Curl through spin on three-manifold

S. Montiel

Published 2022-08-15Version 1

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 $L^2$-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.}