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.