Lower Bounds for the Relative Volume of Poincare-Einstein Manifolds

Abstract

In this paper, we show that for a Poincaré-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 $\gamma \in (0,1)$, $${\left(\frac{Y_{2\gamma }(M,[\hat{g}])}{Y_{2\gamma }({\mathbb{S}}^n,[g_{\mathbb{S}}])}\right)}^{\frac{n}{2\gamma }}\le \frac{V({\Gamma }_t(p),g_{+})}{V({\Gamma }_t(0),g_{\mathbb{H}})}\le \frac{V(B_t(p),g_{+})}{V(B_t(0),g_{\mathbb{H}})}\le 1,0<t<\infty ,$$ 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 $0\in {\mathbb{H}}^{n+1}$.

16 pages