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Uniqueness of complete hypersurfaces with bounded higher order mean curvatures in semi-Riemannian warped products. (English) Zbl 1242.53065
This paper aims uniqueness results for complete hypersurfaces immersed into warped products of type $\pm\bbfR\times_{e^t} M^n$ with certain curvature restrictions. The key assumption is that the hypersurface satisfies $0<\beta\le H_r\le H_{r+1}\le\alpha$ for two constants $\alpha$, $\beta$ where $H_r$ denotes the $r$-th mean curvature. The conclusion is that the hypersurface is a slice in the warped product. This continues previous work by the authors in [Differ. Geom. Appl. 29, No. 4, 590--596 (2011; Zbl 1219.53056)] and by {\it A. Caminha} and the second author in [Gen. Relativ. Gravitation 41, No. 1, 173--189 (2009; Zbl 1162.83304)]. One of the ingredients in each case is Yau’s square operator.

53C42Immersions (differential geometry)
53B30Lorentz metrics, indefinite metrics
53C50Lorentz manifolds, manifolds with indefinite metrics
53Z05Applications of differential geometry to physics
83C99General relativity
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