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

Dandan Ji, Yuguang Shi, Bo Zhu

Published 2015-12-09Version 1

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 $R\geq -6$, we prove that connected isoperimetric region $\{D_i\}$ with $\mathcal{L}_g ^3(D_i)\geq \delta_0>0$ cannot slide off to infinity provided $(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region $\{D_i\}$ with topological sphere $\{\partial D_i\}$ is exhausting regions of $M$ if Hawking mass $m_H(\partial D_i)$ has uniform bound. Under the case of exhausting isoperimetric region , we obtain a formula on expansion of isoperimetric profile in terms of renormalized volume.