ここらの話が、”Teichmuller space”に繋がっているんだね

https://en.wikipedia.org/wiki/Riemann_surface
Riemann surface

Classification of Riemann surfaces
Parabolic Riemann surfaces
If X is a Riemann surface whose universal cover is isomorphic to the complex plane C then it is isomorphic one of the following surfaces:

・ C itself;
・The quotient C/Z;
・A quotient C/(Z +Zτ) where τ ∈ C with Im (τ)>0.
Topologically there are only three types: the plane, the cylinder and the torus. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter τ in the third case gives non-isomorphic Riemann surfaces. The description by the parameter τ gives the Teichmuller space of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmuller space by the mapping class group. In this case it is the modular curve.

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