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 $0\in\mathbb{ H}^{ n+1}$.
Categories: math.DG