Complete Hypersurfaces with Two Distinct Principal Curvatures in a Space Form

Abstract

In this paper, we study complete hypersurfaces Mn immersed in a space form \({\mathbb{Q}_c^{n+1}}\), with \({c \in \{-1,0,1\}}\) and \({n \geq 2}\), having two distinct principal curvatures with multiplicity p and n − p. In the case that such a hypersurface Mn has constant mean curvature, under a suitable restriction on the traceless part of its second fundamental form, we apply a Simons-type formula jointly with the well known generalized maximum principle of Omori–Yau to show that Mn must be either isometric to \({\mathbb{S}^{n - p}(r) \times \mathbb{H}^p(-\sqrt{1 + r^2})}\), when c = −1, \({\mathbb{S}^{n - p}(r) \times \mathbb{R}^p}\), when c = 0, or \({\mathbb{S}^{n - p}(r) \times \mathbb{S}^p(\sqrt{1 - r^2})}\), when c = 1. Afterwards, we use a Cheng–Yau modified operator in order to obtain a sort of extension of this previous result for the context of linear Weingarten hypersurfaces, that is, hypersurfaces whose mean and scalar curvatures are linearly related.