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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
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