Communications in Analysis and Geometry

Volume 26 (2018)

Number 3

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

Pages: 627 – 658

DOI: http://dx.doi.org/10.4310/CAG.2018.v26.n3.a6

Authors

Dandan Ji (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Yuguang Shi (Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China)

Bo Zhu (Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China)

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 $R \geq -6$, we prove that a connected isoperimetric region $\lbrace D_i \rbrace$ with $\mathcal{H} \frac{3}{g} (D_i) \geq \delta_0 \gt 0$ 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 $\lbrace D_i \rbrace$ with topological sphere $\lbrace \partial D_i \rbrace$ as its boundary is exhausting regions of $M$ if the 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 the renormalized volume.

Keywords

isoperimetric regions, Hawking mass, exhuastion

2010 Mathematics Subject Classification

Primary 83C57. Secondary 53C44.

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Dandan Ji was supported by the Beijing Postdoctoral Research Foundation.

The research of Yuguang Shi and Bo Zhu was partially supported by an NSF grant of China (no. 10990013).

Received 26 December 2015