Neste trabalho tratamos das hipersuperfícies fechadas com curvatura média constante e duas curvaturas principais imersas na esfera euclidiana. No caso de hipersuperfícies mínimas, Perdomo (Perdomo 2004) e Wang (Wang 2003) obtiveram uma desigualdade integral envolvendo o quadrado da norma da segunda forma fundamental de M, onde ocorre a igualdade se e somente se M é o toro de Clifford. Neste trabalho, usando o segunda forma fundamental modificada com traço nulo de M, obtemos uma generalização da fórmula integral acima e uma nova caracterização dos H(r)-toros.

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

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).