Sequences of Laplacian cut-off functions

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 $\mathsf{L}^q$-estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole $\mathsf{L}^q$-scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic and Shubin on the nonnegativity of $\mathsf{L}^2$-solutions $f$ of $(-\Delta+1)f\geq 0$. The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.