The metric geometry of paper surfaces under geometric constraints

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.

💡 Research Summary

The paper investigates the metric geometry of a special class of metric surfaces called “paper surfaces.” These objects arise by taking a finite collection of planar polygons and identifying selected boundary segments according to a prescribed pairing scheme. The resulting quotient space inherits a natural length metric from the Euclidean metric on the polygons, but the identification process creates a “scar” – the image of the identified edges – which may contain cone points (where the total angle differs from 2π) and accumulation points of singularities. The main goal is to determine when such spaces can be uniformly parametrized by the standard 2‑sphere via a quasisymmetric homeomorphism.

The authors focus on the class L of paper surfaces, which includes two types of identifications: (i) basic finite pairings of edges and (ii) infinite “type W” pairings, where infinitely many edges are glued together in a geometrically decaying fashion. The presence of type W identifications makes the analysis more delicate because the usual curvature bounds and regularity conditions may fail.

The central result (Theorem 3.1) states that every surface in class L satisfies two crucial geometric properties: (a) Ahlfors 2‑regularity, meaning that the 2‑dimensional Hausdorff measure of any metric ball of radius r is comparable to r² with uniform constants, and (b) linear local contractibility (LLC), i.e., each small ball can be contracted inside a slightly larger ball with a linear bound on the radii. By invoking the Bonk–Kleiner theorem (2002), which asserts that any Ahlfors 2‑regular, LLC metric sphere is quasisymmetrically equivalent to the standard sphere, the authors conclude that every L‑paper surface is quasisymmetrically equivalent to S².

To establish Ahlfors regularity, the paper carefully estimates the contribution of each identified edge to the total area. For finite pairings the estimate is straightforward: the identified edges form a set of measure zero, and the area of the quotient equals the sum of the polygonal areas. For type W identifications the authors exploit the geometric decay of the edge lengths: the series of areas contributed by the glued strips converges, guaranteeing that the total area remains comparable to the square of the diameter. This argument also shows that the scar set has Hausdorff dimension 1 and does not dominate the metric measure.

The LLC property is proved by analyzing the local topology of the scar. Near a regular point of the scar the quotient looks locally like a Euclidean half‑plane, so standard contraction arguments apply. Near a cone point the authors use a fractional power change of coordinates (z↦z^{α}) to flatten the cone, reducing the problem to the Euclidean case. For accumulation points of cone singularities, the decay condition of the type W identifications ensures that in any sufficiently small ball only finitely many singularities affect the geometry, allowing a uniform contraction within a ball of comparable size.

The paper also discusses several motivating examples from dynamics. Translation surfaces and infinite‑type translation surfaces, which appear in billiard dynamics and pseudo‑Anosov theory, can be modeled as paper surfaces. The authors cite De Carvalho and Hall’s work on folding schemes that produce closed Riemann surfaces, and they explain how their metric results complement the complex‑analytic uniformization already known for these objects.

Finally, the authors note limitations. Certain dynamically defined paper surfaces, such as the “tight horseshoe,” fail Ahlfors regularity and therefore lie outside the scope of the Bonk–Kleiner theorem, even though a conformal uniformization may still exist. This highlights a gap between metric and analytic uniformization theories and suggests directions for future research, such as identifying weaker metric conditions that still guarantee quasisymmetric parametrizations.

In summary, the paper provides a thorough metric‑geometric treatment of paper surfaces, proves that a broad class (including those with infinite type W identifications) satisfies Ahlfors 2‑regularity and linear local contractibility, and consequently establishes their quasisymmetric equivalence to the standard 2‑sphere. This bridges constructions from dynamical systems with modern metric uniformization theory.