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.

arXiv commentSome typos corrected and a slightly new title