Pullback of a quasiconformal map between arbitrary metric measure spaces

TL;DR

This paper demonstrates that quasiconformal homeomorphisms between metric measure spaces induce isomorphisms of cotangent modules, preserving geometric structures and Sobolev regularity properties, thus generalizing classical results to broader spaces.

Contribution

It establishes the functorial pullback of cotangent modules for quasiconformal maps in metric measure spaces, extending the theory beyond Euclidean and PI spaces.

Findings

Quasiconformal homeomorphisms induce cotangent module isomorphisms.

Such maps preserve the dimension of Cheeger charts in PI spaces.

Invertibility of pullback implies Sobolev regularity of inverse maps.

Abstract

We prove that every (geometrically) quasiconformal homeomorphism between metric measure spaces induces an isomorphism between the cotangent modules constructed by Gigli. We obtain this by first showing that every continuous mapping $\varphi$ with bounded outer dilatation induces a pullback map $\varphi^*$ between the cotangent modules of Gigli, and then proving the functorial nature of the resulting pullback operator. Such pullback is consistent with the differential for metric-valued locally Sobolev maps introduced by Gigli-Pasqualetto-Soultanis. Using the consistency between Gigli's and Cheeger's cotangent modules for PI spaces, we prove that quasiconformal homeomorphisms between PI spaces preserve the dimension of Cheeger charts, thereby generalizing earlier work by Heinonen-Koskela-Shanmugalingam-Tyson. Finally, we show that if $\varphi$ is a given homeomorphism with bounded outer dilatation, then $\varphi^{-1}$ has bounded outer dilatation if and only if $\varphi^{*}$ is invertible and $\varphi^{-1}$ is Sobolev. In contrast to the setting of Euclidean spaces, Carnot groups, or more generally, Ahlfors regular PI spaces, the Sobolev regularity of $\varphi^{-1}$ needs to be assumed separately.