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Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincar\'{e}-Einstein manifold which is$C^{3,\alpha}$ conformally compact with conformal infinity $(\partial X,[\hat{g}])$. On the conformal compactification $(\overline{X}, \barg=\rho^2g_+)$ via some boundary defining function $\rho$, there are two typesof Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and$Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and(\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between$Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In thispaper, we first show that the equality holds in Gursky-Han's theorem if andonly if $(X^{n},g_+)$ is isometric to the standard hyperbolic space$(\mathbb{H}^{n}, g_{\mathbb{H}})$. Secondly, we derive an inequality between$Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X, [\hat g])$, and showthat 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 ofthe rigidity theorem for Poincar\'{e}-Einstein manifolds with conformalinfinity being conformally equivalent to the standard sphere.

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