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

Abstract

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 ${\mathcal{H}}_g^3(D_i)\ge {\delta }_0>0$ cannot slide off to the infinity of $(M^3,g)$ provided that $(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region $\{D_i\}$ with topological sphere $\{\partial D_i\}$ as boundary is exhausting regions of $M$ if Hawking mass $m_H(\partial D_i)$ has uniform bound. In the case of exhausting isoperimetric region, we obtain a formula on expansion of isoperimetric profile in terms of renormalized volume.

To appear in CAG

References in corpus (1)

• Large isoperimetric regions in asymptotically hyperbolic manifolds

Cited by in corpus (2)

• Rigidity of Hawking mass for surfaces in three manifolds
• Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass