Zolotarev's lemma

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 2m (or equivalently, such that the product ƒg² is holomorphic), and let c₁, c₂, c₃ be constants. Then the surface with coordinates (x₁,x₂,x₃) is minimal, where the x_k are defined using the real part of a complex integral, as follows:

$\begin{array}{rl}{x}_k(\zeta )& =\mathrm{\Re }\{\underset{0}{\overset{\zeta }{\int }}{\phi }_k(z)dz\}+c_k,k=1,2,3\\ {\phi }_1& =f(1-g^2)/2\\ {\phi }_2& =\mathbf{i}f(1+g^2)/2\\ {\phi }_3& =fg\end{array}$

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

For example, Enneper's surface has ƒ(z) = 1, g(z) = z.

See also

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

References

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