Metric and geometric quasiconformality in Ahlfors regular Loewner spaces
Metric and geometric quasiconformality in Ahlfors regular Loewner spaces
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DOI：
10.1090/s1088-4173-01-00064-9
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发表时间：
2001-08
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通讯作者：
J. Tyson
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作者：
J. Tyson
几何学的最新发展强调了对拟共形映射经典理论的抽象表述的需求。我们修改 Pansu 的广义模来研究具有与欧几里得空间足够相似的度量和测度理论属性的空间中的拟共形几何。我们的基本研究对象是配备 Borel 测度的局部紧度量空间，该测度是维数 Q > 1 的 Ahlfors-David 正则，并且满足 Heinonen-Koskela 的 Loewner 条件。对于两个这样的空间中的开集之间的同胚，我们证明了三个条件的等价性：度量拟共形性、局部拟对称性和几何拟共形性。我们从这些结果中得出几个推论。首先，我们证明 Loewner 条件是局部紧致 Ahlfors 正则空间中的拟对称不变量。接下来，我们证明一个真 Q-正则 Loewner 空间，Q > 1，并不准共形等价于任何子域。 （在欧几里得的情况下，这个结果是由 Loewner 得出的。）最后，我们将雪花曲线的乘积表征为拟对称/双利普希茨等价：两个这样的乘积当且仅当它们是等轴时才是双利普希茨等价的，并且当且仅当它们共形等价时才是拟对称等价的。
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.