De Sitter space

complete space-like submanifolds in a

Let $ \mathbf R _ {p} ^ {n + p + 1 } $ be an $ ( n + p + 1 ) $-dimensional Minkowski space of index $ p $, i.e., $ \mathbf R _ {p} ^ {n + p + 1 } = \{ ( x _ {1}, \dots, x _ {n + p + 1 } ) \in \mathbf R ^ {n + p + 1 } \} $ and is equipped with the Lorentz metric $ \sum _ {i = 1 } ^ {n + 1 } ( dx _ {i} ) ^ {2} - \sum _ {j = 1 } ^ {p} ( dx _ {n + 1 + j } ) ^ {2} $. For $ c > 0 $, let

$$ S _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p} ^ {n + p + 1 } } : $$

$$ \ {} {x _ {1} ^ {2} + \dots + x _ {n + 1 } ^ {2} - x _ {n + 2 } ^ {2} - \dots - x _ {n + p + 1 } ^ {2} = {1 / c } } \} . $$

Thus, $ S _ {p} ^ {n + p } ( c ) $ is an $ ( n + p ) $-dimensional indefinite Riemannian manifold of index $ p $ and of constant curvature $ c $. It is called an $ ( n + p ) $-dimensional de Sitter space of constant curvature $ c $ and of index $ p $. E. Calabi, S.Y. Cheng and S.T. Yau proved that a complete maximal space-like hypersurface in a Minkowski space $ \mathbf R _ {1} ^ {n + 1 } $ possesses a remarkable Bernstein property. As a generalization of the Bernstein-type problem, S. Ishihara proved that a complete maximal space-like submanifold in a de Sitter space $ S _ {p} ^ {n + p } ( c ) $ is totally geodesic (cf. Totally-geodesic manifold). It was proved by K. Akutagawa [a1], Q.M. Cheng [a2] and K.G. Ramanathan that complete space-like submanifolds with parallel mean curvature vector in a de Sitter space $ S _ {p} ^ {n + p } ( c ) $ are totally umbilical (cf. also Differential geometry) if

1) $ H ^ {2} \leq c $, when $ n = 2 $;

2) $ n ^ {2} H ^ {2} < 4 ( n - 1 ) c $, when $ n \geq 3 $. The conditions 1) and 2) are best possible. When $ n = 2 $, Akutagawa and Ramanathan constructed many examples of space-like submanifolds in $ S _ {1} ^ {3} ( c ) $ that are not totally umbilical. When $ n \geq 3 $, $ H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} ) $, where $ c _ {1} = ( 2 - n ) c $ and $ c _ {2} = [ { {( n - 2 ) } / {( n - 1 ) } } ] c $, is a complete space-like hypersurface in $ S _ {1} ^ {n + 1 } ( c ) $ of constant mean curvature $ H $ that is not totally umbilical and satisfies $ n ^ {2} H ^ {2} = 4 ( n - 1 ) c $. Cheng gave a characterization of complete non-compact hypersurfaces in $ S _ {1} ^ {n + 1 } ( c ) $ with $ n ^ {2} H ^ {2} = 4 ( n - 1 ) c $: a complete non-compact hypersurface in $ S _ {1} ^ {n + 1 } ( c ) $ with $ n ^ {2} H ^ {2} = 4 ( n - 1 ) c $ is either isometric to $ H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} ) $ or its Ricci curvature is positive and the squared norm of the second fundamental form is a subharmonic function. Therefore, the Cheeger–Gromoll splitting theorem implies that a complete non-compact hypersurface $ M $ in $ S _ {1} ^ {n + 1 } ( c ) $ with $ n ^ {2} H ^ {2} = 4 ( n - 1 ) c $ is isometric to $ H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} ) $ if the number of its ends is not less than $ 2 $. S. Montiel [a4] has proved that a compact space-like hypersurface in $ S _ {1} ^ {n + 1 } ( c ) $ of constant mean curvature is totally umbilical, and Aiyama has generalized this to compact space-like submanifolds in $ S _ {p} ^ {n + p } ( c ) $ with parallel mean curvature vector and flat normal bundle. Complete space-like hypersurfaces in $ S _ {1} ^ {n + 1 } ( c ) $ with constant mean curvature have also been characterized under conditions on the squared norm of the second fundamental form.

Cf. also Anti-de Sitter space.

References

[a1]	K. Akutagawa, "On space-like hypersurfaces with constant mean curvature in the de Sitter space" Math. Z. , 196 (1987) pp. 13–19
[a2]	Q. M. Cheng, "Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector" Math. Z. , 206 (1991) pp. 333–339
[a3]	Q. M. Cheng, "Hypersurfaces of a Lorentz space form" Arch. Math. , 63 (1994) pp. 271–281
[a4]	S. Montiel, "An integral inequality for compact space-like hypersurfaces in a de Sitter space and application to the case of constant mean curvature" Indiana Univ. Math. J. , 37 (1988) pp. 909–917