arXiv2017
Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of 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 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 $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X, [\hat g])$, 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 Poincaré-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.