The metric geometry of paper surfaces under geometric constraints

TL;DR

This paper studies the geometric structure of paper surfaces, a class of metric surfaces with singularities, and proves they can be parametrized quasisymmetrically onto the sphere under certain conditions.

Contribution

It establishes conditions under which paper surfaces with complex geometry are quasisymmetrically equivalent to the standard 2-sphere.

Findings

Paper surfaces satisfy Ahlfors 2-regularity.

Paper surfaces are linearly locally contractible.

Existence of quasisymmetric parametrization onto the sphere.

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.