Minimal surface

Last updated November 07, 2023

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

Definitions

Minimal surfaces can be defined in several equivalent ways in $\mathbb {R} ^{3}$ . The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.^[1]

Local least area definition: A surface

M\subset \mathbb {R} ^{3}

is minimal if and only if every point p ∈ M has a neighbourhood, 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 \mathbb {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, which are analogously defined as critical points of the length functional.

Mean curvature definition: A surface

M\subset \mathbb {R} ^{3}

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 with equal and opposite principal curvatures. Additionally, this makes minimal surfaces into the static solutions of mean curvature flow. By the Young–Laplace equation, 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 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 \mathbb {R} ^{3}

is minimal if and only if it can be locally expressed as the graph of a solution of

(1+u_{x}^{2})u_{yy}-2u_{x}u_{y}u_{xy}+(1+u_{y}^{2})u_{xx}=0

The partial differential equation in this definition was originally found in 1762 by Lagrange,^[2] and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.^[3]

Energy definition: A conformal immersion

X:M\rightarrow \mathbb {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 and potential theory.

Harmonic definition: If

X=(x_{1},x_{2},x_{3}):M\rightarrow \mathbb {R} ^{3}

is an isometric immersion of a Riemann surface 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 complete minimal surfaces in $\mathbb {R} ^{3}$ .

Gauss map definition: A surface

M\subset \mathbb {R} ^{3}

is minimal if and only if its stereographically projected Gauss map

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

is meromorphic with respect to the underlying Riemann surface 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 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 $\mathbb {R} ^{3}$ .

History

Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface $z=z(x,y)$ of least area stretched across a given closed contour. He derived the Euler–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_{x}z_{y}z_{xy}+\left(1+z_{y}^{2}\right)z_{xx}=0

Gaspard Monge 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 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 complex analysis and harmonic functions. 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 and Tibor Radó 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 $\mathbb {R} ^{3}$ 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, 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 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 $\mathbb {R} ^{3}$
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 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 $\mathbb {R} ^{3}$ , such as hyperbolic space, 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.^[4]

In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions.^[5] 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.^[6]

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.^[7] The endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.^[8]

In the fields of general relativity and Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as apparent horizons, are significant.^[9] In contrast to the event horizon, they represent a curvature-based approach to understanding black hole boundaries.

Circus tent approximates a minimal surface. CircusTent02.jpg — Circus tent approximates a minimal surface.

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 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.^[10] Another notable example, also by Frei Otto, is the German Pavilion at Expo 67 in Montreal, Canada.^[11]

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.

Related Research Articles

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<span class="mw-page-title-main">Curvature</span> Mathematical measure of how much a curve or surface deviates from flatness

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.

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

<span class="mw-page-title-main">Catenoid</span> Surface of revolution of a catenary

In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis. It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

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<span class="mw-page-title-main">Enneper surface</span>

In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by:

<span class="mw-page-title-main">Scherk surface</span>

In mathematics, a Scherk surface 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 periodic. They were the third non-trivial examples of minimal surfaces. The two surfaces are conjugates of each other.

In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of a Riemannian or pseudo-Riemannian manifold.

In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly. Except in special cases, the mean curvature flow develops singularities.

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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: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics.

<span class="mw-page-title-main">Constant-mean-curvature surface</span>

In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.

<span class="mw-page-title-main">Associate family</span>

In differential geometry, the associate family of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation

In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A biharmonic submanifold refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by James Eells and Luc Lemaire in 1983. The study of harmonic maps, of which the study of biharmonic maps is an outgrowth, had been an active field of study for the previous twenty years. A simple case of biharmonic maps is given by biharmonic functions.

References

↑ Meeks, William H. III; Pérez, Joaquín (2011). "The classical theory of minimal surfaces". Bull. Amer. Math. Soc. 48 (3): 325–407. doi: 10.1090/s0273-0979-2011-01334-9 . MR 2801776.
↑ J. 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.
↑ 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.
↑ "Isothermal surface - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-09-04.
↑ Pinkall, Ulrich; Polthier, Konrad (1993). "Computing Discrete Minimal Surfaces and Their Conjugates". Experimental Mathematics . 2 (1): 15–36. doi:10.1080/10586458.1993.10504266. MR 1246481.
↑ Neel, Robert (2009). "A martingale approach to minimal surfaces". Journal of Functional Analysis. 256 (8): 2440–2472. arXiv: 0805.0556 . doi:10.1016/j.jfa.2008.06.033. MR 2502522. S2CID 15228691.
↑ Han, Lu; Che, Shunai (April 2018). "An Overview of Materials with Triply Periodic Minimal Surfaces and Related Geometry: From Biological Structures to Self-Assembled Systems". Advanced Materials. 30 (17): 1705708. doi:10.1002/adma.201705708. PMID 29543352. S2CID 3928702.
↑ Terasaki, Mark; Shemesh, Tom; Kasthuri, Narayanan; Klemm, Robin W.; Schalek, Richard; Hayworth, Kenneth J.; Hand, Arthur R.; Yankova, Maya; Huber, Greg (2013-07-18). "Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs". Cell. 154 (2): 285–296. doi:10.1016/j.cell.2013.06.031. ISSN 0092-8674. PMC 3767119 . PMID 23870120.
↑ Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. ISBN 978-0-19-923072-3 (page 417)
↑ "AD Classics: Olympiastadion (Munich Olympic Stadium) / Behnisch and Partners & Frei Otto". ArchDaily. 2011-02-11. Retrieved 2022-09-04.
↑ "Expo 67 German Pavilion". Architectuul. Retrieved 2022-09-04.

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[1] Meeks, William H. III; Pérez, Joaquín (2011). "The classical theory of minimal surfaces". Bull. Amer. Math. Soc. 48 (3): 325–407. doi: 10.1090/s0273-0979-2011-01334-9 . MR 2801776.

[Lagrange1760-2] J. 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.

[Meusnier1785-3] 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.

[4] "Isothermal surface - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-09-04.

[5] Pinkall, Ulrich; Polthier, Konrad (1993). "Computing Discrete Minimal Surfaces and Their Conjugates". Experimental Mathematics . 2 (1): 15–36. doi:10.1080/10586458.1993.10504266. MR 1246481.

[6] Neel, Robert (2009). "A martingale approach to minimal surfaces". Journal of Functional Analysis. 256 (8): 2440–2472. arXiv: 0805.0556 . doi:10.1016/j.jfa.2008.06.033. MR 2502522. S2CID 15228691.

[7] Han, Lu; Che, Shunai (April 2018). "An Overview of Materials with Triply Periodic Minimal Surfaces and Related Geometry: From Biological Structures to Self-Assembled Systems". Advanced Materials. 30 (17): 1705708. doi:10.1002/adma.201705708. PMID 29543352. S2CID 3928702.

[8] Terasaki, Mark; Shemesh, Tom; Kasthuri, Narayanan; Klemm, Robin W.; Schalek, Richard; Hayworth, Kenneth J.; Hand, Arthur R.; Yankova, Maya; Huber, Greg (2013-07-18). "Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs". Cell. 154 (2): 285–296. doi:10.1016/j.cell.2013.06.031. ISSN 0092-8674. PMC 3767119 . PMID 23870120.

[9] Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. ISBN 978-0-19-923072-3 (page 417)

[10] "AD Classics: Olympiastadion (Munich Olympic Stadium) / Behnisch and Partners & Frei Otto". ArchDaily. 2011-02-11. Retrieved 2022-09-04.

[11] "Expo 67 German Pavilion". Architectuul. Retrieved 2022-09-04.

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