Characterization of quasispheres via smooth approximation

TL;DR

This paper demonstrates that all two-dimensional quasispheres can be approximated by smooth spheres with uniform geometric properties, and establishes conditions characterizing metric quasispheres with finite area.

Contribution

It proves the approximation of quasispheres by smooth uniform quasispheres and identifies geometric conditions characterizing finite-area quasispheres.

Findings

Every two-dimensional quasisphere is the limit of smooth uniform quasispheres.

Necessary and sufficient conditions for metric quasispheres of finite area involve doubling, LLC, Loewner, and reciprocity.

Any quasisphere can be approximated by uniform quasispheres with controlled geometric properties.

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.