Inspired by a discussion with Neil Strickland I am very interested to know examples of the following type.
A compact Riemann surface can be presented in many different ways. For example:
- A smoothly embedded embedded surface in the three-sphere $S^3$.
- A smooth projective curve (say cut out of $\mathbb{CP}^2$ by a single equation).
- A quotient of the hyperbolic plane by some fuchsian group.
In each of these we can accept some minor modifications. In (1) we accept embeddings into three-space $\mathbb{R}^3$ or the three-torus $\mathbb{T}^3$. In (3) we accept quotients of $\mathbb{C}$ by a lattice $\mathbb{Z} + \mathbb{Z}\omega$. We also accept tilings of the upper half-plane by a tiling without "moduli" (from which the fuchsian group can be deduced, with sufficient amount of hyperbolic trig). (We could also replace (3) in various other ways - for example by square-tiled surfaces or more generally by gluing explicitly described polygons in $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$.) I am not algebraic enough to deform the condition given in (2) - I hope some reader will suggest the correct modifications.
Question: Give explicit examples of pairs (or triples) of isomorphic Riemann surfaces of the above types.
As a motivating example, we have tori. For (3), we specify $\omega$ and thus the lattice $\mathbb{Z} + \mathbb{Z}\omega$. For (2), we have the Eisenstein series giving the modular invariants. For (1) we have the Hopf tori embedded in $S^3$ (and thus, after stereographic projection, embedded in three-space). Pinkall [Inventiones, 1985] defines these, and has explicit constructions (see Figure 4b (reproduced below) and Figure 6b). See also Figure 5 of Sullivan [Bridges, 2011]. I give further examples in the comments here.
