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Parabolicity and rigidity of two-sided hypersurfaces in Killing warped products

  • Eudes L. de Lima
  • Henrique F. de Lima
  • Eraldo A. LimaJr
  • Adriano A. Medeiros
Original Paper

Abstract

We extend a technique due to Romero et al. (Class Quantum Gav 30:1–13, 2013) establishing sufficient conditions to guarantee the parabolicity of a complete two-sided hypersurface immersed into a Killing warped product \(M\times _{\rho }{\mathbb {R}}\), whose base \(M^n\) has parabolic universal Riemannian covering. Afterwards, we apply our parabolicity criterium in order to obtain a rigidity result concerning these hypersurfaces, whose mean curvature is not supposed a priori be constant. Finally, parametric uniqueness results are applied to obtain suitable non-parametric ones, i.e., to the case of entire Killing graphs.

Keywords

Killing warped product Complete two-sided hypersurfaces Parabolic hypersurfaces Entire Killing graphs 

Mathematics Subject Classification

Primary 53C42 Secondary 53B30 53C50 

Notes

Acknowledgements

The authors would like to thank the referee for reading the manuscript in great detail and giving several valuable suggestions and useful comments which improved the paper. The second author is partially supported by CNPq, Brazil, Grant 303977/2015-9.

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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  • Eudes L. de Lima
    • 1
  • Henrique F. de Lima
    • 2
  • Eraldo A. LimaJr
    • 3
  • Adriano A. Medeiros
    • 3
  1. 1.Unidade Acadêmica de Ciências Exatas e da NaturezaUniversidade Federal de Campina GrandeCajazeirasBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

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