A simple generalization of Garsia's conjecture

TL;DR

This paper generalizes Garsia's conjecture on conformal classes of genus-1 surfaces, offering new parameterizations of their moduli space through connections on principal bundles and providing solutions in key cases.

Contribution

It introduces a natural generalization of Garsia's conjecture within the framework of principal bundle connections and offers new geometric parameterizations of the moduli space.

Findings

Solutions provided for several natural cases of the conjecture

New parameterizations of the moduli space of genus-1 surfaces

Geometric interpretation of the parameterizations

Abstract

We propose a natural generalization of a conjecture by Garsia, originally concerning the realization of conformal classes of genus-1 surfaces via embeddings in three-dimensional Euclidean space. This generalized conjecture is formulated within the framework of connections on principal bundles. We address this conjecture by providing solutions in several natural cases. As an outcome, our work yields novel parameterizations of the moduli space of conformal classes of compact surfaces of genus 1, each endowed with a clear geometric interpretation.