Conformal Geometry and Dynamics

ISSN 1088-4173



Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Author: Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 5 (2001), 21-73
MSC (2000): Primary 30C65; Secondary 28A78, 46E35, 43A85
Published electronically: August 8, 2001
MathSciNet review: 1872156
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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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Additional Information

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651

Keywords: Quasiconformal/quasisymmetric map, conformal modulus, Loewner condition, Hausdorff/packing measure, Poincar\'{e} inequality
Received by editor(s): May 31, 2000
Received by editor(s) in revised form: June 4, 2001
Published electronically: August 8, 2001
Additional Notes: The results of this paper form part of the author’s Ph.D. thesis completed at the University of Michigan in 1999. Research supported by an NSF Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
Article copyright: © Copyright 2001 American Mathematical Society