In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero

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 ...

(see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a

soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plate ...

, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see

minimal surface of revolution In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in ...

): the standard definitions only relate to a

local optimum In applied mathematics and computer science, a local optimum of an optimization problem is a solution that is optimal (either maximal or minimal) within a neighboring set of candidate solutions. This is in contrast to a global optimum, which ...

, not a

global optimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

Definitions

Minimal surfaces can be defined in several equivalent ways in R³. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially

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 multilin ...

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely grav ...

, complex analysis and

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developmen ...

. :Local least area definition: A surface ''M'' ⊂ R³ is minimal if and only if every point ''p'' ∈ ''M'' has a

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area ...

, bounded by a simple closed curve, which has the least area among all surfaces having the same boundary. This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area. :Variational definition: A surface ''M'' ⊂ R³ is minimal if and only if it is a critical point of the area functional for all compactly supported

variations Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation, the difference in DNA among individual ...

. This definition makes minimal surfaces a 2-dimensional analogue to

geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

, which are analogously defined as critical points of the length functional. Minimal surface curvature planes-en

:Mean curvature definition: A surface ''M'' ⊂ R³ is minimal if and only if its

is equal to zero at all points. A direct implication of this definition is that every point on the surface is a

saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. ...

with equal and opposite

principal curvatures In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...

. Additionally, this makes minimal surfaces into the static solutions of

mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of su ...

. By the

Young–Laplace equation In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or ...

, the

of a soap film is proportional to the difference in pressure between the sides. If the soap film does not enclose a region, then this will make its mean curvature zero. By contrast, a spherical

soap bubble 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 w ...

encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature. :Differential equation definition: A surface ''M'' ⊂ R³ is minimal if and only if it can be locally expressed as the graph of a solution of ::

(1+u_x^2)u_ - 2u_xu_yu_ + (1+u_y^2)u_=0

The partial differential equation in this definition was originally found in 1762 by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJ. L. Lagrange. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. Miscellanea Taurinensia 2, 325(1):173{199, 1760. and

Jean Baptiste Meusnier Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...

discovered in 1776 that it implied a vanishing mean curvature.J. B. Meusnier. Mémoire sur la courbure des surfaces. Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans, 10:477–510, 1785. Presented in 1776. :Energy definition: A conformal immersion ''X'': ''M'' → R³ is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point ''p'' ∈ ''M'' has a neighbourhood with least energy relative to its boundary. This definition ties minimal surfaces to

harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : ...

and

. :Harmonic definition: If ''X'' = (''x''₁, ''x''₂, ''x''₃): ''M'' → R³ is an

isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...

immersion of a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versi ...

into 3-space, then ''X'' is said to be minimal whenever ''x_i'' is a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : ...

on ''M'' for each ''i''. A direct implication of this definition and the maximum principle for harmonic functions is that there are no

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...

complete minimal surfaces in R³. :Gauss map definition: A surface ''M'' ⊂ R³ is minimal if and only if its stereographically projected

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 '' ...

''g'': ''M'' → C ∪ {∞} is

meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pol ...

with respect to the underlying

structure, and ''M'' is not a piece of a sphere. This definition uses that the mean curvature is half of the

trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...

of the

shape operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives ...

, which is linked to the derivatives of the Gauss map. If the projected Gauss map obeys the

Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and different ...

then either the trace vanishes or every point of ''M'' is

umbilic In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...

, in which case it is a piece of a sphere. The local least area and variational definitions allow extending minimal surfaces to other

Riemannian manifolds In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

than R³.

History

Minimal surface theory originates with

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaEuler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

for the solution :

\frac{d}{dx}\left(\frac{z_x}{\sqrt{1+z_x^2+z_y^2\right ) + \frac{d}{dy}\left(\frac{z_y}{\sqrt{1+z_x^2+z_y^2\right )=0

He did not succeed in finding any solution beyond the plane. In 1776

Jean Baptiste Marie Meusnier Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...

discovered that the

helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similari ...

and

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 ...

satisfy the equation and that the differential expression corresponds to twice the

of the surface, concluding that surfaces with zero mean curvature are area-minimizing. By expanding Lagrange's equation to :

\left(1 + z_x^2\right)z_{yy} - 2z_xz_yz_{xy} + \left(1 + z_y^2\right)z_{xx} = 0

Gaspard Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...

and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by Heinrich Scherk in 1830 to derive his

surfaces A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surf ...

, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface. Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. The "first golden age" of minimal surfaces began. Schwarz found the solution of the

Plateau problem In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem i ...

for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 (allowing the construction of his periodic surface families) using complex methods.

Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and

. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten. Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by

Jesse Douglas Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem. Life and career He was born to a Jewish family in New York City, the son of Sarah (née ...

and

Tibor Radó Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I. Biography Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...

was a major milestone. Bernstein's problem and

Robert Osserman Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces. Raised in Bronx, he went to Bronx High School o ...

's work on complete minimal surfaces of finite total curvature were also important. Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R³ of finite topological type. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). Another cause was the verification by H. Karcher that the

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 originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them. Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. the

Smith conjecture In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of ''f'' cannot be a nontrivial knot. showed that a non-trivial orientation-preserving diffeomorphism of ...

, the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by H ...

, the

Thurston Geometrization Conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...

Examples

Classical examples of minimal surfaces include: * the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

, which is a

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

case *

s: minimal surfaces made by rotating a

catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficia ...

once around its directrix *

s: A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity Surfaces from the 19th century golden age include: *

Schwarz minimal surface In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named ...

s: triply periodic surfaces that fill R³ *

Riemann's minimal surface In differential geometry, Riemann's minimal surface is a one-parameter family of minimal surfaces described by Bernhard Riemann in a posthumous paper published in 1867. Surfaces in the family are singly periodic minimal surfaces with an infinite n ...

: A posthumously described periodic surface * the

Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\right ...

* the Henneberg surface: the first non-orientable minimal surface * Bour's minimal surface * the Neovius surface: a triply periodic surface Modern surfaces include: * the

Gyroid A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. History and properties The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces. I ...

: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for liquid crystal structure * the Saddle tower family: generalisations of Scherk's second surface * Costa's minimal surface: Famous conjecture disproof. Described in 1982 by Celso Costa and later visualized by Jim Hoffman. Jim Hoffman, David Hoffman and William Meeks III then extended the definition to produce a family of surfaces with different rotational symmetries. * the Chen–Gackstatter surface family, adding handles to the Enneper surface.

Generalisations and links to other fields

Minimal surfaces can be defined in other

manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

than R³, such as

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...

, higher-dimensional spaces or

. The definition of minimal surfaces can be generalized/extended to cover

constant-mean-curvature surface In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2 ...

s: surfaces with a constant mean curvature, which need not equal zero. The curvature lines of an isothermal surface form an isothermal net. In

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 In mathematics, a simplicial complex is a set c ...

discrete minimal surfaces are studied:

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

es of triangles that minimize their area under small perturbations of their vertex positions. Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known.

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position ins ...

on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces. Minimal surfaces have become an area of intense scientific study, especially in the areas of

molecular engineering Molecular engineering is an emerging field of study concerned with the design and testing of molecular properties, behavior and interactions in order to assemble better materials, systems, and processes for specific functions. This approach, in whi ...

and materials science, due to their anticipated applications in

self-assembly Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...

of complex materials. The

endoplasmic reticulum The endoplasmic reticulum (ER) is, in essence, the transportation system of the eukaryotic cell, and has many other important functions such as protein folding. It is a type of organelle made up of two subunits – rough endoplasmic reticulum ( ...

, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface. In the fields of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...

and Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as apparent horizons, are significant. In contrast to the

event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...

, they represent a

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...

-based approach to understanding

black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can defo ...

boundaries. CircusTent02

Structures with minimal surfaces can be used as tents. Minimal surfaces are part of the generative design toolbox used by modern designers. In architecture there has been much interest in

tensile structure A tensile structure is a construction of elements carrying only tension and no compression or bending. The term ''tensile'' should not be confused with tensegrity, which is a structural form with both tension and compression elements. Tensile ...

s, which are closely related to minimal surfaces. Notable examples can be seen in the work of

Frei Otto Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for t ...

Shigeru Ban Biography
, The Hyatt Foundation, retrieved 26 March 2014 is a Japanese architect, known for his i ...

, and

Zaha Hadid Dame Zaha Mohammad Hadid ( ar, زها حديد ''Zahā Ḥadīd''; 31 October 1950 – 31 March 2016) was an Iraqi-British architect, artist and designer, recognised as a major figure in architecture of the late 20th and early 21st centu ...

. The design of the Munich Olympic Stadium by Frei Otto was inspired by soap surfaces. Another notable example, also by Frei Otto, is the German Pavilion at

Expo 67 The 1967 International and Universal Exposition, commonly known as Expo 67, was a general exhibition from April 27 to October 29, 1967. It was a category One World's Fair held in Montreal, Quebec, Canada. It is considered to be one of the most su ...

in Montreal, Canada. In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927–2018), Robert Longhurst (1949– ), and Charles O. Perry (1929–2011), among others.

References

External links

*
3D-XplorMath-J Homepage — Java program and applets for interactive mathematical visualisation

{{Authority control Differential geometry Differential geometry of surfaces

Definitions

History

Examples

Generalisations and links to other fields

See also

References

Further reading

External links