On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces

Abstract:

A theorem of Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalar curvature $R\geq \frac {2}{3}H^2$, then $R=H^2$, $R=\frac {8}{9}H^2$ or $R=\frac {2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $R=H^2$ and $R=\frac {8}{9}H^2$, and provide an example in the case $R=\frac {2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well-known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.