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A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold

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Abstract

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.

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Authors and Affiliations

  1. Universidade Federal de Pelotas—IFM, Campus Universitário—Capão do Leão, Caixa Postal 354, 96010-900, Pelotas, RS, Brazil

    Giovanni Nunes

  2. Universidade Federal do R.G. do Sul, Instituto de Matemática, Av. Bento Gonçalves 9500, 91501-570, Porto Alegre, RS, Brazil

    Jaime Ripoll

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  1. Giovanni Nunes
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  2. Jaime Ripoll
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Correspondence to Jaime Ripoll.

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Nunes, G., Ripoll, J. A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold. J Geom Anal 18, 1088–1097 (2008). https://doi.org/10.1007/s12220-008-9046-7

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  • DOI: https://doi.org/10.1007/s12220-008-9046-7

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