mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 Plat ...

, 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): the standard definitions only relate to a local optimum, not a global optimum.

Definitions

Minimal surfaces can be defined in several equivalent ways in

\R^3

. 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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

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 the two fundamental forces of nature known at the time, namely g ...

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

and

mathematical physics Mathematical physics is 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 the de ...

. :Local least area definition: A surface

M \subset \R^3

is minimal if and only if every point ''p'' ∈ ''M'' has a

neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...

, 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 \subset \R^3

is minimal if and only if it is a critical point of the area functional for all compactly supported variations. This definition makes minimal surfaces a 2-dimensional analogue to

geodesics In geometry, a geodesic () is a curve representing in some sense the locally 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 connec ...

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

:Mean curvature definition: A surface

M \subset \R^3

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 (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

with equal and opposite principal curvatures. 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 sur ...

. By the

Young–Laplace equation In physics, the Young–Laplace equation () is an 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 wall tensi ...

, 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 (commonly referred to as simply a bubble) is an extremely thin soap film, 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 b ...

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 \subset \R^3

formed by the image of a region

X \subset \R^2

under function

\mathbf : X \to M

(x, y) \mapsto (x, y, u(x, y))

, where

u: X \to \R

is a real valued function, is minimal if and only if

u

satisfies ::

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

:The

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

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. This equation gives an asymmetric definition in the sense that the position on the

z

-axis is determined as a function

u

x

and

y

. Not all surfaces are conveniently represented this way. An alternative definition based on the more general representation

\mathbf{x} : \R^{2} \to \R^{3}, (u,v) \mapsto (x,y,z)

is :

\frac{\partial}{\partial u} \frac{\frac{\partial \mathbf{x{\partial v} \boldsymbol{\times} (\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} )}{\sqrt{(\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} ) \boldsymbol{\cdot} (\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} )} } = \frac{\partial}{\partial v} \frac{\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} (\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} )}{\sqrt{(\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} ) \boldsymbol{\cdot} (\frac{\partial \mathbf{x{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x{\partial v} )} }

. :Energy definition: A conformal immersion

X: M \rightarrow \R^3

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 \in 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\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that ...

and

. :Harmonic definition: If

X = (x_1, x_2, x_3) : M \rightarrow \R^3

is an isometric

immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux * ''Immersion'' (album), the third album by Australian group Pendulum * ''Immersion'' (film), a 2021 ...

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

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

complete minimal surfaces in

\R^3

. :Gauss map definition: A surface

M \subset \R^3

is minimal if and only if its stereographically projected Gauss map

g: M \rightarrow \C \cup {\infty}

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 of the shape operator, 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-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...

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

\R^3

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

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. Dur ...

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, 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 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 Karl Theodor Wilhelm 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 trained as a school t ...

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 (July 3, 1897 – September 7, 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 ...

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 c ...

was a major milestone. Bernstein's problem 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

\R^3

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 surfaces 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, 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 b ...

, the Thurston Geometrization Conjecture).

Examples

Classical examples of minimal surfaces include: * the plane, which is a trivial case * catenoids: minimal surfaces made by rotating a

catenary In physics and geometry, a catenary ( , ) is the curve that an idealized hanging chain or wire rope, 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, ...

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 surfaces: triply periodic surfaces that fill

\R^3

* Riemann's minimal surface: A posthumously described periodic surface * the Enneper surface * 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: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for

liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules may be oriented in a common direction as i ...

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 than

\R^3

, 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 symme ...

, higher-dimensional spaces or Riemannian manifolds. The definition of minimal surfaces can be generalized/extended to cover constant-mean-curvature surfaces: 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 minimal surfaces are studied:

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

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 is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

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 Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...

, 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 a part of a transportation system of the eukaryote, eukaryotic cell, and has many other important functions such as protein folding. The word endoplasmic means "within the cytoplasm", and reticulum is Latin for ...

, 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 as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

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 outside 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 c ...

, they represent a

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

-based approach to understanding

black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

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 uses software to generate outputs that fulfill a set of constraints iteratively adjusted by a designer. Whether a human, test program, or artificial intelligence, the designer algorith ...

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, Shigeru Ban, and Zaha Hadid. 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 28 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 s ...

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