I’m interested in studying how many possible different complex structures there are on the real plane.
With complex structure, I mean any element of the quotient of the set of Riemann surfaces with respect to the equivalence relation given by biholomorphisms.
I know Riemann theorem about simply connected Riemann surfaces classification. I know that there are at least two complex structure (one is simply given by the usual complex plane, and the other is induced by Poincaré disk). Is there any other complex structure? Or if these two are the unique possibilities, how can I prove this fact?
Thank you in advance.
If you know the Riemann mapping theorem, then the answer to your question can be derived from it.
The Riemann mapping theorem says that up to biholomorphism there are exactly three simply connected Riemann surfaces: the complex plane $\mathbb C$; the Poincaré disc $\mathbb D$; and the unit sphere in 3-space $\mathbb S$.
The real plane $\mathbb R^2$ is homeomorphic to $\mathbb C$ and to $\mathbb D$ (by construction of appropriate homeomorphisms) but not to $\mathbb S$ (by using compactness).
That immediately tells you that there are at most 2 complex structure on $\mathbb R^2$ up to biholomorphism.
To prove that both are achieved, you choose homeomorphisms and transport the structures. For example, to obtain a complex structure on $\mathbb R^2$ bihomolorphic to $\mathbb D$, you choose a homeomorphism $h : \mathbb R^2 \to \mathbb D$ and then use $h$ to pull the structure back from $\mathbb D$ to $\mathbb R^2$, obtaining a complex structure on $\mathbb R^2$ for which $h$ serves as a biholomorphism to $\mathbb D$.