Hopf tori and standard tori
Leonardo A. Cano García
TL;DR
This work provides an explicit classification of the conformal classes of standard and product tori in genus-1 moduli space ${\mathscr{M}}={\mathbb{IH}}/SL(2,{\mathbb{Z}})$, showing these classes lie on the positive imaginary axis and can be realized as Hopf tori arising from circles on $S^2$. By constructing holomorphic coordinates on standard and product tori, the authors derive period lattices ${\Lambda}=2\pi\mathbb{Z}+i\omega\mathbb{Z}$ with $\omega=\frac{2\pi}{\sqrt{(R/r)^2-1}}$ and $\tau=i\omega/(2\pi)$, and similarly describe product tori via lattices ${\mathbb{Z}}+(bi/a)\mathbb{Z}$, thereby proving standard and product tori are conformally equivalent. They then connect these to Hopf tori: for circles on $S^2$ the associated complex torus is $\mathbb{C}/(2\pi\mathbb{Z}+(\frac{A}{2}+i\frac{L}{2})\mathbb{Z})$, with $A(p)$ and $L$ the area and length, and show that Hopf tori exhaust all genus-1 conformal classes. In particular, a standard torus $T_{R,r}$ with $R/r>\sqrt{2}$ is conformally equivalent to the Hopf torus $H_t$ with $t=\frac{r}{R}$, via the chain $T_{R,r}\sim S_b\sim H_t$, thus unifying the three torus families through explicit conformal data. The results illuminate the structure of the genus-1 moduli space from a differential-geometric viewpoint and open avenues to relate these geometric classifications to algebraic-geometry perspectives on elliptic curves and moduli.
Abstract
This article provides a complete characterization of the conformal classes of product tori and standard flat tori in complex dimension 1 (real dimension 2). Utilizing basic differential geometry methods, our approach contrasts with techniques employing Hopf tori for the conformal classification of Riemann surfaces of genus 1. While the results may be familiar to experts in complex analysis and Riemann surface theory, we contend that this work offers a clear and insightful perspective on the conformal properties of these geometrically distinct appearing tori.