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In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to (see definitions below) having a mean curvature of zero. 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 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, calculus of variations, potential theory, complex analysis and mathematical physics. :Local least area definition: A surface ''M'' ⊂ R3 is minimal if and only if every point ''p'' ∈ ''M'' has a neighbourhood with least-area relative to its boundary. Note that this property is local: there might exist other surfaces that minimize area better with the same global boundary. :Variational definition:A surface ''M'' ⊂ R3 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. :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. Note that 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. :Mean curvature definition: A surface ''M'' ⊂ R3 is minimal if and only if its mean curvature vanishes identically. A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. :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,〔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.〕 and Jean Baptiste Meusnier 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 and potential theory. :Harmonic definition: If ''X'' = (''x''1, ''x''2, ''x''3): ''M'' → R3 is an isometric immersion of a Riemann surface into 3-space, then ''X'' is said to be minimal whenever ''xi'' 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 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 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. :Mean curvature flow definition: Minimal surfaces are the critical points for the mean curvature flow. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「minimal surface」の詳細全文を読む スポンサード リンク
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