Geometric and analytic quasiconformality in metric measure spaces

2010/08/21 by Williams, Marshall · 2 citations
#30L10 #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1008.3588

Abstract

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism f\colon X→ Y between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either space. When X and Y have locally Q-bounded geometry and Y is contained in an Alexandrov space of curvature bounded above, the sharpness of our results implies that, as in the classical case, the modular and pointwise outer dilatations of \map are related by K_O(f)= esssup H_O(x,f).

Cited by

• Pull-back of differential forms by multi-valued Sobolev maps, and the quasiregularity of the multi-valued inverse of a quasiregular map
• Quasisymmetric Koebe Uniformization of metric surfaces

Related

• Quasiconformal maps in metric spaces with controlled geometry
• Definitions of quasiconformality on metric surfaces
• Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces
• Pullback of a quasiconformal map between arbitrary metric measure spaces
• Nonexistence of quasiconformal maps between certain metric measure\n spaces