Generalizing a theorem of Huang, Cheng and Wan classified the completehypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature andconstant scalar curvature. In our work, we obtain results of this nature inhigher dimensions. In particular, we prove that if a complete hypersurface of$\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalarcurvature $R\geq\frac{2}{3}H^2$, 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 ofSmyth-Xavier and a well known formula for the Laplacian of the squared norm ofthe second fundamental form of a hypersurface in a space form.
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机译：根据黄定理，郑和万的一个定理，将\ mathbb R ^ 4 $的完整超曲面分类为具有非零的恒定平均曲率和恒定的标量曲率。在我们的工作中，我们在更高维度上获得了这种性质的结果。特别地，我们证明如果$ \ mathbb R ^ 5 $的完整超曲面具有恒定的平均曲率$ H \ neq 0 $和恒定的标量曲率$ R \ geq \ frac {2} {3} H ^ 2 $，则$ R = H ^ 2 $，$ R = \ frac {8} {9} H ^ 2 $或$ R = \ frac {2} {3} H ^ 2 $。此外，我们在$ R = H ^ 2 $和$ R = \ frac {8} {9} H ^ 2 $的情况下表征超曲面，并在$ R = \ frac {2} {3 } H ^ 2 $。证明基于Smyth-Xavier的主曲率定理和空间形式的超曲面的第二基本形式的平方范数的平方模的拉普拉斯算子的众所周知的公式。
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