Quasisymmetric structures on surfaces

TL;DR

This paper proves that certain metric surfaces with specific regularity and contractibility properties are locally quasisymmetrically equivalent to the disk, aiding in characterizing surfaces in Euclidean space.

Contribution

It establishes a local quasisymmetric equivalence for metric surfaces with Ahlfors regularity and linear local contractibility, advancing surface classification.

Findings

Metric surfaces are locally quasisymmetrically equivalent to the disk.

Application to characterizing surfaces in Euclidean space.

Provides conditions for bi-Lipschitz equivalence to the plane.

Abstract

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces in some Euclidean space that are locally bi-Lipschitz equivalent to an open subset of the plane.