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Minimal surface
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==History== Minimal surface theory originates with [[Lagrange]] who in 1762 considered the variational problem of finding the surface <math>z=z(x,y)</math> of least area stretched across a given closed contour. He derived the [[Euler–Lagrange equation]] for the solution :<math>\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</math> 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 :<math>\left(1 + z_x^2\right)z_{yy} - 2z_xz_yz_{xy} + \left(1 + z_y^2\right)z_{xx} = 0</math> [[Gaspard Monge]] and [[Adrien-Marie Legendre|Legendre]] in 1795 derived representation formulas for the solution surfaces. While these were successfully used by [[Heinrich Scherk]] in 1830 to derive his [[Scherk surface|surfaces]], they were generally regarded as practically unusable. [[Eugène Charles Catalan|Catalan]] proved in 1842/43 that the helicoid is the only [[ruled surface|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. [[Hermann Schwarz|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 [[Schwarz minimal surface|surface families]]) using complex methods. [[Weierstrass]] and [[Alfred Enneper|Enneper]] developed more useful [[Weierstrass–Enneper parameterization|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 [[Costa's minimal surface|a surface]] that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in <math>\R^3</math> 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 [[Associate family|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]]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 [[Riemannian Penrose inequality|Penrose conjecture]]) and three-manifold geometry (e.g. the [[Smith conjecture]], the [[Poincaré conjecture]], the [[Thurston Geometrization Conjecture]]).
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