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A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

Volume 125 / 2020

Fabiano G. B. Brito, Icaro Gonçalves, Adriana V. Nicoli Annales Polonici Mathematici 125 (2020), 203-213 MSC: Primary 57R25; Secondary 47H11, 58E20. DOI: 10.4064/ap200203-14-5 Published online: 9 November 2020

Abstract

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere $\mathbb {S}^{2n+1}$, immersed with degree 1, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $\mathcal {B}_k$ on a compact Riemannian manifold $M^{m}$, $1\leq k\leq m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $\mathcal {B}_n$ on $\mathbb {S}^{2n+1}$.

Authors

  • Fabiano G. B. BritoCentro de Matemática, Computação e Cognição
    Universidade Federal do ABC
    09.210-170 Santo André, Brazil
    e-mail
  • Icaro GonçalvesCentro de Matemática, Computação e Cognição
    Universidade Federal do ABC
    09.210-170 Santo André, Brazil
    e-mail
  • Adriana V. NicoliDepartamento de Matemática
    Instituto de Matemática e Estatística
    Universidade de São Paulo
    05508-900 São Paulo, SP, Brazil
    e-mail

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