On the curvatures of complete spacelike hypersurfaces in de Sitter space (Q1573681)

Let \(M^n\) be a complete spacelike hypersurface in the de Sitter space \(S^{n+1}_1\subset L^{n+2}\), whose sectional curvature are bounded away from \(-\infty\). Let \(M^n\) be contained in the region \(\Omega(a,r)= \{x\in S^{n+1}_1:\langle a,x\rangle\leq -\sinh(r)< 0\}\) for a timelike direction \(a\in L^{n+2}\) and a positive real number \(r>0\). It is proved that then \(\sup H_j\geq \tanh(r)^j\) and \(\text{inf Ric}(v,v)\leq {n-1\over\cosh^2(r)}\), where \(H_j\) is the \(j\)th mean curvature, \(\text{Ric}(X,Y)\) is the Ricci curvature and \(v\) is a tangent to \(M^n\) vector with \(|v|= 1\). In the proofs a generalized maximum principle is used, due to \textit{H. Omori} [J. Math. Soc. Jap. 19, 205-214 (1967; Zbl 0154.21501)].