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

Dandan Ji, Yuguang Shi+1 more|arXiv (Cornell University)|Dec 9, 2015

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes that isoperimetric regions in 3-dimensional asymptotically hyperbolic manifolds with scalar curvature $ R \geq -6 $ and positive mass parameter $ m > 0 $ either exhaust the manifold or converge to a limit surface of mean curvature $ H = 2 $, which must be conformally equivalent to $ \mathbb{C} $. If the Hawking mass is uniformly bounded, the regions must exhaust the manifold, and the isoperimetric profile expansion is shown to relate to the renormalized volume via $ \lim_{i\to\infty}(A_g(v_i) - A_{\mathbb{H}}(v_i)) = -2V(M,g) $.

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 connected isoperimetric region $\{D_i\}$ with $\mathcal{H}_g ^3(D_i)\geq δ_0>0$ 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.

Motivation & Objective

• To understand the asymptotic behavior of isoperimetric regions in 3-dimensional asymptotically hyperbolic manifolds with scalar curvature $ R \geq -6 $.
• To determine whether large isoperimetric regions can 'slide off' to infinity or must exhaust the manifold.
• To establish conditions under which isoperimetric regions form an exhaustion of the manifold.
• To derive an expansion formula for the isoperimetric profile in terms of the renormalized volume for exhausting regions.

Proposed method

• Analyzes the limit behavior of sequences of isoperimetric regions $ \{D_i\} $ with volume $ \mathcal{H}^3_g(D_i) \to \infty $ using geometric and analytic techniques.
• Applies the Hawking mass bound $ m_H(\partial D_i) \leq C $ as a key condition to rule out non-exhausting behavior.
• Employs curvature estimates involving the trace-free second fundamental form $ \AA $, showing $ \int_{\Sigma_i} |\AA|^2 \to 0 $ under the Hawking mass bound.
• Uses the Gauss-Bonnet theorem on the limit surface $ \mathbb{S} $, deriving $ \int_{\mathbb{S}} K = 0 $, which leads to a contradiction unless $ (M,g) \cong \mathbb{H}^3 $.
• Compares the isoperimetric profile $ A_g(v) $ to that of hyperbolic space $ A_{\mathbb{H}}(v) $, using volume and area asymptotics in the renormalized volume framework.
• Derives the key identity $ \lim_{i\to\infty}(A_g(v_i) - A_{\mathbb{H}}(v_i)) = -2V(M,g) $ via asymptotic expansions of $ \sinh^2 \rho $ and volume differences.

Experimental results

Research questions

• RQ1 Under what conditions do isoperimetric regions in an asymptotically hyperbolic 3-manifold with $ R \geq -6 $ fail to exhaust the manifold?
• RQ2 Can a sequence of isoperimetric regions with bounded Hawking mass avoid exhausting the manifold?
• RQ3 What is the asymptotic expansion of the isoperimetric profile in terms of the renormalized volume for exhausting isoperimetric regions?
• RQ4 Does the existence of a limit surface with $ H = 2 $ and $ \int K \leq 0 $ force the manifold to be isometric to $ \mathbb{H}^3 $?
• RQ5 How does the renormalized volume $ V(M,g) $ relate to the difference between the actual and hyperbolic isoperimetric profiles?

Key findings

• If the Hawking mass of the boundary $ \partial D_i $ is uniformly bounded, then the isoperimetric regions $ \{D_i\} $ must exhaust the manifold $ (M,g) $.
• Any non-exhausting sequence of isoperimetric regions must converge to a limit surface $ \mathbb{S} $ that is a complete, stable, constant mean curvature surface with $ H = 2 $, conformally equivalent to $ \mathbb{C} $.
• If such a limit surface exists and satisfies $ \int_{\mathbb{S}} K \leq 0 $, then $ (M,g) $ must be isometric to $ \mathbb{H}^3 $, contradicting $ m > 0 $, so non-exhausting behavior is ruled out under this condition.
• For exhausting isoperimetric regions, the isoperimetric profile satisfies $ \lim_{i\to\infty}(A_g(v_i) - A_{\mathbb{H}}(v_i)) = -2V(M,g) $, linking the profile to the renormalized volume.
• The trace-free second fundamental form satisfies $ \int_{\Sigma_i} |\AA|^2 \to 0 $ as $ i \to \infty $ under the Hawking mass bound, indicating the surfaces become increasingly round.
• The volume difference $ \mathcal{H}^3_g(D_i) - \mathcal{H}^3_{\mathbb{H}}(D_i) \to V(M,g) $, and the area difference is asymptotically controlled by this volume term.