Unit vector fields on antipodally punctured spheres: big index, big volume

Brito, Fabiano G. B.; Chacón, Pablo M.; Johnson, David L.

doi:10.24033/bsmf.2551

Unit vector fields on antipodally punctured spheres: big index, big volume
[Champs unitaires dans les sphères antipodalement trouées : grand indice entraîne grand volume]

Brito, Fabiano G. B. ; Chacón, Pablo M. ; Johnson, David L.

Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 1, pp. 147-157.

Résumé
Abstract

Nous établissons une borne inférieure pour le volume d’un champ de vecteurs $\vec{v}$ défini dans $𝐒^{n} ∖ {\pm x}$ , $n = 2, 3$ . Cette borne inférieure dépend de la somme des valeurs absolues des indices de $\vec{v}$ en $x$ et en $- x$ .

We establish in this paper a lower bound for the volume of a unit vector field $\vec{v}$ defined on $𝐒^{n} ∖ {\pm x}$ , $n = 2, 3$ . This lower bound is related to the sum of the absolute values of the indices of $\vec{v}$ at $x$ and $- x$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/bsmf.2551

Classification : 53C20, 57R25, 53C12
Keywords: unit vector fields, volume, singularities, index
Mot clés : champs vectoriels unitaires, volume, singularités, indice

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     title = {Unit vector fields on antipodally punctured spheres: big index, big volume},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {147--157},
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     mrnumber = {2415338},
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Brito, Fabiano G. B.; Chacón, Pablo M.; Johnson, David L. Unit vector fields on antipodally punctured spheres: big index, big volume. Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 1, pp. 147-157. doi : 10.24033/bsmf.2551. https://www.numdam.org/articles/10.24033/bsmf.2551/

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Albuquerque, Rui Vector fields with big and small volume on the 2-sphere, Hiroshima Mathematical Journal, Volume 53 (2023) no. 2 | DOI:10.32917/h2022009
Albuquerque, Rui Calibrations for the Volume of Unit Vector Fields in Dimension 2, Results in Mathematics, Volume 78 (2023) no. 3 | DOI:10.1007/s00025-023-01891-w
Brito, Fabiano; Conrado, Jackeline; Gonçalves, Icaro; Nicoli, Adriana; Nunes, Giovanni Minimally Immersed Klein Bottles in the Unit Tangent Bundle of the Unit 2-Sphere Arising from Area-Minimizing Unit Vector Fields on $S^{2} ∖ {N, S}$ , The Journal of Geometric Analysis, Volume 33 (2023) no. 5 | DOI:10.1007/s12220-023-01190-4
Gil-Medrano, Olga Vector Fields of Constant Length on Punctured Spheres, The Volume of Vector Fields on Riemannian Manifolds, Volume 2336 (2023), p. 89 | DOI:10.1007/978-3-031-36857-8_5
Brito, Fabiano G. B.; Conrado, Jackeline; Gonçalves, Icaro; Nicoli, Adriana V. Area minimizing unit vector fields on antipodally punctured unit 2-sphere, Comptes Rendus. Mathématique, Volume 359 (2022) no. 10, p. 1225 | DOI:10.5802/crmath.258
Brito, Fabiano G. B.; Gomes, André O.; Gonçalves, Icaro Poincaré index and the volume functional of unit vector fields on punctured spheres, manuscripta mathematica, Volume 161 (2020) no. 3-4, p. 487 | DOI:10.1007/s00229-019-01107-y

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