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We construct exhaustion and cut-off functions with controlled gradient andLaplacian on manifolds with Ricci curvature bounded from below by a (possiblyunbounded) nonpositive function of the distance from a fixed reference point,without any assumptions on the topology or the injectivity radius. Along theway we prove a generalization of the Li-Yau gradient estimate which is ofindependent interest. We then apply our cut-offs to the study of the fast andporous media diffusion, of $L^q$-properties of the gradient and of theself-adjointness of Schroedinger-type operators.

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications