×

On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds. (English) Zbl 1102.53044

The paper deals with \(n\)-dimensional space-like hypersurfaces of constant sectional curvature Lorentz manifolds given by an immersion of the form \(x: M^n \to \widetilde M^{n+1}.\) The author first computes \(L_r(S_r)\) for such immersions where \(L_r: \mathcal D(M) \to \mathcal D(M)\) is the second order differential operator defined as \[ L_r(f)=\text{tr}(P_r \text{Hess} f) \] where \(P_r\) is the \(r\)-th Newton transformation on \(M.\) Then the author makes use of the formula not only to consider \(r\)-maximal space-like hypersurfaces of \(\widetilde M\) but also to obtain conditions for the sectional curvature of \(M\) to make it umbilic in the case of a constant higher order mean curvature. One of the main results of the paper in terms of applications of the formula for \(L_r(S_r)\) is Theorem 1, i.e, if \(x: M^n \to \widetilde M^{n+1}_c\) is a closed space-like hypersurface of a time-oriented Lorentz manifold of constant sectional curvature \(c \geq 0\) and \(M\) is of constant scalar curvature \(R\) such that
(a) \(c(\frac{n-2}{n})< R \leq c\), then \(M\) is totally umbilical,
(b) \(c(\frac{n-2}{n})\leq R \leq c\) and \(S_3 \neq 0,\) then \(M\) is totally umbilical,
(c) \(c(\frac{n-2}{n})\leq R < c\), then \(M\) has constant mean curvature.
Moreover, the case of \(c(\frac{n-2}{n})< R < c\) is considered under some assumptions on the mean curvature \(H\) of \(M\) in Theorem 2.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C99 Global differential geometry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aledo, J.A.; Alías, L.J.; Romero, A., Integral formulas for compact spacelike hypersurfaces in the de Sitter space: applications to the case of constant higher order Mean curvature, J. geom. phys., 31, 195-208, (1999) · Zbl 0969.53031
[2] Alencar, H.; do Carmo, M.; Colares, G., Stable hypersurfaces with constant scalar curvature, Math. Z., 213, 117-131, (1993) · Zbl 0792.53057
[3] Alías, L.J.; Brasil, A.; Colares, A.G., Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications, Proc. Edinburgh math. soc., 46, 465-488, (2003) · Zbl 1053.53038
[4] Alías, L.J.; Romero, A.; Sánchez, M., Uniqueness of complete spacelike hypersurfaces of constant Mean curvature in generalized Robertson-Walker spacetimes, Gen. relat. gravit., 27, 71-84, (1995) · Zbl 0908.53034
[5] Barbosa, J.L.M.; Colares, A.G., Stability of hypersurfaces with constant r-Mean curvature, Ann. global anal. geom., 15, 277-297, (1997) · Zbl 0891.53044
[6] Barbosa, J.L.M.; Oliker, V., Spacelike hypersurfaces with constant Mean curvature in Lorentz spaces, Matem. contemporânea, 4, 27-44, (1993) · Zbl 0852.53045
[7] Brasil, A.; Colares, A.G.; Palmas, O., Complete spacelike hypersurfaces with constant Mean curvature in the de Sitter space: A gap theorem, Ill. J. math., 47, 847-866, (2003) · Zbl 1047.53031
[8] Cheng, S.Y.; Yau, S.T., Maximal spacelike hypersurfaces in the lorentz – minkowski space, Ann. math., 104, 407-419, (1976) · Zbl 0352.53021
[9] Cheng, S.Y.; Yau, S.T., Hypersurfaces with constant scalar curvature, Math. ann., 225, 195-204, (1977) · Zbl 0349.53041
[10] Dajczer, M., Submanifolds and isometric immersions, (1990), Publish or Perish Houston · Zbl 0705.53003
[11] Goddard, A.J., Some remarks on the existence of spacelike hypersurfaces of constant Mean curvature, Math. proc. Cambridge phil. soc., 82, 489-495, (1977) · Zbl 0386.53042
[12] Hardy, G.; Littlewood, J.E.; Pólya, G., Inequalities, (1989), Cambridge Mathematical Library Cambridge · Zbl 0634.26008
[13] Hounie, J.; Leite, M.L., The maximum principle for hypersurfaces with vanishing curvature functions, J. diff. geom., 41, 247-258, (1995) · Zbl 0821.53007
[14] Hounie, J.; Leite, M.L., Two-ended hypersurfaces with zero scalar curvature, Ind. univ. math. J., 48, 867-882, (1999) · Zbl 0929.53033
[15] Montiel, S., An integral inequality for compact spacelike hypersurfaces in the de Sitter space and applications to the case of constant Mean curvature, Ind. univ. math. J., 37, 909-917, (1988) · Zbl 0677.53067
[16] Montiel, S., Uniqueness of spacelike hypersurfaces of constant Mean curvature in foliated spacetimes, Math. ann., 314, 529-553, (1999) · Zbl 0965.53043
[17] O’Neill, B., Semi-Riemannian geometry with applications to relativity, (1983), Academic Press New York · Zbl 0531.53051
[18] Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. sci. math., 117, 217-239, (1993) · Zbl 0787.53046
[19] Reilly, R., Variational properties of functions of the Mean curvatures for hypersurfaces in space forms, J. diff. geom., 8, 465-477, (1973) · Zbl 0277.53030
[20] Simons, J., Minimal varieties in Riemannian manifolds, Ann. math., 88, 62-105, (1968) · Zbl 0181.49702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.