Ngô Quốc Anh

January 10, 2010

A nice family of cut-off functions due to Lars Andersson

Filed under: Các Bài Tập Nhỏ, Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Ngô Quốc Anh @ 18:43

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 (M, g, \rho) be an asymptotically hyperbolic manifold (that is (M, g) a conformally compact manifold with defining function \rho such that g has asymptotically sectional curvature -1). For R \in \mathbb R large enough, there exits a cut-off function \Psi_R : M \to [0, 1] depending only on \rho, supported in the annulus

\displaystyle\left\{ {\frac{1}{{{e^{8R}}}} < \rho< \frac{1}{{{e^R}}}} \right\},

equal to 1 in

\displaystyle\left\{ {\frac{1}{{{e^{4R}}}} < \rho< \frac{1}{{{e^{2R}}}}} \right\},

and which satisfies for R large

\displaystyle\left| {\frac{{{d^k}{\Psi _R}}}{{d{\rho ^k}}}(\rho )} \right| \leqslant \frac{{{C_k}}}{{R{\rho ^k}}},

for all k \in \mathbb N \backslash \left\{ 0 \right\}, where C_k is independent of R.

Proof. Let \chi : \mathbb R \to [0,1] be a smooth function equal to 1 on \left( { - \infty ,1} \right] and 0 on \left[ {2, + \infty } \right). The existence of such functions will be discussed in an earlier entry. We define

\displaystyle {\chi _R}(x) = \chi \left( {\frac{{\log \rho (x)}}{{ - R}}} \right)

we then have \chi _R:M \to [0,1] is equal to 1 on \rho \geq e^{-R} and 0 on \rho \leq e^{-2R}. Now we define

\displaystyle {\Psi _R} = {\chi _{4R}}\left( {1 - {\chi _R}} \right)

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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