How useful is the Weierstrass representation of minimal surfaces?

Question

Weierstrass representation of minimal surfaces says that if I have a holomorphic function $f: U \rightarrow \mathbb{C}$ and a meromorphic function $g: U \rightarrow \mathbb{C}$ such that $f g^2$ is holomorphic with $U$ being simply connected, then we have an integral representation of the minimal surface.

1. Now, this means that for $U = B (0,1)$ for example, we have a million possible pairs of homolomorphic functions $f,g$ defined on this set. So this should give us a million different minimal surfaces, is this correct?

2. We cannot control the shape of our surface or the domain its range in $\mathbb{R}^3$, as it is totally arbitrary how $f,g$ can be chosen. The only thing we said is that for a given minimal surface, we can find such a pair $f,g$, but normally you would be interested in the other question: Given some boundary in $\mathbb{R}^3$: How can I find a minimal surface? - This seems to be unanswered. - So I see no point in this Weierstraß representation.

3. Is it important that $U$ is simply connected? Let's consider the following "stupid" example $U = \mathbb{C}\backslash \{0\}$ $f=g=1$ . Can I still construct a minimal surface by the Weierstraß representation?

Answer 1

1. Yes, the Weierstrass–Enneper representation gives us infinitely many explicit examples of parametric minimal surfaces. This is what it does.

2. Then there are some things that it does not. It does not solve Plateau's problem. Not surprising, since boundary value problems for nonlinear PDE are not expected to admit explicit solutions.

3. The representation still works in non-simply-connected domains, although the integrals involved in it may be multiply valued, making the representation more difficult to use. For example, the helicoid has Weierstrass–Enneper representation defined on punctured plane, and involving the complex logarithm (see these notes). As a result, the image of parametrization is topologically different from the domain: it is simply-connected.

You can find the general form of W-E representation in section 9.3 of Harmonic Mappings in the Plane by Duren.

In the specific case you mentioned, the functions $f,g$ have holomorphic extension to $\mathbb C$, which is simply-connected; the puncture at $0$ has no effect other than removing one point from the surface.