Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems

Let $(X^n,g_{+})$ $(n\ge 3)$ be a Poincar'{e}-Einstein manifold which is $C^{3,\alpha }$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$\mathrm{.}Ontheconformalcompactification$(\overline{X}, \bar g=\rho^2g_+)$viasomeboundarydefiningfunction$\rho$, there are two types of Yamabe constants: $Y(\bar{X},\mathrm{\partial }X,[\bar{g}])$ and $Q(\bar{X},\mathrm{\partial }X,[\bar{g}])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\bar{X},\mathrm{\partial }X,[\bar{g}])$ and $Y(\mathrm{\partial }X,[\widehat{g}])$. In this paper, we first show that the equality holds in Gursky-Han’s theorem if and only if $(X^n,g_{+})$ is isometric to the standard hyperbolic space $({\mathbb{H}}^n,g_{\mathbb{H}})$. Secondly, we derive an inequality between $Q(\bar{X},\mathrm{\partial }X,[\bar{g}])$ and $Y(\mathrm{\partial }X,[\widehat{g}])$, and show that the equality holds if and only if $(X^n,g_{+})$ is isometric to $({\mathbb{H}}^n,g_{\mathbb{H}})$. Based on this, we give a simple proof of the rigidity theorem for Poincar'{e}-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.