On the conformal volume of 2-tori
Robert L. Bryant
TL;DR
The paper proves a strengthened version of Montiel–Ros's conjecture for the conformal volume of 2-tori by first establishing a sharp lower bound via a balanced-volume/Fourier-analytic framework and then deriving an upper bound for the fundamental immersion $f_\tau$, showing that $V_c(T_\tau,[|dz|^2])=\frac{4\pi^{2} y}{y^{2}+x^{2}-x+1}$ when $|\tau-\tfrac{1}{2}|\le \tfrac{3}{2}$; this exact value is achieved by the extremal finite Fourier-mode immersion, making the bound tight. The method hinges on decomposing balanced maps into Fourier modes on the dual lattice $\Lambda^{*}$, employing a convex-hull argument to identify the extremal support, and reducing the optimization to three principal modes, followed by delicate elliptic-integral estimates to control the remaining terms. The results connect conformal volume to Willmore energy bounds and illuminate equality conditions in a region of torus conformal structures, with an explicit upper-bound analysis provided for the remaining parameter range. A Maple-assisted computation supplements the analytic steps, yielding a concise and robust verification of the conjecture in the stated domain.
Abstract
This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of tori in the cases not covered by the conjecture of Montiel and Ros. Originally, I did not think that this claim was of enough interest to warrant including the (somewhat involved) proof, but time has shown otherwise. (Also, the proof included here is shorter than my original proof; instead, it relies on a MAPLE computation.)