Escobar–Yamabe compactifications for Poincaré–Einstein manifolds and rigidity theorems

Abstract

Let (Xn,g+)(n≥3) be a Poincaré–Einstein manifold which is C3,α conformally compact with conformal infinity (∂X,[gˆ]). On the conformal compactification (X‾,g¯=ρ2g+) via a boundary defining function ρ, there are two types of Yamabe constants: Y(X‾,∂X,[g¯]) and Q(X‾,∂X,[g¯]). (See definitions (1) and (2).) In [13], Gursky and Han obtained an inequality between Y(X‾,∂X,[g¯]) and Y(∂X,[gˆ]). In this paper, we first show that the equality holds in Gursky–Han's inequality if and only if (Xn,g+) is isometric to the standard hyperbolic space (Hn,gH). Secondly, we derive a conformal invariant inequality between Q(X‾,∂X,[g¯]) and Y(∂X,[gˆ]), and show the equality holds if and only if (Xn,g+) is isometric to (Hn,gH). Based on this, we give a simple proof of the rigidity theorem for Poincaré–Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.