Existence of cut-off function in Riemannian manifold

Question

Please give an reference of the construction of the cut-off function $\varphi_r\in C^2_0(B(p,2r))\subset M$ for $r>0$ such that \begin{cases} 0\leq \varphi_r\leq 1 &\text{ in }B(p,2r)\\ \varphi_r=1 & \text{ in }B(p,r) \\ |\nabla \varphi_r|^2\leq\frac{C}{r^2}& \text{ in }B(p,2r) \\ \Delta \varphi_r\leq \frac{C}{r^2} & \text{ in }B(p,2r), \end{cases} where $M$ is complete Riemannian manifold.

Answer 1

There could be easier constructions, but at least you can take a look at the paper

Cheeger and Colding, "Lower bounds on Ricci curvature and the almost rigidity of warped products", Ann. of Math. 144 (1996) 189-237.

See Theorem 6.33; it can be read independently of other parts of the paper. Rescale the metric to get what you need.