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The Journal of Geometric Analysis

, Volume 26, Issue 1, pp 171–184 | Cite as

Sequences of Laplacian Cut-Off Functions

  • Batu GüneysuEmail author
Article

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.

Keywords

Riemannian manifolds Sobolev spaces Brownian motion 

Mathematics Subject Classification

53C20 53C21 46E35 58J65 58J50 

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Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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