Complex structure on a punctured torus giving a complex structure on the torus?

Question

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus? In other words, the Teichmuller space of the punctured torus is the same as that of the Torus (=upper half plane)?

Answer 1

There's a subtlety you may be missing. By definition of a punctured Riemann surface, your "complex structure on the once-punctured torus" extends to a complex structure on the torus (it is not just a complex structure on the torus minus a point). So there's nothing to show. I suppose a less confusing terminology would be "a punctured complex structure on the torus minus a point" instead of a "complex structure on the punctured torus".

However that does not imply trivially that the Teichmüller space of the punctured torus is the same as the Teichmüller space of the torus, even though it is true. Otherwise by the same reasoning, the Teichmüller space of a punctured surface would always be the same as the Teichmüller space of the same surface without the punctures (the closed surface), but that is not true. This is because the equivalence relation between complex structures for the punctured surface and for the closed surface are not the same a priori. It turns out that for the torus and the once-punctured torus, you can actually equate the Teichmüller spaces. I'll let you think about that, what you need to write down in order to prove it is going to depend on your precise definition of Teichmüller space. But ultimately, the key ingredient will be the fact that the group of (homotopically trivial) automorphisms of the torus with a complex structure acts transitively on that torus.

Edit: To clarify further: it is not true that any complex structure on $T^2 \setminus \{x_0\}$ extends to a complex structure on $T^2$. By definition of a "puncture", you only consider the complex structures that do extend.

Answer 2

Let $S$ be a closed surface and $F\subset S$ a finite subset. Then for each $f\in F$ there exists an open cylinder $Z_f\subset S\setminus F$ such that $S\setminus (\{f\}\cup Z_f)$ is compact. A complex structure on $S\setminus F$ extends over $f\in F$ if and only if the cylinder $Z_f$ has necessarily infinite conformal modulus. In other words it does not extend over $f$, if there is some cylinder $Z_f\subset S\setminus F$ of finite conformal modulus such that $S\setminus (\{f\} \cup Z_f)$ is compact. If $S\setminus F$ admits a conformal hyperbolic metric, a complex structure on $S\setminus F$ extends over $f$ if and only if the end of $S$ associated to $f$ has finite area (and is therefore a cusp).

But as mentioned by seub, in the context you consider, the complex structure is (usually) assumed to be defined on the whole of $S$ (and the set $F$ may be viewed as marked points on $S$). If $S\setminus F$ admits a conformal hyperbolic metric, this is equivalent to the assumption that $S\setminus F$ has finite area and hence all ends are cusps/punctures.

Fix a marking surface (a torus) $S_0$ together with $p_0\in S_0$ for $(T^2,p)$. The fact that the Teichmüller spaces of $T^2$ and $(T^2,p)$ can be identified follows from the bijectivity of the following forgetful map

$\{\text{equivalence classes of markings of $(T^2,p)$ by } (S_0,p_0)\}\rightarrow\{ \text{equivalence classes of markings of $T^2$ by }S_0\}$

for suitable notions of 'equivalence' and 'markings' (which are slightly different on left and right hand side). To show both the injectivity and the surjectivity of this map one uses the key ingredient mentioned by seub.