Lower Bounds for the Relative Volume of Poincare-Einstein Manifolds

In this paper, we show that for a Poincar\'{e}-Einstein manifold $(X^{n+1},g_+)$ with conformal infinity $(M,[\hat{g}])$ of nonnegative Yamabe type, the fractional Yamabe constants of the boundary provide lower bounds for the relative volume. More explicitly, for any $\gamma\in (0,1)$, $$\left(\frac{Y_{2\gamma} (M,[\hat{g}])}{Y_{2\gamma} (\mathbb{S}^n , [g_{\mathbb{S}}])}\right)^{\frac{n}{2\gamma}} \leq \frac{V(\Gamma_t(p),g_+)}{V(\Gamma_t(0),g_{\mathbb{H}})} \leq \frac{V( B_t(p),g_+)}{V( B_t(0), g_{\mathbb{H}})} \leq 1,\quad 0<t<\infty,$$ where $B_t(p)$, $\Gamma_t(p)$ are the the geodesic ball and geodesic sphere of radius $t$ in $(X,g_+)$ with center at $p\in X^{n+1}$; and $B_t(0)$, $\Gamma_t(0)$ are the the geodesic ball and geodesic sphere in $\mathbb{H}^{n+1}$ with center at $0\in\mathbb{H}^{n+1}$.