Curl through spin on three-manifold

TL;DR

This paper uncovers a surprising link between the curl operator and the Dirac operator on three-manifolds, leading to new spectral bounds and insights into geometric problems like the isoperimetric problem.

Contribution

It reveals an unexpected relation between curl and Dirac operators on three-manifolds, providing new spectral bounds and proving the non-existence of certain solutions for the curl-based isoperimetric problem.

Findings

Eigenvalues of curl are bounded below by those of Dirac, with equality characterizing the round three-sphere.

No mean-convex L^2 solutions exist for the curl isoperimetric problem.

The relation simplifies proofs of known results and yields new insights into the spectrum of curl.

Abstract

In the last decades, many mathematicians have studied the {\em curl operator} on compact (both with or without empty boundary) three-manifolds, mainly the behaviour of its spectrum and some iso\-pe\-ri\-me\-tric problems associated with it. In this paper, we reveal an (unexpected?) relation between this curl operator and the Dirac operator correspondingto any of the spin$^c$ structures on the manifold. Then, we make the ellipticity of $D$ (curl is not) and the many facts already known about the spectrum of $D$ to recuperate with almost immediate proofs some results above curl and obtain others unknown for me. {\em For example, we will find that the eigenvalues of curl, removing the point spectrum zero, are always, up to a fixed constant, lower bounded by those of the Dirac and the equality characterize the round three-sphere}. Also, we also show that {\em there do not exist mean-convex $L^2$-solutions for the isoperimetric problem associated to curl}, as Cantarella, de Turck, Gluck y Teytel \cite{CdTGT} had conjectured, while other authors proved properties for these unknown solutions (adding always whether optimal domains exist) perhaps by thinking in the case of the successful maximization of the {\em helicity.}