#### Vol. 297, No. 1, 2018

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A characterization of round spheres in space forms

### Francisco Fontenele and Roberto Alonso Núñez

Vol. 297 (2018), No. 1, 67–78
##### Abstract

Let ${ℚ}_{c}^{n+1}$ be the complete simply connected $\left(n+1\right)$-dimensional space form of curvature $c$. We obtain a new characterization of geodesic spheres in ${ℚ}_{c}^{n+1}$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in ${ℚ}_{c}^{n+1}$, $c\le 0$, with constant mean curvature and constant scalar curvature. The proof relies on the well known Omori–Yau maximum principle, a formula of Walter for the Laplacian of the $r$-th mean curvature of a hypersurface in a space form, and a classical inequality of Gårding for hyperbolic polynomials.

##### Keywords
hypersurfaces in space forms, scalar curvature, Laplacian of the $r$-th mean curvature, hyperbolic polynomials
##### Mathematical Subject Classification 2010
Primary: 14J70, 53C42
Secondary: 53A10, 53C40