Curvature integral estimates for complete hypersurfaces

Abstract

We consider the integrals of $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in space forms ${\mathcal{Q}}_c^{n+1}$ which generalize volume $(r=0)$, total mean curvature $(r=1)$, total scalar curvature $(r=2)$ and total curvature $(r=n)$. Among other results we prove that a complete properly immersed hypersurface of a space form with $S_r\ge 0$, $S_r\not\equiv 0$ and $S_{r+1}\equiv 0$ for some $r\le n-1$ has ${\int }_M{S}_{r}dM=\infty .$