Minimal surfaces

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.

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.

Note that this property is local: there might exist other surfaces that minimize area better with the same global boundary.

This definition makes minimal surfaces a 2-dimensional analogue to geodesics.

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.

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