A note on the Compactness of Poincare-Einstein manifolds

Abstract

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 $(l,\beta)$.