Sequences of Laplacian Cut-Off Functions

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 $$\mathsf {L}^q$$ -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 $$(-\Delta +1)f\ge 0$$ . The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.