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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2208.07317 |
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| _version_ | 1866912032127713280 |
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| author | Montiel, S. |
| author_facet | Montiel, S. |
| contents | 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.} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07317 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Curl through spin on three-manifold Montiel, S. Differential Geometry Analysis of PDEs 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.} |
| title | Curl through spin on three-manifold |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2208.07317 |