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Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 57–86 | Cite as

Stability of constant mean curvature surfaces in three-dimensional warped product manifolds

  • Gregório Silva NetoEmail author
Article
  • 65 Downloads

Abstract

In this paper, we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three-dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space.

Keywords

Stability Warped product manifolds Constant mean curvature 

Mathematics Subject Classification

53C42 49Q10 53A10 

Notes

Acknowledgements

The author would like to thank Hilário Alencar by helpful conversations during the preparation of this paper and to the anonymous referee by the useful observations.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal de AlagoasMaceióBrazil

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