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Surface that locally minimizes its area
A helicoid minimal surface formed by a soap film on a helical frame
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.
Saddle tower minimal surface. While any small change of the surface increases its area, there exist other surfaces with the same boundary with a smaller total area.
This definition makes minimal surfaces a 2-dimensional analogue to geodesics.
Soap film definition: A surface M ? R3 is minimal if and only if every point p ? M has a neighbourhood Dp which is equal to the unique idealized soap film with boundary ?Dp
By the Young-Laplace equation the curvature of a soap film is proportional to the difference in pressure between the sides: if it is zero, the membrane has zero mean curvature. Spherical bubbles are not minimal surfaces as per this definition: while they minimize total area subject to a constraint on internal volume, they have a positive pressure.
Minimal surface curvature planes. On a minimal surface, the curvature along the principal curvature planes are equal and opposite at every point. This makes the mean curvature zero.
Mean curvature definition: A surface M ? R3 is minimal if and only if its mean curvature vanishes identically.
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
The partial differential equation in this definition was originally found in 1762 by Lagrange, and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.
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 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 R3.
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
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
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.
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 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 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.
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 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.
In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes 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, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.
Osserman, Robert (1986). A Survey of Minimal Surfaces (Second ed.). New York: Dover Publications, Inc. ISBN978-0-486-64998-6. MR0852409.(Introductory text for surfaces in n-dimensions, including n=3; requires strong calculus abilities but no knowledge of differential geometry.)
Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Grundlehren der Mathematischen Wissenschaften. 339. With assistance and contributions by A. Küster and R. Jakob (Second ed.). Heidelberg: Springer. doi:10.1007/978-3-642-11698-8. ISBN978-3-642-11697-1. MR2566897.(Review of minimal surface theory, in particularly boundary value problems. Contains extensive references to the literature.)