The quasiconformal Jacobian problem
The quasiconformal Jacobian problem
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拟共形雅可比问题
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
E. Saksman
中科院分区：
文献类型：
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作者：
M. Bonk;J. Heinonen;E. Saksman
作为拟共形映射 f :R n → R, n ≥ 2 的雅可比行列式 Jf(x) = det(Df(x))，可以出现哪些非负函数，直至有界乘法误差？哪些度量空间是 bi-Lipschitz 等价于 R，n ≥ 2？我们首先调查了关于这些问题的已知信息，并指出如何从 B. Kleiner 和第一作者 [BoK] 最近的工作中得出结论，这两个问题实际上在 n = 2 维度上是等价的。然后，我们继续展示一类迄今为止未知的函数，它们满足第一个问题的要求（也在维度 n = 2 中）。更具体地说，我们利用两个问题之间的联系，并从 Fu 定理推导出以下结果 [F]。
Which nonnegative functions can arise, up to a bounded multiplicative error, as Jacobian determinants Jf(x) = det(Df(x)) of quasiconformal mappings f :R n → R, n ≥ 2? Which metric spaces are bi-Lipschitz equivalent to R, n ≥ 2? We first survey what is known about these problems, and point out how it follows from recent work of B. Kleiner and the first author [BoK] that the two problems are in fact equivalent in dimension n = 2. Then we proceed to exhibit a hitherto unknown class of functions which satisfy the requirement of the first question (also in dimension n = 2). To be more specific, we exploit the connection between the two problems, and deduce the following result from a theorem of Fu [F].