant mean curvature, then it is the standard sphere. Subsequently, A. D. Alexandrov proved that a compact embedded surface in formula_2 with constant mean curvature formula_4 must be a sphere. Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in formula_5must be a standard embedded formula_6 sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in formula_7. In 1984 Henry C. Wente constructed the Wente torus, an immersion into formula_2 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. Meeks showed that there are no embedded CMC surfaces with just one end in formula_2. Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids. Each end carries a formula_10 "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 formula_11 coplanar "k"-unduloids of genus 0 satisfy formula_12 for odd "k", and formula_13 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 formula_1 in formula_2 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 formula_16 be an open simply connected subset of formula_17 and formula_18 be an arbitrary non-zero real constant. Suppose formula_19 is a harmonic function into the Riemann sphere. If formula_20 then formula_21 defined by with for formula_24 is a regular surface having formula_25 as Gauss map and mean curvature formula_18. For formula_27 and formula_28 this produces the sphere. formula_29 and formula_30 gives a cylinder where formula_31. Conjugate cousin method. Lawson showed in 1970 that each CMC surface in formula_2has an isometric "cousin" minimal surface in formula_33. This allows constructions starting from geodesic polygons in formula_33, 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 formula_35 and formula_36 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 formula_37 into formula_35 and formula_36, and spectral data of the form formula_40 where formula_41 is  a hyperelliptic curve called the spectral curve, formula_42 is a meromorphic function on formula_41, formula_44 and formula_45 are points on formula_46, formula_47 is an antiholomorphic involution and formula_48 is a line bundle on formula_41 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. Applications. CMC surfaces are natural for representations of soap bubbles, since they have the curvature corresponding to a nonzero pressure difference. 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.