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.