Weierstrass–Enneper parameterization

In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.

Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.

be functions on either the entire complex plane or the unit disk, where

are defined using the real part of a complex integral, as follows:

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.

[1] For example, Enneper's surface has f(z) = 1, g(z) = zm.

The Weierstrass-Enneper model defines a minimal surface

space), the Jacobian matrix of the surface can be written as a column of complex entries:

represents the two orthogonal tangent vectors of the surface:[2]

leads to a number of important properties:

The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.

[3] The derivatives can be used to construct the first fundamental form matrix:

and the second fundamental form matrix

{\displaystyle {\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}}

on the complex plane maps to a point

for all minimal surfaces throughout this paper except for Costa's minimal surface where

The classical examples of embedded complete minimal surfaces in

with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface.

Costa's surface involves Weierstrass's elliptic function

, a one parameter family of minimal surfaces is obtained.

Choosing the parameters of the surface as

( s , ϕ ) = cos ⁡ ( α )

At the extremes, the surface is a catenoid

represents a mixing angle.

The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the

axis in a helical fashion.

One can rewrite each element of second fundamental matrix as a function of

And consequently the second fundamental form matrix can be simplified as

{\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}}

which represents the principal direction in the complex domain.