Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On complete hypersurfaces with constant mean and scalar curvatures in
  Euclidean spaces   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.

37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int