Polyhedral approximation and uniformization for non-length surfaces

TL;DR

This paper establishes that metric surfaces with finite Hausdorff 2-measure can be approximated by polyhedral surfaces and admits quasiconformal parametrizations, extending classical uniformization results to more general metric surfaces.

Contribution

It proves polyhedral approximation and uniformization for metric surfaces with finite Hausdorff measure, generalizing previous results to non-length metric surfaces.

Findings

Any metric surface with finite Hausdorff 2-measure is a Gromov-Hausdorff limit of polyhedral surfaces.

Any metric surface homeomorphic to the 2-sphere admits a quasiconformal parametrization by the Riemann sphere.

New proofs of uniformization theorems for quasispheres and reciprocal surfaces are provided.

Abstract

We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala-Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk-Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.