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On stable constant mean curvature surfaces in $$\mathbb S^2\times \mathbb R$$ and $$\mathbb H^2\times \mathbb R$$. (English) Zbl 1195.53089
Author’s 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 $$H\geq 1/\sqrt {2}$$ and that it has genus at most one if $$1/\sqrt{3} < H < 1/\sqrt {2}$$ and genus at most two if $$H=1/\sqrt{3}.$$

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 49Q10 Optimization of shapes other than minimal surfaces
##### Keywords:
constant mean curvature; stability
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