Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Tyson, Jeremy

doi:10.1090/S1088-4173-01-00064-9

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces
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Conform. Geom. Dyn. 5 (2001), 21-73 Request permission

Abstract:

Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu’s generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension $Q>1$, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.

We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper $Q$-regular Loewner space, $Q>1$, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

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