Vol. 297, No. 1, 2018

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ISSN: 0030-8730
A characterization of round spheres in space forms

Francisco Fontenele and Roberto Alonso Núñez

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

Let cn+1 be the complete simply connected (n + 1)-dimensional space form of curvature c. We obtain a new characterization of geodesic spheres in cn+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 cn+1, c 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
Authors
Francisco Fontenele
Departamento de Geometria
Universidade Federal Fluminense
Niterói, RJ
Brazil
Roberto Alonso Núñez
Arequipa
Peru