In this entry, we recall a nice family of cut-off functions which can be found in a paper published in the *Indiana Univ. Math. J.* due to Lars Andersson [see Definition 2.1, p. 1362].

Let be an asymptotically hyperbolic manifold (that is a conformally compact manifold with defining function such that has asymptotically sectional curvature ). For large enough, there exits a cut-off function depending only on , supported in the annulus

,

equal to in

,

and which satisfies for large

,

for all , where is independent of .

*Proof*. Let be a smooth function equal to on and on . The existence of such functions will be discussed in an earlier entry. We define

we then have is equal to on and on . Now we define

which satisfies the desired properties. The use of such cut-off functions can be found in either the original paper by Anderson or a paper due to Erwann Delay published in *J. Geom. Phys.* in 2002.

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