Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as therefore satisfies Lagrange's
equation,
(1)
(Gray 1997, p. 399).
Finding a minimal surface of a boundary with specified constraints is a problem in the calculus of variations and is sometimes
known as Plateau's problem. Minimal surfaces
may also be characterized as surfaces of minimal surface
area for given boundary conditions. A plane is a trivial
minimal surface, and the first nontrivial examples (the catenoid
and helicoid) were found by Meusnier in 1776 (Meusnier
1785). The problem of finding the minimum bounding surface of a skew
quadrilateral was solved by Schwarz in 1890 (Schwarz 1972).
Note that while a sphere is a "minimal surface" in the sense that it minimizes the surface area-to-volume ratio, it does not qualify
as a minimal surface in the sense used by mathematicians.
Euler proved that a minimal surface is planar iff its Gaussian curvature is zero at every point so that
it is locally saddle-shaped. The existence
of a solution to the general case was independently proven by Douglas (1931) and
Radó (1933), although their analysis could not exclude the possibility of
singularities. Osserman (1970) and Gulliver (1973) showed that a minimizing solution
cannot have singularities.
The only known complete (boundaryless), embedded (no self-intersections) minimal surfaces of finite topology known for 200 years were the catenoid,
helicoid, and plane. Hoffman
discovered a three-ended genus 1 minimal embedded surface,
and demonstrated the existence of an infinite number of such surfaces. A four-ended
embedded minimal surface has also been found. L. Bers proved that any finite
isolated singularity of a single-valued parameterized
minimal surface is removable.
A surface can be parameterized using an isothermal parameterization. Such a parameterization is minimal if the coordinate functions
are harmonic,
i.e.,
are analytic. A minimal surface can therefore
be defined by a triple of analytic functions
such that
A minimal surface known as "Karcher's Jacobi elliptic saddle towers" appeared on the cover of the June/July 1999 issue of Notices of the American Mathematical
Society (Karcher and Palais 1999).