# Optimal metrics for the first curl eigenvalue on 3-manifolds

## TL;DR

This paper investigates optimal metrics on 3D manifolds that minimize the first curl eigenvalue, linking it to properties of the Hodge Laplacian.

## Contribution

The paper establishes necessary and sufficient conditions for locally optimal metrics and proves local minimality for specific 3-manifolds.

## Key findings

- S³ and RP³ with the round metric are C¹-local minimizers for the first curl eigenvalue in their conformal and volume class.
- The canonical metrics of S³ and RP³ are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms.
- The results contrast with the behavior observed in four-dimensional manifolds.

## Abstract

In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the \documentclass[12pt]{minimal}
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\begin{document}$$L^{\frac{3}{2}}$$\end{document}L32-norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {S}^3$$\end{document}S3 and \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}\mathbb{P}^3$$\end{document}RP3 endowed with the round metric are \documentclass[12pt]{minimal}
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\begin{document}$$C^1$$\end{document}C1-local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {S}^3$$\end{document}S3 and \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}\mathbb{P}^3$$\end{document}RP3 are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in four dimensions.