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The metric geometry of paper surfaces under geometric constraints

Luciana Menezes Vasconcelos

TL;DR

This work develops a metric-geometric framework for paper surfaces obtained from planar multipolygons through segment pairings and Type W identifications, linking quotient metric geometry with quasisymmetric uniformization. By analyzing intrinsic metrics, quotient constructions, and the preimage structure of metric balls, it proves that the class of paper surfaces $\mathcal{L}$ is Ahlfors 2-regular and linearly locally contractible, hence quasisymmetrically equivalent to the standard sphere $S^2$ for the subclass $\mathcal{L}^*$. The results accommodate conic singularities and accumulation points, providing a unified route to uniformize many dynamically motivated spaces, including those arising from generalized pseudo-Anosov maps. The tight horseshoe case is discussed as a boundary example where Ahlfors regularity fails, showing the necessity of the LLC/regularity hypotheses in the main theorem. Overall, the paper connects metric uniformization theory with dynamical systems constructions to yield concrete quasisymmetric parametrizations of a broad class of singular surfaces.

Abstract

We investigate the quasisymmetric uniformization of a special class of metric surfaces known as paper surfaces, constructed as quotients of planar multipolygons via segment pairings, including infinite Type W identifications. These spaces, which arise naturally in dynamical settings, exhibit conic singularities and complex geometric structure. Our goal is to prove that a broad class of such surfaces satisfies Ahlfors 2-regularity and linear local contractibility, which together ensure the existence of a quasisymmetric parametrization onto the standard 2-sphere.

The metric geometry of paper surfaces under geometric constraints

TL;DR

This work develops a metric-geometric framework for paper surfaces obtained from planar multipolygons through segment pairings and Type W identifications, linking quotient metric geometry with quasisymmetric uniformization. By analyzing intrinsic metrics, quotient constructions, and the preimage structure of metric balls, it proves that the class of paper surfaces is Ahlfors 2-regular and linearly locally contractible, hence quasisymmetrically equivalent to the standard sphere for the subclass . The results accommodate conic singularities and accumulation points, providing a unified route to uniformize many dynamically motivated spaces, including those arising from generalized pseudo-Anosov maps. The tight horseshoe case is discussed as a boundary example where Ahlfors regularity fails, showing the necessity of the LLC/regularity hypotheses in the main theorem. Overall, the paper connects metric uniformization theory with dynamical systems constructions to yield concrete quasisymmetric parametrizations of a broad class of singular surfaces.

Abstract

We investigate the quasisymmetric uniformization of a special class of metric surfaces known as paper surfaces, constructed as quotients of planar multipolygons via segment pairings, including infinite Type W identifications. These spaces, which arise naturally in dynamical settings, exhibit conic singularities and complex geometric structure. Our goal is to prove that a broad class of such surfaces satisfies Ahlfors 2-regularity and linear local contractibility, which together ensure the existence of a quasisymmetric parametrization onto the standard 2-sphere.
Paper Structure (22 sections, 13 theorems, 80 equations, 11 figures)

This paper contains 22 sections, 13 theorems, 80 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Metric spaces: quotient spaces and intrinsic metric
  4. Paper Surfaces
  5. Metric structure of paper spaces
  6. Quasisymmetric uniformization
  7. Quasisymmetric maps
  8. Hausdorff measure
  9. Definitions of regularity
  10. Quasisymmetric Uniformization of paper spaces
  11. The construction
  12. Infinite Type $W$ Identifications.
  13. Finite Type $W$ Identifications.
  14. Preimages of the balls under the projection map
  15. Accumulation case without intersection with other sides.
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $\mathcal{G}$ be an upper semi-continuous and monotone decomposition of a topological sphere $S$ such that no element of $\mathcal{G}$ separates $S$. Then, the quotient space obtained by collapsing each decomposition element to a point is a topological sphere.

Figures (11)

  • Figure 1: Segment pairing.
  • Figure 2: Paper Space of Example \ref{['ex13']}.
  • Figure 3: Quasisymmetric map.
  • Figure 5: Paper space obtained as the quotient of four rectangles.
  • Figure 6: An example of a paper space, belonging to the class of surfaces denoted by $\mathcal{L}^*$.
  • ...and 6 more figures

Theorems & Definitions (28)

  • Theorem 2.1: moore1925upper
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 18 more