Minimal Surface

A minimal surface is the image of an immersion into a 3-dimensional Riemannian manifold whose mean curvature is zero. Equivalently, it is a stationary point of surface area under variations with compact support. This definition includes surfaces in 3-dimensional Euclidean space, hyperbolic minimal surfaces, and minimal surfaces in other 3-dimensional ambient geometries.

A minimal surface in 3-dimensional Euclidean space parametrized as satisfies Lagrange's equation,

(1)

(Gray 1997, p. 399).

Finding an area-minimizing surface spanning a specified boundary is a problem in the calculus of variations known as Plateau's problem. Every smooth area-minimizing surface is minimal, but a minimal surface need only be a stationary point and need not minimize surface area globally. 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. Bers (1951) proved that a finite isolated singularity of a minimal graph is a removable singularity.

For isothermal parameters , define . For a minimal surface, these functions are analytic and satisfy

(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 Karcher JE saddle tower appeared on the cover of the June/July 1999 issue of Notices of the American Mathematical Society (Karcher and Palais 1999).

Minimal surfaces such as the helicoid, catenoid, and surfaces similar to Schwarz's minimal surface also occur in architectural shell forms (Bock Hyeng et al. 2025).