Quasiconformal Mappings and Curvatures on Metric Measure Spaces

TL;DR

This paper explores the properties of quasiconformal mappings in higher-dimensional metric measure spaces with curvature constraints, establishing new geometric and analytical principles for such spaces.

Contribution

It demonstrates that non-collapsed RCD(0,n) spaces with Euclidean volume growth are Loewner spaces and satisfy the infinitesimal-to-global principle, advancing the understanding of curvature and mapping theory.

Findings

RCD(0,n) spaces with Euclidean growth are Loewner spaces

These spaces satisfy the infinitesimal-to-global principle

Extension of quasiconformal mapping theory to curved metric measure spaces

Abstract

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexsandrov, we show that a non-collapsed $\mathrm{RCD}(0,n)$ space ($n\geq2$) with Euclidean growth volume is an $n$-Loewner space and satisfies the infinitesimal-to-global principle.