mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 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, 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, 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 ran ...

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 multili ...

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 gravi ...

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and

mathematical physics Mathematical physics refers to the development of mathematics, 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 t ...

. :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 are ...

, 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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

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 individuals ...

. 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 mean curvature 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 surf ...

. 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 mean curvature 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 wi ...

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, : \f ...

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 vers ...

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

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

minimal surfaces in R³. :Gauss map definition: A surface ''M'' ⊂ R³ is minimal if and only if its stereographically projected Gauss map ''g'': ''M'' → C ∪ {∞} is meromorphic 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), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The ...

of the shape operator, which is linked to the derivatives of the Gauss map. If the projected Gauss map obeys the Cauchy–Riemann equations 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 than R³.

History

Minimal surface theory originates with

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaEuler–Lagrange equation 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 discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature 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. Durin ...

and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by

Heinrich Scherk Heinrich Ferdinand Scherk (27 October 1798 – 4 October 1885) was a German mathematician notable for his work on minimal surfaces and the distribution of prime number A prime number (or a prime) is a natural number greater than 1 that ...

in 1830 to derive his surfaces, they were generally regarded as practically unusable.

Catalan Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #1 ...

proved in 1842/43 that the helicoid is the only

ruled ''Ruled'' is the fifth full-length LP by The Giraffes. Drums, bass and principal guitar tracks recorded at The Bunker in Brooklyn, NY. Vocals and additional guitars recorded at Strangeweather in Brooklyn, NY. Mixed at Studio G in Brooklyn, NY ...

minimal surface. Progress had been fairly slow until the middle of the century when the

Björling problem In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björl ...

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 is ...

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

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 In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear? This is true in dimensions ''n'' at most 8, but false in dimens ...

and Robert Osserman'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 positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...

, 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 Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), knot. showed that a non-trivial or ...

, the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

, 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 (gen ...

, 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 * catenoids: minimal surfaces made by rotating a catenary once around its directrix * helicoids: 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 b ...

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 ...

: 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\righ ...

* the

Henneberg surface In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation :\begin x(u,v) &= 2\cos(v)\sinh(u) - (2/3)\cos(3v)\sinh(3u)\\ y(u,v) &= 2\sin(v)\sinh(u) + (2/3)\sin( ...

: the first non-orientable minimal surface *

Bour's minimal surface In mathematics, Bour's minimal surface is a two-dimensional minimal surface, embedded with self-crossings into three-dimensional Euclidean space. It is named after Edmond Bour, whose work on minimal surfaces won him the 1861 mathematics prize ...

* the

Neovius surface In differential geometry, the Neovius surface is a triply periodic minimal surface originally discovered by Finnish mathematician Edvard Rudolf Neovius (the uncle of Rolf Nevanlinna). The surface has genus 9, dividing space into two infinite non ...

: a triply periodic surface Modern surfaces include: * the

Gyroid A gyroid is an infinitely connected Triply periodic minimal surface, 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 ...

: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for liquid crystal structure * the

Saddle tower In differential geometry, a saddle tower is a minimal surface family generalizing the singly periodic Scherk's second surface so that it has ''N''-fold (''N'' > 2) symmetry around one axis. These surfaces are the only properly embedded ...

family: generalisations of

Scherk's second surface In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly perio ...

Costa's minimal surface In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact s ...

: Famous conjecture disproof. Described in 1982 by

Celso Costa Celso José da Costa (born April 7, 1949 in Congonhinhas) is a Brazilian mathematician working in differential geometry. His research activity has focused in the construction and classification of minimal surfaces embedded in three-dimensional Eu ...

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 In differential geometry, the Chen–Gackstatter surface family (or the Chen–Gackstatter–Thayer surface family) is a family of minimal surfaces that generalize the Enneper surface by adding handles, giving it nonzero topological genus. They ...

family, adding handles to the Enneper surface.

Generalisations and links to other fields

Minimal surfaces can be defined in other manifolds than R³, such as hyperbolic space, higher-dimensional spaces or Riemannian manifolds. 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. It is used in the study of computer graphics, ge ...

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 set ...

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 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 and materials science, due to their anticipated applications in self-assembly 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 ...

and

Lorentzian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...

, certain extensions and modifications of the notion of minimal surface, known as

apparent horizon In general relativity, an apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards and those directed outward but moving inward. Apparent horizons are not invariant properties of spacetim ...

s, are significant. In contrast to the event horizon, 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 canonic ...

-based approach to understanding

black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...

boundaries. CircusTent02

Structures with minimal surfaces can be used as tents. Minimal surfaces are part of the

generative design Generative design is an iterative design process that involves a program that will generate a certain number of outputs that meet certain constraints, and a designer that will fine tune the feasible region by selecting specific output or chang ...

toolbox used by modern designers. In architecture there has been much interest in tensile structures, 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 ...

, Shigeru Ban, 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 Olympiastadion () is a stadium located in Munich, Germany. Situated at the heart of the '' Olympiapark München'' in northern Munich, the stadium was the main venue for the 1972 Summer Olympics. The original capacity was maximally and officiall ...

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 Robert Engman (April 29, 1927 – July 4, 2018)
T ...

(1927–2018),

Robert Longhurst Robert Longhurst is an American sculptor who was born in Schenectady, New York in 1949. At an early age he was fascinated by his father's small figurative woodcarvings. Longhurst received a Bachelor of Architecture from Kent State University in ...

(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