Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R\ge -6$

In this paper, aimed at exploring the fundamental properties of isoperimetric region in $3$-manifold $(M^3,g)$ which is asymptotic to Anti-de Sitter-Schwarzschild manifold with scalar curvature $R\ge -6$, we prove that connected isoperimetric region $\{D_i\}$ with $$\mathcal{H}$_g ^3(D_i)\geq \delta_0>0$cannotslideofftotheinfinityof$(M^3,g)$providedthat$(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region $\{D_i\}$ with topological sphere $\{\mathrm{\partial }{D}_i\}$ as boundary is exhausting regions of $M$ if Hawking mass $m_H(\mathrm{\partial }{D}_i)$ has uniform bound. In the case of exhausting isoperimetric region, we obtain a formula on expansion of isoperimetric profile in terms of renormalized volume.