A simple generalization of Garsia's conjecture
arXiv:2507.22240
The paper extends Garsia's conjecture on embedding genus‑1 surfaces in Euclidean 3‑space by formulating it in terms of connections on principal bundles and solves it in several natural cases, providing new geometric parameterizations of the moduli space of conformal classes of compact genus‑1 surfaces.
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.