For a conformally compact Poincar\'{ e}-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{ g}=\rho^2g_+$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{ g}_s=\rho^2_sg_+$ with a continuous parameter $s>\frac{ n}{ 2}$. In this paper, we mainly prove that for a set of conformally compact Poincar\'{ e}-Einstein manifolds $\{ (X, g_{ +}^{ (i)})\}$ with conformal infinity of positive Yamabe type, $\{ \bar{ g}^{ (i)}\}$ is compact in $C^{ k,\alpha}(\overline{ X})$ topology if and only if $\{ \bar{ g}_s^{ (i)}\}$ is compact in some $C^{ l,\beta}(\overline{ X})$ topology, provided that $\bar{ g}^{ (i)}|_{ TM}=\bar{ g}_s^{ (i)}|_{ TM}=\hat{ g}^{ (i)}$ and $\hat{ g}^{ (i)}$ has positive scalar curvature for each $i$. See Theorem 1.1 and Corollary 1.1 for the exact relation of $(k,\alpha)$ and $(l,\beta)$.
Categories: math.DG