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 R 3 {\displaystyle \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.
Local least area definition: A surface M ⊂ R 3 {\displaystyle 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 ⊂ R 3 {\displaystyle 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 ⊂ R 3 {\displaystyle 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 ⊂ R 3 {\displaystyle M\subset \mathbb {R} ^{3}} formed by the image of a region X ⊂ R 2 {\displaystyle X\subset \mathbb {R} ^{2}} under function f : X → M {\displaystyle \mathbf {f} :X\to M} , ( x , y ) ↦ ( x , y , u ( x , y ) ) {\displaystyle (x,y)\mapsto (x,y,u(x,y))} , where u : X → R {\displaystyle u:X\to \mathbb {R} } is a real valued function, is minimal if and only if u {\displaystyle u} satisfies
( 1 + u x 2 ) u y y − 2 u x u y u x y + ( 1 + u y 2 ) u x x = 0 {\displaystyle (1+u_{x}^{2})u_{yy}-2u_{x}u_{y}u_{xy}+(1+u_{y}^{2})u_{xx}=0}
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