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Characterization of quasispheres via smooth approximation

Dimitrios Ntalampekos

TL;DR

This work addresses how to characterize 2D quasispheres and how to approximate non-smooth quasispheres by smooth, uniformly controlled models. It develops a robust framework of metric-space approximations, metric gluing, and quasiconformal simplicial complexes, combined with smoothing, to transfer quasisymmetric structure between spaces. The main contributions include (i) a finite-area characterization linking LLC and modulus to quasisphere structure and (ii) a general theorem that every 2D quasisphere is the Gromov–Hausdorff limit of uniform quasispheres, with quantitative control on distortion; plus a bi-Lipschitz extension. These results bridge non-smooth metric sphere theory with smooth, uniform geometries, offering tools for applications in dynamics and geometric group theory and providing a pathway to uniformize broad classes of metric spheres with controlled geometry.

Abstract

We prove that every two-dimensional quasisphere is the limit of a sequence of smooth spheres that are uniform quasispheres. In the case of metric spheres of finite area we provide necessary and sufficient geometric conditions for a quasisphere, involving the doubling property, linear local connectivity, the Loewner property, conformal modulus, and reciprocity. In particular, although an arbitrary quasisphere does not satisfy necessarily all of those geometric conditions, we prove that every quasisphere can be approximated by uniform quasispheres that are uniformly doubling, linearly locally connected, 2-Loewner, reciprocal and satisfy a uniform bound for the modulus of annuli.

Characterization of quasispheres via smooth approximation

TL;DR

This work addresses how to characterize 2D quasispheres and how to approximate non-smooth quasispheres by smooth, uniformly controlled models. It develops a robust framework of metric-space approximations, metric gluing, and quasiconformal simplicial complexes, combined with smoothing, to transfer quasisymmetric structure between spaces. The main contributions include (i) a finite-area characterization linking LLC and modulus to quasisphere structure and (ii) a general theorem that every 2D quasisphere is the Gromov–Hausdorff limit of uniform quasispheres, with quantitative control on distortion; plus a bi-Lipschitz extension. These results bridge non-smooth metric sphere theory with smooth, uniform geometries, offering tools for applications in dynamics and geometric group theory and providing a pathway to uniformize broad classes of metric spheres with controlled geometry.

Abstract

We prove that every two-dimensional quasisphere is the limit of a sequence of smooth spheres that are uniform quasispheres. In the case of metric spheres of finite area we provide necessary and sufficient geometric conditions for a quasisphere, involving the doubling property, linear local connectivity, the Loewner property, conformal modulus, and reciprocity. In particular, although an arbitrary quasisphere does not satisfy necessarily all of those geometric conditions, we prove that every quasisphere can be approximated by uniform quasispheres that are uniformly doubling, linearly locally connected, 2-Loewner, reciprocal and satisfy a uniform bound for the modulus of annuli.

Paper Structure

This paper contains 25 sections, 30 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Quasispheres of finite area
  4. Approximation by smooth quasispheres
  5. Bi-Lipschitz surfaces
  6. Proof sketch
  7. Approximations of metric spaces and quasisymmetries
  8. Preliminaries
  9. Gluing metrics
  10. Approximations of metric spaces
  11. Approximations and connectedness assumptions
  12. Images of approximations of metric spaces
  13. Approximations and quasisymmetric homeomorphisms
  14. Simplicial complexes
  15. Metric simplicial complexes
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $X$ be a Riemannian $2$-sphere. The following statements are quantitatively equivalent.

Figures (4)

  • Figure 1: The first stage of the construction of the snowsphere.
  • Figure 2: Sketch of the proof of Theorem \ref{['theorem:main:approximation']}.
  • Figure 3: Shown is the simplex $\tau(S)$ and the shaded simplex represents $\phi(\tau(S))$.
  • Figure 4: The simplicial complex $K$ constructed in Lemma \ref{['lemma:triangle']}.

Theorems & Definitions (53)

  • Theorem 1.1: Smooth quasispheres
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.8: Smooth approximation of quasisymmetric $2$-manifolds
  • Theorem 1.9
  • Corollary 1.10
  • Theorem 1.11
  • Lemma 2.1: Heinonen:metric*Proposition 10.8
  • Lemma 2.2
  • ...and 43 more