Minimal surfaces, how to convert different Enneper-Weierstrass representation?

Question

I don't know much about Enneper-Weierstrass representation, but it seems in general, for a surface, we provide a holomorphic function $f$ and a meromorphic function $g$. For the catenoïd for instance, this would be : $(f,g) = (-\frac{e^{-z}}{2},-e^z)$. But in this documents at page 15, the author gives a representation $(G,dh)=(z,\frac{1}{z})$ for the catenoïd. Are these the same? How can one in general pass from one to the other representation?

Answer 1

Let $(f,g)$ a Weierstrass pair defined on a simply connected Riemann Surface $M$. Now consider $\widetilde{M}$ another simply connected Riemann Surface and $w : \widetilde{M} \rightarrow M$, $w=w(\xi)$, a conformal map. Define $$\left\{\begin{aligned}\widetilde{f}(\xi) & =f(w(\xi))w'(\xi) \\ \widetilde{g}(\xi) &= g(w(\xi))\end{aligned}\right.$$ Then the new pair $(\widetilde{f},\widetilde{g})$ is a Weierstrass pair defining the same minimal immersion.