A new characterization of complete linear Weingarten hypersurfaces in real space forms

Cícero P. Aquino, Henrique F. de Lima and Marco A. L. Velásquez

Vol. 261 (2013), No. 1, 33–43

DOI: 10.2140/pjm.2013.261.33

Abstract

We apply the Hopf’s strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in a real space form ℚ_cⁿ⁺¹ of constant sectional curvature c. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a Clifford torus, if c = 1, a circular cylinder, if c = 0, or a hyperbolic cylinder, if c = −1.

Keywords

space forms, linear Weingarten hypersurfaces, totally umbilical hypersurfaces, Clifford torus, circular cylinder, hyperbolic cylinder

Mathematical Subject Classification 2010

Primary: 53C42

Secondary: 53A10, 53C20, 53C50

Milestones

Received: 29 June 2012

Revised: 24 August 2012

Accepted: 4 September 2012

Published: 28 February 2013