Minimal Surface

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

(2)

The real parameterization is then obtained as

(3)

But, for an analytic function and a meromorphic function , the triple of functions

			(4)
			(5)
			(6)

are analytic as long as has a zero of order at every pole of of order . This gives a minimal surface in terms of the Enneper-Weierstrass parameterization

(7)

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).