
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, constant-mean-curvature (CMC) surfaces are surfaces with constant
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
.
[Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E1]
/ref> This includes minimal surfaces
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
as a subset, but typically they are treated as special case.
Note that these surfaces are generally different from constant Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
surfaces, with the important exception of the sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.
History
In 1841 Delaunay Delaunay is a French surname. Notable people with the surname include:
People
Arts
* Catherine Delaunay (born 1969), French jazz clarinet player and composer
* Charles Delaunay (1911–1988), French author and jazz expert
* Joseph-Charles Delau ...
proved that the only surfaces of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on wh ...
with constant mean curvature were the surfaces obtained by rotating the roulettes
The Roulettes are the Royal Australian Air Force's formation aerobatic display team.
They provide about 150 flying displays a year, in Australia and in friendly countries around the Southeast Asian region. The Roulettes form part of the RAAF ...
of the conics. These are the plane, cylinder, sphere, the catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...
, the unduloid
In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulti ...
and nodoid
In differential geometry, a nodoid is a surface of revolution with Constant mean curvature surface, constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the Focus (geometry), focus, and revolving the resulti ...
.
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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
with constant mean curvature.
Up until this point it had seemed that CMC surfaces were rare; new techniques produced a plethora of examples.[Karsten Grosse-Brauckmann, Robert B. Kusner, John M. Sullivan. Coplanar constant mean curvature surfaces. Comm. Anal. Geom. 15:5 (2008) pp. 985–1023. ArXiv math.DG/0509210]
/ref> 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 . 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 space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s. 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
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...
is a harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
.[ Shoichi Fujimori, Shimpei Kobayashi and Wayne Rossman, Loop Group Methods for Constant Mean Curvature Surfaces. Rokko Lectures in Mathematics 2005 ] Kenmotsu’s representation formula is the counterpart to the Weierstrass–Enneper parameterization
In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.
Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Let f and g be functions on either ...
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
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, ge ...
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
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact ...
, 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
Ultrahydrophobic (or superhydrophobic) surfaces are highly hydrophobic, i.e., extremely difficult to wet. The contact angles of a water droplet on an ultrahydrophobic material exceed 150°. This is also referred to as the lotus effect, after the ...
surface.
Like triply periodic minimal surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations.
These surfaces have the symmetries of a crystallographic group. Numerous examples are known ...
s there has been interest in periodic CMC surfaces as models for block copolymers
In polymer chemistry, a copolymer is a polymer derived from more than one species of monomer. The polymerization of monomers into copolymers is called copolymerization. Copolymers obtained from the copolymerization of two monomer species are some ...
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
An air-supported (or air-inflated) structure is any building that derives its structural integrity from the use of internal pressurized air to inflate a pliable material (i.e. structural fabric) envelope, so that air is the main support of the str ...
such as inflatable domes and enclosures, as well as a source of flowing organic shapes.[Helmut Pottmann, Yang Liu, Johannes Wallner, Alexander Bobenko, Wenping Wang. Geometry of Multi-layer Freeform Structures for Architecture. ACM Transactions on Graphics – Proceedings of ACM SIGGRAPH 2007 Volume 26 Issue 3, July 2007 Article No. 65]
/ref>
See also
* Double bubble conjecture
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a ''standard double bubble'': three spherical surfaces meet ...
* Free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
* Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
References
{{reflist
External links
* CMC surfaces at the Scientific Graphics Projec
* GeometrieWerkstatt surface galler
* GANG gallery of CMC surface
* Noid, software for computing ''n''-noid CMC surface
Differential geometry of surfaces