Geometric and analytic quasiconformality in metric measure spaces

Details Geometric and analytic quasiconformality in metric measure spaces Geometric and analytic quasiconformality in metric measure spaces

2010-08-21

arxiv.org/abs/1008.3588v2

Mathematics

Complex Variables

15 pages. Flaw in covering argument for Theorem 1.2 corrected. Flaws in Remarks 4.3 and 4.4 corrected. Accepted for publicatio

Scientific paper

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ 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 $K_O(f)= \operatorname{esssup} H_O(x,f)$.

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