# 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.

Let ƒ and *g* be functions on either the entire complex plane or the unit disk, where *g* is meromorphic and ƒ is analytic, such that wherever *g* has a pole of order *m*, *f* has a zero of order 2*m* (or equivalently, such that the product ƒ*g*^{2} is holomorphic), and let *c*_{1}, *c*_{2}, *c*_{3} be constants. Then the surface with coordinates (*x*_{1},*x*_{2},*x*_{3}) is minimal, where the *x*_{k} 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 ƒ(*z*) = 1, *g*(*z*) = *z^m*.

## Contents

## Parametric surface of complex variables[edit]

The Weierstrass-Enneper model defines a minimal surface () on a complex plane (). Let (the complex plane as the space), we write the Jacobian matrix of the surface as a column of complex entries:

Where and are holomorphic functions of .

The Jacobian represents the two orthogonal tangent vectors of the surface:^{[2]}

The surface normal is given by

The Jacobian 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

Finally, a point on the complex plane maps to a point on the minimal surface in by

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

## Embedded minimal surfaces and examples[edit]

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 :^{[4]}

where is a constant ^{[5]} (Chapter 22).

## Lines of curvature[edit]

One can rewrite each element of second fundamental matrix as a function of and , for example

And consequently we can simplify the second fundamental form matrix as

One of its eigenvectors is

which represents the principal direction in the complex domain.^{[6]} Therefore, the two principal directions in the space turn out to be

## See also[edit]

- Associate family
- Bryant surface, found by an analogous parameterization in hyperbolic space

## References[edit]

**^**Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.*Minimal surfaces*, vol. I, p. 108. Springer 1992. ISBN 3-540-53169-6**^**Andersson, S., Hyde, S.T., Larsson, K., Lidin, S.: Minimal surfaces and structures: from inorganic and metal crystals to cell membranes and biopolymers. Chem. Rev. 88(1), 221–242 (1988)**^**Sharma, R.: The weierstrass representation always gives a minimal surface. arXiv preprint arXiv:1208.5689 (2012)**^**Lawden, D.F.: Elliptic Functions and Applications, vol. 80. Springer, Berlin (2013)**^**Abbena, E., Salamon, S., Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton (2006)**^**Hua, H. and Jia, T., 2018. Wire cut of double-sided minimal surfaces; the Visual Computer, 34(6-8), pp.985-995.