Explicit elliptic estimates for nowhere vanishing harmonic 1-forms

TL;DR

This paper derives explicit elliptic estimates for harmonic 1-forms on the 3-Torus, providing concrete constants and demonstrating the existence of nowhere vanishing harmonic 1-forms under metric perturbations.

Contribution

It computes explicit constants for injectivity estimates on the 3-Torus and extends these results to perturbed metrics, establishing the existence of nowhere vanishing harmonic 1-forms.

Findings

Explicit injectivity constant for the Laplace operator on the 3-Torus

Extension of estimates to perturbed metrics

Existence proof of nowhere vanishing harmonic 1-forms

Abstract

We compute an explicit constant for an injectivity estimate on the 3-Torus involving the Laplace Operator. First, we provide motivation for such explicit estimates. We perform the computation for the 3-Torus endowed with the flat metric before generalising to perturbed metrics. Finally, we apply these results to show existence of a nowhere vanishing harmonic 1-form on the 3-Torus endowed with a perturbed metric.