Integral inequalities for maximal spacelike hypersurfaces in the indefinite space form. (English) Zbl 0954.53038

Cao, Xi-Fang

Differ. Geom. Appl. 10, No. 2, 155-159 (1999).

The author gives two integral inequalities for \(n\)-dimensional compact maximal spacelike hypersurfaces in an \((n+1)\)-dimensional indefinite space form \(N_1^{n+1}(c)\). We note that the following result is well-known: any \(n\)-dimensional complete maximal spacelike hypersurface in an \((n+1)\)-dimensional indefinite space form \(N_1^{n+1}(c)\) \((c\geq 0)\) is totally geodesic [see S. Y. Cheng and S.-T. Yau, Ann. Math. 104, 407-419 (1976; Zbl 0352.53021); also see reviewer’s generalization: Soochow J. Math. 23, 79-89 (1997; Zbl 0870.53048)]. We also know that “complete” implies “compact”. Therefore, the results of the paper under review only make sense when \(c<0\). For the results of complete maximal spacelike hypersurfaces in \(N_1^{n+1}(c)\) \((c<0)\), readers can consult T. Ishihara’s paper [Mich. Math. J. 35, 345-352 (1988; Zbl 0682.53055)] and the above reviewer’s paper.

Reviewer: Haizhong Li (Beijing)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

complete maximal spacelike hypersurface; indefinite space form

Citations:

Zbl 0352.53021; Zbl 0870.53048; Zbl 0682.53055

References:

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