Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Fabiano G. B. Brito; Jackeline Conrado; Icaro Gonçalves; Adriana V. Nicoli

doi:10.5802/crmath.258

Geometry and Topology

Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Fabiano G. B. Brito ¹; Jackeline Conrado ²; Icaro Gonçalves ¹; Adriana V. Nicoli ²

¹ Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Santo André, 09210-170, Brazil
² Dpto. de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão 1010, São Paulo-SP, 05508-900, Brazil

Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1225-1232.

Abstract

We provide a lower value for the volume of a unit vector field tangent to an antipodally punctured Euclidean sphere $𝕊^{2}$ depending on the length of an ellipse determined by the indexes of its singularities. We also exhibit minimizing vector fields ${\vec{v}}_{k}$ within each index class and show that they are the only ones that are sharp for the volume. These fields have areas given essentially by the length of ellipses depending just on the indexes in $N$ and $S$ .

Received: 2021-02-18
Revised: 2021-05-20
Accepted: 2021-08-11
Published online: 2022-01-04

DOI: 10.5802/crmath.258

Author's affiliations:

Fabiano G. B. Brito ¹; Jackeline Conrado ²; Icaro Gonçalves ¹; Adriana V. Nicoli ²

¹ Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Santo André, 09210-170, Brazil
² Dpto. de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão 1010, São Paulo-SP, 05508-900, Brazil

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2021__359_10_1225_0,
     author = {Fabiano G. B. Brito and Jackeline Conrado and Icaro Gon\c{c}alves and Adriana V. Nicoli},
     title = {Area minimizing unit vector fields on antipodally punctured unit 2-sphere},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1225--1232},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {10},
     year = {2021},
     doi = {10.5802/crmath.258},
     language = {en},
}

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Fabiano G. B. Brito; Jackeline Conrado; Icaro Gonçalves; Adriana V. Nicoli. Area minimizing unit vector fields on antipodally punctured unit 2-sphere. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1225-1232. doi : 10.5802/crmath.258. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.258/

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