Characterizing linear Weingarten submanifolds in a Riemannian space form via $L$-parabolicity

We obtain a parabolicity criterion related to a modified Cheng—Yau's operator, and we apply it to prove that a complete linear Weingarten submanifold, immersed with parallel normalized mean curvature vector in a Riemannian space form $\mathbb Q_c^{n+p}$ of constant sectional curvature $c\in\{-1,0,1\}$ must be either totally umbilical or isometric to a hyperbolic cylinder, when $c=-1$, a circular cylinder, when $c=0$, and a Clifford torus, when $c=1$.