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2004•Journal of Southwest China Normal UniversityRequires access

Global Pinching Theorems of Space-Like Submanifolds in the de Sitter Space

Shen Xue-wen

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Abstract

The author proves: Let M~n be a compact space-like submanifold with unit parallel mean curvature vector in an de Sitter space S~(n+p)_p(1)(p1) and σ the square of the length of the second fundamental form of M~n, c and M are isoperimetric constant and volume of M~n respectively, then there is a constant A, depending only on n,c and M, such that if ∫σ~~~(n2)dV~~(2n)A, then M~n is a totally umbilic submanifold. Let M~n be a maximal compact space-like submanifold in an de Sitter space S~(n+p)_p(1), then M~n is a totally geodesic.

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The author proves: Let M~n be a compact space-like submanifold with unit parallel mean curvature vector in an de Sitter space S~(n+p)_p(1)(p1) and σ the square of the length of the second fundamental form of M~n, c and M are isoperimetric constant and volume of M~n respectively, then there is a constant A, depending only on n,c and M, such that if ∫σ~~~(n2)dV~~(2n)A, then M~n is a totally umbilic submanifold. Let M~n be a maximal compact space-like submanifold in an de Sitter space S~(n+p)_p(1), then M~n is a totally geodesic.

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Available abstract

The author proves: Let M~n be a compact space-like submanifold with unit parallel mean curvature vector in an de Sitter space S~(n+p)_p(1)(p1) and σ the square of the length of the second fundamental form of M~n, c and M are isoperimetric constant and volume of M~n respectively, then there is a constant A, depending only on n,c and M, such that if ∫σ~~~(n2)dV~~(2n)A, then M~n is a totally umbilic submanifold. Let M~n be a maximal compact space-like submanifold in an de Sitter space S~(n+p)_p(1), then M~n is a totally geodesic.

Key concepts: Submanifold, Isoperimetric inequality, Anti-de Sitter space, De Sitter space, Mathematics, Space (punctuation), Constant (computer programming), Space form

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