Knotted structures  in high-energy Beltrami fields  on the torus and the sphere

Enciso, Alberto; Peralta-Salas, Daniel; Torres de Lizaur, Francisco

doi:10.24033/asens.2337

Knotted structures in high-energy Beltrami fields on the torus and the sphere
[Structures nouées dans les champs de Beltrami à hautes énergies sur le tore et la sphère]

Enciso, Alberto ; Peralta-Salas, Daniel ; Torres de Lizaur, Francisco

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 995-1016.

Résumé
Abstract

Soit $𝒮$ une collection finie de courbes et de tubes fermés, disjoints deux à deux mais pouvant être noués et entrelacés, dans la sphère ronde $𝕊^{3}$ ou dans le tore plat $𝕋^{3}$ . Dans le cas du tore, on suppose davantage que $𝒮$ est contenu dans un sous-ensemble contractile de $𝕋^{3}$ . Dans cet article on montre que, pour tout entier impair $λ$ suffisamment grand, il existe un champ de Beltrami dans $𝕊^{3}$ ou $𝕋^{3}$ satisfaisant $curl u = λ u$ et qui a une collection de lignes et tubes de vorticité donnés par $𝒮$ , modulo un difféomorphisme ambiant.

Let $𝒮$ be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere $𝕊^{3}$ or in the flat torus $𝕋^{3}$ . In the case of the torus, $𝒮$ is further assumed to be contained in a contractible subset of $𝕋^{3}$ . In this paper we show that for any sufficiently large odd integer $λ$ there exists a Beltrami field on $𝕊^{3}$ or $𝕋^{3}$ satisfying $curl u = λ u$ and with a collection of vortex lines and vortex tubes given by $𝒮$ , up to an ambient diffeomorphism.

MR Zbl

DOI : 10.24033/asens.2337

Classification : 58J50; 34K19; 35Q35
Keywords: Beltrami fields, high energy spectrum, vortex lines, knots and links.
Mot clés : Champs de Beltrami, spectre aux hautes énergies, lignes de tourbillons, nœuds et entrelacs.

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     author = {Enciso, Alberto and Peralta-Salas, Daniel and Torres de Lizaur, Francisco},
     title = {Knotted structures  in high-energy {Beltrami} fields  on the torus and the sphere},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {995--1016},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {4},
     year = {2017},
     doi = {10.24033/asens.2337},
     mrnumber = {3679619},
     zbl = {1379.35226},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2337/}
}

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Enciso, Alberto; Peralta-Salas, Daniel; Torres de Lizaur, Francisco. Knotted structures  in high-energy Beltrami fields  on the torus and the sphere. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 995-1016. doi : 10.24033/asens.2337. http://www.numdam.org/articles/10.24033/asens.2337/

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