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Geometric barriers for the existence of hypersurfaces with prescribed curvatures in M\(^n\times \) R

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Abstract

We show the existence of a deformation process of hypersurfaces from a product space \(\mathbb {M}_1\times \mathbb {R}\) into another product space \(\mathbb {M}_2\times \mathbb {R}\) such that the relation of the principal curvatures of the deformed hypersurfaces can be controlled in terms of the sectional curvatures or Ricci curvatures of \(\mathbb {M}_1\) and \(\mathbb {M}_2.\) In this way, we obtain barriers which are used for proving existence or non existence of hypersurfaces with prescribed curvatures in a general product space \(\mathbb {M}\times \mathbb {R}\).

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, E-18071, Granada, Spain

    José A. Gálvez

  2. Departamento de Matemática Aplicada, UNED, E-13600, Alcázar de San Juan, Ciudad Real, Spain

    Victorino Lozano

Authors
  1. José A. Gálvez
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  2. Victorino Lozano
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Corresponding author

Correspondence to José A. Gálvez.

Additional information

Communicated by L. Ambrosio.

The authors were partially supported by MICINN-FEDER, Grant No. MTM2013-43970-P, by Junta de Andalucía Grant No. FQM325, and by Junta de Comunidades de Castilla La Mancha Grant No. PEII-2014-001-A.

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Gálvez, J.A., Lozano, V. Geometric barriers for the existence of hypersurfaces with prescribed curvatures in M\(^n\times \) R. Calc. Var. 54, 2407–2419 (2015). https://doi.org/10.1007/s00526-015-0869-3

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  • DOI: https://doi.org/10.1007/s00526-015-0869-3

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