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Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric

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Abstract.

Given an orientable hypersurface M of a Lie group 𝔾 with a bi-invariant metric we consider the map N : M → 𝕊n that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of 𝔾 are also obtained.

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Authors and Affiliations

  1. UFRJ, Instituto de Matemática, Cidade Universitária, Rio de Janeiro – RJ, Brazil

    N. do EspĂ­rito-Santo

  2. Departamento de Matemática, ICEX, UFMG, Av. Antônio Carlos, 6627 C.P. 702, 30123-970, Belo Horizonte – MG, Brazil

    S. Fornari

  3. UFF, Instituto de Matemática, Av. São Paulo s/n, 24020-005, Niterói – RJ, Brazil

    K. Frensel

  4. UFRGS, Instituto de Matemática, Av. Bento Gonçalves, 9500, 91501-970, Porto Alegre – RS, Brazil

    J. Ripoll

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  1. N. do EspĂ­rito-Santo
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  2. S. Fornari
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  3. K. Frensel
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  4. J. Ripoll
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Correspondence to N. do EspĂ­rito-Santo.

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do Espírito-Santo, N., Fornari, S., Frensel, K. et al. Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric. manuscripta math. 111, 459–470 (2003). https://doi.org/10.1007/s00229-003-0357-5

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  • DOI: https://doi.org/10.1007/s00229-003-0357-5

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