A note on the Compactness of Poincare-Einstein manifolds

Abstract

For a conformally compact Poincaré-Einstein manifold $(X,g_{+})$, we consider two types of compactifications for it. One is $\bar{g}={\rho }^2{g}_{+}$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification ${\bar{g}}_s={\rho }_s^2{g}_{+}$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré-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 $(l,\beta )$.

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