The structure of hypersurfaces with some curvature conditions

Abstract:

Let $M$ be a hypersurface in $\mathbf {R}^{n+1}$, and let $H, R$ denote the mean curvature and the scalar curvature of $M$ respectively. We show that if $M$ is compact and $R>\frac {n-2}{n-1}H^{2}$, then $M$ is diffeomorphic to $S^{n}$. Also we prove that if $M$ is complete, $H$ is constant and $R\geq \frac {n-2}{n-1}H^{2}$, then $M$ is $\mathbf {R}^{n}$ or $S^{n}$ or $S^{n-1}\times \mathbf {R}^{1}$.