A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds

Abstract

Let $(X^{n+1},g_+)$ be a Poincaré--Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe constant $Y(M,[h])$. Then a relative comparison inequality \[ \frac{Y_1(X,M,[\bar g])}{Y_1(\mathbb{S}^{n+1}_+,\mathbb{S}^n,[g_{\mathbb{S}_+^{n+1}}])} \geq \left(\frac{Y(M,[h])}{Y(\mathbb{S}^n,[g_{\mathbb{S}^n}])}\right)^{\frac{n}{n+1}} \] holds for the \textbf{type-I} Escobar--Yamabe compactification $Y_1(X,M,[\bar g]$, with equality if and only if $\left(X, g_{+}\right)$ is isometric to the hyperbolic space $\left(\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}}\right)$. This confirms a conjecture raised by Sun-Yung A. Chang.