I am currently learning about how to generate minimal surfaces using complex analysis — namely by calculating the Weierstraß parametrization (here's the article on Wikipedia).
Now in my lecture notes it is written that given a holomorphic function $f$ and a meromorphic function $g$ (such that at each pole of $g$ of order $k$, $f$ has a zero of order at least $2k$), the resulting Weierstraß parametrization will certainly locally be an embedding, but it might not be one globally.
Can you please help me out why that is the case? And maybe give an example where we don't get a global embedding?
Thank you very much!
The easiest answer is this: Enneper's surface. The Weierstraß representation is as simple as it gets, but the surface clearly crosses itself.