A note on the compactness of Poincare-Einstein manifolds

For a conformally compact Poincare-Einstein manifold (X, g(+)), we consider two types of compactifications for it. One is (g) over bar = rho(2)g(+), where rho is a fixed smooth defining function; the other is the adapted (including Fetterman-Graham) compactification (g) over bar (s) = rho(2)(s)g(+) with a continuous parameter s > n/2. In this paper, we mainly prove that for a set of conformally compact Poincare-Einstein manifolds {(X, g(+)((i)))} with conformal infinity of positive Yamabe type, {(g) over bar ((i))} is compact in C-k,C-alpha ((X) over bar) topology if and only if {(g) over bar ((i))(s)} is compact in some C-l,C-beta ((X) over bar) topology, provided that (g) over bar ((i) )(s)vertical bar(TM) = (g) over cap (()(i)) and (g) over cap ((i)) and (g) over cap ((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).