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:

${\begin{aligned}x_{k}(\zeta )&{}=\Re \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}={\mathbf {i}}f(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}$

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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