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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On stable constant mean curvature surfaces in $ \mathbb{S}^2\times \mathbb{R}$ and $ \mathbb{H}^2\times \mathbb{R} $


Author: Rabah Souam
Journal: Trans. Amer. Math. Soc. 362 (2010), 2845-2857
MSC (2010): Primary 53C42, 49Q10
Published electronically: January 20, 2010
MathSciNet review: 2592938
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Abstract: We study the stability of immersed compact constant mean curvature (CMC) surfaces without boundary in some Riemannian 3-manifolds, in particular the Riemannian product spaces $ \mathbb{S}^2 \times \mathbb{R}$ and $ \mathbb{H}^2\times\mathbb{R}.$ We prove that rotational CMC spheres in $ \mathbb{H}^2\times\mathbb{R}$ are all stable, whereas in $ \mathbb{S}^2\times\mathbb{R}$ there exists some value $ H_0\approx 0.18$ such that rotational CMC spheres are stable for $ H\geq H_0$ and unstable for $ 0<H<H_0.$ We show that a compact stable immersed CMC surface in $ \mathbb{S}^2\times \mathbb{R}$ is either a finite union of horizontal slices or a rotational sphere. In the more general case of an ambient manifold which is a simply connected conformally flat 3-manifold with nonnegative Ricci curvature we show that a closed stable immersed CMC surface is either a sphere or an embedded torus. Under the weaker assumption that the scalar curvature is nonnegative, we prove that a closed stable immersed CMC surface has genus at most three. In the case of $ \mathbb{H}^2\times \mathbb{R}$ we show that a closed stable immersed CMC surface is a rotational sphere if it has mean curvature $ H\ge 1/\sqrt {2}$ and that it has genus at most one if $ 1/\sqrt{3} < H < 1/\sqrt {2}$ and genus at most two if $ H=1/\sqrt{3}.$


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Rabah Souam
Affiliation: Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris Diderot - Paris 7, Géométrie et Dynamique, Site Chevaleret, Case 7012, 75205 Paris Cedex 13, France
Email: souam@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04826-9
PII: S 0002-9947(10)04826-9
Keywords: Constant mean curvature, stability
Received by editor(s): October 23, 2007
Published electronically: January 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.