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

Understand

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 $Y(\partial X,[\hat{g}])$.