Characterizations of linear Weingarten submanifolds

In this paper, we deal with complete linear Weingarten submanifolds Mn immersed with parallel normalized mean curvature vector field in a Riemannian space form Qcn+p of constant sectional curvature c and with n≥4. We recall that a submanifold is called linear Weingarten when its mean and scalar curvatures are linearly related. In this setting, we establish a suitable extension of the generalized maximum principle of Omori-Yau in order to show that such a submanifold Mn must be either totally umbilical or isometric to a Clifford torus S1(1−r2)×Sn−1(r), when c=1, a circular cylinder R×Sn−1(r), when c=0, or a hyperbolic cylinder H1(−1+r2)×Sn−1(r), when c=−1. We also study the parabolicity of these submanifolds with respect to a Cheng-Yau modified operator.