A characterization of round spheres in space forms

Francisco Fontenele and Roberto Alonso Núñez

Vol. 297 (2018), No. 1, 67–78

DOI: 10.2140/pjm.2018.297.67

Abstract

Let ${\mathbb{Q}}_c^{n+1}$ be the complete simply connected $\left(n+1\right)$-dimensional space form of curvature $c$. We obtain a new characterization of geodesic spheres in ${\mathbb{Q}}_c^{n+1}$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in ${\mathbb{Q}}_c^{n+1}$, $c\le 0$, 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

Milestones

Received: 18 May 2017

Revised: 11 September 2017

Accepted: 2 March 2018

Published: 7 October 2018