  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 $(\overline{X}, \bar
g=\rho^2g_+)$ via some boundary defining function $\rho$, 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 Poincar\'{e}-Einstein manifolds with conformal
infinity being conformally equivalent to the standard sphere.

Differential Geometry