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Complete space-like submanifolds with parallel mean curvature vector of an indefinite space form. (English) Zbl 0804.53089
The authors prove the following theorems:
Theorem 1. Let \(M\) be an \(n\)-dimensional complete space-like submanifold with parallel mean curvature vector of an indefinite space form \(M^{n+p}_ p(c)\). If one of the following conditions is satisfied: (1) \(c\leq 0\), (2) \(c>0\) and \(n^ 2 H^ 2\geq 4(n-1)c\) then \(S\leq S_ +(p)+ k(p)\), where \(k(p)\) is a constant defined by \[ k(p)= (p-1) H\bigl\{nH+ \sqrt{n(n-1)\{S_ +(1)- n^ 2 H^ 2\}}\bigr\}. \] [\(S\) is the squared norm of the second fundamental form.]
Theorem 2. The hyperbolic cylinder \(H^ 1(c_ 1)\times \mathbb{R}^{n-1}\) in \(\mathbb{R}^{n+1}_ 1\) is the only complete connected space-like \(n\)- dimensional submanifold with parallel mean curvature vector of \(\mathbb{R}^{n+p}_ p\) satisfying \(S= S_ +(p)+ k(p)\).
Theorem 3. The hyperbolic cylinder \(H^ 1(c_ 1)\times H^{n-1}(c_ 2)\) of \(H^{n-1}_ 1(c)\) and the maximal submanifolds \(H^{n_ 1}(c_ 1)\times\cdots \times H^{n_{p+1}}(c_{p+1})\) of \(H^{n+p}_ p(c)\) are the only complete connected space-like \(n\)- dimensional submanifolds with parallel mean curvature vector satisfying \(S= S_ +(p)+ k(p)\), where \(c_ r= (n/n_ r)c\) and \(\sum^{p+1}_{r=1} n_ r= n\) in the latter case.
Reviewer: B.Rouxel (Quimper)

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