A note on the compactness of Poincaré–Einstein manifolds

Abstract

For a conformally compact Poincaré–Einstein manifold [Formula: see text], we consider two types of compactifications for it. One is [Formula: see text], where [Formula: see text] is a fixed smooth defining function; the other is the adapted (including Fefferman–Graham) compactification [Formula: see text] with a continuous parameter [Formula: see text]. In this paper, we mainly prove that for a set of conformally compact Poincaré–Einstein manifolds [Formula: see text] with conformal infinity of positive Yamabe type, [Formula: see text] is compact in [Formula: see text] topology if and only if [Formula: see text] is compact in some [Formula: see text] topology, provided that [Formula: see text] and [Formula: see text] has positive scalar curvature for each [Formula: see text]. See Theorem 1.1 and Corollary 1.1 for the exact relation of [Formula: see text] and [Formula: see text].