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Smooth Approximations of Quasispheres

Spencer Cattalani

TL;DR

The paper proves that any metric space (X,d) quasisymmetric to a compact connected Riemannian manifold (X,d_g) is the Gromov-Hausdorff limit of spaces locally isometric to Riemannian manifolds and uniformly quasisymmetric to (X,d_g), with a parallel bi-Lipschitz version. The central technique introduces a dimension-free conformal rescaling using a scale function λ_ε, coupled with a gluing construction and a smoothing step, to produce a sequence of smooth approximants that converge to d. A key local-to-global proposition, together with ε-net discretizations, replaces earlier dimension-dependent steps, yielding uniform control over quasisymmetries across dimensions. The results extend Ntalampekos’ framework to arbitrary dimension and offer a simpler, potentially widely applicable approach to smooth approximations of quasispheres and related metric spaces.

Abstract

We prove that every quasisphere is the Gromov-Hausdorff limit of a sequence of locally smooth uniform quasispheres. We also prove an analogous result in the bi-Lipschitz setting. This extends recent results of D. Ntalampekos from dimension 2 to arbitrary dimension. In the process, we replace the second half of his argument by a completely different, more efficient approach, which should be applicable to other problems.

Smooth Approximations of Quasispheres

TL;DR

The paper proves that any metric space (X,d) quasisymmetric to a compact connected Riemannian manifold (X,d_g) is the Gromov-Hausdorff limit of spaces locally isometric to Riemannian manifolds and uniformly quasisymmetric to (X,d_g), with a parallel bi-Lipschitz version. The central technique introduces a dimension-free conformal rescaling using a scale function λ_ε, coupled with a gluing construction and a smoothing step, to produce a sequence of smooth approximants that converge to d. A key local-to-global proposition, together with ε-net discretizations, replaces earlier dimension-dependent steps, yielding uniform control over quasisymmetries across dimensions. The results extend Ntalampekos’ framework to arbitrary dimension and offer a simpler, potentially widely applicable approach to smooth approximations of quasispheres and related metric spaces.

Abstract

We prove that every quasisphere is the Gromov-Hausdorff limit of a sequence of locally smooth uniform quasispheres. We also prove an analogous result in the bi-Lipschitz setting. This extends recent results of D. Ntalampekos from dimension 2 to arbitrary dimension. In the process, we replace the second half of his argument by a completely different, more efficient approach, which should be applicable to other problems.

Paper Structure

This paper contains 4 sections, 16 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Basic metric and Riemannian geometry
  3. Nets and approximations
  4. Smooth approximations

Key Result

Theorem 1

Let $(X,d)$ be a metric space which is quasisymmetric to a compact connected Riemannian manifold $(X,d_g)$. Then, $(X,d)$ is the Gromov-Hausdorff limit of a sequence of metric spaces which are locally isometric to Riemannian manifolds and uniformly quasisymmetric to $(X,d_g)$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Lemma 3: H01
  • Lemma 4: H01
  • Lemma 5: B03
  • Corollary 6
  • proof
  • Lemma 7: B03
  • Lemma 8: N25
  • Lemma 9: H01
  • ...and 15 more