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.
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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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G. Silva Neto was partially supported by the National Council for Scientific and Technological Development—CNPq of Brazil.
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Silva Neto, G. Stability of constant mean curvature surfaces in three-dimensional warped product manifolds. Ann Glob Anal Geom 56, 57–86 (2019). https://doi.org/10.1007/s10455-019-09656-x
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DOI: https://doi.org/10.1007/s10455-019-09656-x