Abstract
In this paper we generalize the explicit formulas for constant mean curvature (CMC) immersion of hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces given in Perdomo (Asian J Math 14(1):73–108, 2010; Rev Colomb Mat 45(1):81–96, 2011) to provide explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-Riemannian manifolds with constant sectional curvature. In particular, we prove that every \(h\in[-1,-\frac{2\sqrt{n-1}}{n})\) can be realized as the constant curvature of a complete immersion of \(S_1^{n-1}\times \mathbb{R}\) in the (n + 1)-dimensional de Sitter space \(\hbox{\bf S}_1^{n+1}\). We provide 3 types of immersions with CMC in the Minkowski space, 5 types of immersion with CMC in the de Sitter space and 5 types of immersion with CMC in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.
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This work was partially financed by a CCSU research grant.
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Perdomo, O.M. CMC Hypersurfaces on Riemannian and Semi-Riemannian Manifolds. Math Phys Anal Geom 15, 17–37 (2012). https://doi.org/10.1007/s11040-011-9101-7
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DOI: https://doi.org/10.1007/s11040-011-9101-7