Abstract. We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature): In particular, we prove that this existence implies L q -estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole L q -scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic and Shubin on the nonnegativity of L 2 -solutions f of (−∆ + 1)f ≥ 0. The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.