[Paper Review] Weak Capacity and Critical Exponents

Jeff Lindquist|arXiv (Cornell University)|Oct 25, 2017

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper introduces weak capacity-based critical exponents in Ahlfors regular metric measure spaces to characterize quasisymmetric equivalence to the 2-sphere 𝕊². By leveraging hyperbolic fillings and weak p-capacity, it proves that local quasisymmetric equivalence to 𝕊² implies global equivalence, and under group actions, finite weak capacity for finitely many separated sets suffices for global quasisymmetric equivalence to 𝕊², with critical exponents linked to the Ahlfors regular conformal dimension.

ABSTRACT

We investigate critical exponents relating to weak capacity in Ahlfors regular metric measure spaces. This allows a proof of a weak capacity version of a result by Bonk and Kleiner about the uniformization of metric $2$-spheres. Using our result, we promote local quasisymmetric equivalence with $\mathbb{S}^2$ to global quasisymmetric equivalence. We also use our weak capacity version to derive conditions for quasisymmetric equivalence to $\mathbb{S}^2$ in the presence of a group action. We investigate the relation between our defined critical exponents and Ahlfors regular conformal dimension, particularly in the cases where the Combinatorial Loewner Property is present or the space attains its Ahlfors regular conformal dimension.

Motivation & Objective

• To establish a weak capacity framework for characterizing quasisymmetric equivalence to the 2-sphere in Ahlfors regular metric measure spaces.
• To generalize Bonk and Kleiner’s result on uniformization of metric 2-spheres using weak capacity instead of modulus.
• To show that under a cocompact group action, finite weak capacity for finitely many pairs of separated sets implies global quasisymmetric equivalence to 𝕊².
• To investigate the relationship between critical exponents $ Q_w $, $ Q_w' $, and the Ahlfors regular conformal dimension (ARCdim), particularly under the Combinatorial Loewner Property.
• To prove that $ Q_w' = \operatorname{ARCdim} $ when the Combinatorial Loewner Property holds or when ARCdim is attained.

Proposed method

• Define weak p-capacity $ \operatorname{wcap_p}(A,B) $ using hyperbolic fillings of the metric space, replacing paths in the space with paths in the filling.
• Introduce critical exponents $ Q_w(Z) $ and $ Q_w'(Z) $ as infima over $ p > 1 $ such that weak capacity is finite or controlled by relative distance for all positively separated open sets.
• Use hyperbolic filling geometry and combinatorial modulus estimates to bound weak capacity from below, relying on vertex functions and level-by-level analysis across scales.
• Apply Lemma 4.8 with $ \delta = 1/5 $ and $ \ell_0 $ from Lemma 4.5 to control path lengths and ensure admissible paths cannot connect sets with small capacity.
• Construct sequences of sets $ A_k, B_k $ with controlled relative distance and show $ \operatorname{wcap_p}(A_k, B_k) \to \infty $ as $ k \to \infty $ for $ p < Q $, implying $ Q_w \geq Q $.
• Use the fact that $ \|\tau\|_{p,\infty}^p \leq a $ implies $ \|\tau\|_Q^Q < C(a) $, and bound the number of levels with high mass or large edge values to find a level where no admissible path exists.

Experimental results

Research questions

• RQ1 Under what conditions on weak capacity does a compact, connected, Ahlfors Q-regular metric space homeomorphic to 𝕊² become quasisymmetrically equivalent to 𝕊²?
• RQ2 How do the critical exponents $ Q_w $ and $ Q_w' $ relate to the Ahlfors regular conformal dimension (ARCdim) in spaces satisfying the Combinatorial Loewner Property?
• RQ3 Can the local quasisymmetric equivalence of a space to subsets of 𝕊² be promoted to global quasisymmetric equivalence using weak capacity control?
• RQ4 Does a cocompact group action on a hyperbolic filling reduce the number of required capacity checks for global equivalence to 𝕊²?
• RQ5 Is $ Q_w' = \operatorname{ARCdim} $ when the Combinatorial Loewner Property holds or when ARCdim is attained?

Key findings

• The paper proves that if $ \varphi_2(t_0) < \infty $ for some $ t_0 > 0 $, and the space is linearly locally connected and Ahlfors regular, then it is quasisymmetrically equivalent to $ \mathbb{S}^2 $.
• Under a cocompact group action, it suffices to verify finite weak 2-capacity for finitely many pairs of separated sets to conclude global quasisymmetric equivalence to $ \mathbb{S}^2 $.
• The critical exponent $ Q_w'(Z) $ equals the Ahlfors regular conformal dimension $ \operatorname{ARCdim}(Z) $ when the Combinatorial Loewner Property holds or when ARCdim is attained.
• For $ p < Q $, the weak capacity $ \operatorname{wcap_p}(A_k, B_k) \to \infty $ as $ k \to \infty $, implying $ Q_w \geq Q $, and thus $ Q_w' \geq \operatorname{ARCdim} $.
• The critical exponent $ Q_w $ is equal to $ Q_w' $ under the presence of a cocompact group action, and the infimum is attained in both definitions if it is attained in $ Q_w $.
• The result provides a new proof of a special case of a result in [Su] on locally quasisymmetrically modeled metric 2-spheres, showing global equivalence to $ \mathbb{S}^2 $.