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Compact stable constant mean curvature surfaces in homogeneous 3-manifolds. (English) Zbl 1278.53065
In 1984, J. L. Barbosa and M. P. do Carmo [Math. Z. 185, 339–353 (1984; Zbl 0513.53002)] proved that any compact, stable orientable hypersurface in Euclidean space with nonzero constant mean curvature is a sphere. There have been many results on such hypersurfaces in several ambient spaces, for example, real space forms, and homogeneous 3-manifolds. In 2004, U. Abresch and H. Rosenberg [Acta Math. 193, No. 2, 141–174 (2004; Zbl 1078.53053)] obtained that the Berger spheres, the Heisenberg group, the special linear group, and the Riemannian product \(S^{2} \times \mathbb{R}\) and \(\mathbb{H}^{2} \times \mathbb{R}\), where \(S^{2}\) and \(\mathbb{H}^{2}\) are the 2-dimensional sphere and hyperbolic plane with their standard metric, are the cases for the homogeneous 3-manifolds. \smallskip In this paper, the authors investigate compact orientable constant mean curvature surfaces in the Berger spheres, the special linear group, and the Heisenberg group. They show that all surfaces in the last two are stable, whereas some in the Berger spheres are unstable. As a result, they also solve the isoperimetric problem: Given a Berger sphere \(S^{3}_{b}(\kappa,\tau)\) and a number \(V\) with \(0<V<\mathrm{vol}(S^{3}_{b}(\kappa,\tau))=\frac{32\tau\pi^{2}}{\kappa^{2}}\), find the embedded compact surfaces of least area enclosing a domain of volume \(V\).

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
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