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The Infimum of the Energy of Unit Vector Fields on Odd-Dimensional Spheres

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Abstract

We construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere \(\mathbb{S}\) 2k+1 for k ≥ 2, with energyconverging to the energy of the unit radial vector field, which isdefined on the complementary of two antipodal points. So we prove thatthe infimum of the energy of globally defined unit smooth vector fieldsis

$$\left( {\frac{{2k + 1}}{2} + \frac{k}{{2k - 1}}} \right){\text{ vol (}}\mathbb{S}^{2k + 1} ).$$

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Authors and Affiliations

  1. Institut Girard Desargues, Université Claude Bernard, Lyon 1, 43, boulevard du 11 Novembre 1918, 69622, Villeurbanne Cedex, France

    Vincent Borrelli

  2. Departamento de Matemática, IME-USP, Caixa Postal 66281, CEP 05315-970, São Paulo, SP, Brazil

    Fabiano Brito

  3. Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Valencia, 46100, Burjassot, Valencia, Spain

    Olga Gil-Medrano

Authors
  1. Vincent Borrelli
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  2. Fabiano Brito
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  3. Olga Gil-Medrano
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Borrelli, V., Brito, F. & Gil-Medrano, O. The Infimum of the Energy of Unit Vector Fields on Odd-Dimensional Spheres. Annals of Global Analysis and Geometry 23, 129–140 (2003). https://doi.org/10.1023/A:1022404728764

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