On the conformal volume of 2-tori

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.)