Polyhedral approximation of metric surfaces and applications to uniformization

TL;DR

This paper demonstrates that length metric surfaces can be approximated by polyhedral surfaces and uses this to establish a quasiconformal uniformization theorem, providing new insights into the geometry of 2-manifolds and related surfaces.

Contribution

It introduces a polyhedral approximation framework for length surfaces and applies classical uniformization to prove a new quasiconformal uniformization result.

Findings

Length surfaces are Gromov-Hausdorff limits of polyhedral surfaces.

Established a quasiconformal uniformization theorem for length surfaces.

Provided a new proof of the Bonk-Kleiner theorem for Ahlfors 2-regular quasispheres.

Abstract

We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general "one-sided" quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular quasispheres.