# A characterization of round spheres in space forms

## TL;DR

This paper provides a new characterization of geodesic spheres in space forms using higher order mean curvatures, establishing uniqueness results for certain hypersurfaces with constant curvatures.

## Contribution

It introduces a novel characterization of geodesic spheres in space forms based on higher order mean curvatures, extending previous understandings.

## Key findings

- Geodesic spheres are uniquely characterized by higher order mean curvatures.
- Complete bounded hypersurfaces with constant mean and scalar curvatures are geodesic spheres in non-positive curvature space forms.
- The proof employs the Omori-Yau maximum principle, a Laplacian formula, and Gårding's inequality.

## Abstract

Let $\mathbb Q^{n+1}_c$ be the complete simply-connected $(n+1)$-dimensional space form of curvature $c$. In this paper we obtain a new characterization of geodesic spheres in $\mathbb Q^{n+1}_c$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in $\mathbb Q^{n+1}_c,\;c\leq 0,$ 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 G\aa rding for hyperbolic polynomials.