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

TL;DR

This paper extends classification results for complete hypersurfaces with constant mean and scalar curvatures from four to five dimensions, identifying possible scalar curvatures and characterizing the hypersurfaces in specific cases.

Contribution

It generalizes previous classifications to higher dimensions and provides new characterizations and examples for hypersurfaces in 9-dimensional Euclidean space.

Findings

Possible scalar curvatures are R=H^2, 8/9 H^2, or 2/3 H^2.

Characterizations are provided for R=H^2 and R=8/9 H^2.

An example is constructed for R=2/3 H^2.

Abstract

Generalizing a theorem of Huang, 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.