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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 $±ℝ{×}_{{e}^{t}}{M}^{n}$ 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 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.
##### MSC:
 53C42 Immersions (differential geometry) 53B30 Lorentz metrics, indefinite metrics 53C50 Lorentz manifolds, manifolds with indefinite metrics 53Z05 Applications of differential geometry to physics 83C99 General relativity