Let Mn be a complete spacelike hypersurface with constant normalized scalar curvature R in the de Sitter Space S1n + 1. Let H the mean curvature and suppose that <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> = (R - 1) > 0 and <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> <= sup H² <= C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$">, where C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$"> is a constant depending only on R and n. It is proved that either sup H² = <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> and Mn is totally umbilical, or sup H² = C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$"> and Mn is the hyperbolic cylinder H¹(1 - coth²r) x Sn - 1 (1 - tanh²r).
hyperbolic cylinder; spacelike hypersurfaces; de Sitter space
