On the conformal volume of 2-tori

2015/07/06 by Robert L. Bryant, Bryant, Robert L. · 2 citations
Mathematics · #53A30 #53C42 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1507.01485

openalex publication_date 2015/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.)

Citations

• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Cited by

• On Berger's Isoperimetric Problem
• An improved upper bound for the second eigenvalue on tori

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