Conformal invariants and quasiregular mappings
Conformal invariants and quasiregular mappings
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DOI：
10.1007/bf02792546
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发表时间：
1985-12
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影响因子：
--
通讯作者：
M. Vuorinen
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文献类型：
--
作者：
M. Vuorinen
R ′，n _--2中的拟共形映射和拟正则映射理论是推广一复变解析函数和共形映射理论的结果（参见[1]）。调查文章[50]）。n维拟共形映射，即一对一拟正则映射的系统研究始于1960年，由FW Gehring和J. V/iis~ il/i.非同胚n维拟正则映射的研究是由余启超开始的。G. 1966年的雷舍特尼亚克。经典复分析的常用方法，如基于复数的几何或代数性质（幂级数、无穷或有限和与积、复积分）的方法，不适用于研究R”，n> 3中的拟正则映射。n= 2的情况有些例外，因为在这种情况下可以应用一些强有力的结果，这些结果在高维情况n _-> 3中没有对应的结果。其中的结果是黎曼映射定理[25，第14页]，斯托洛分解定理[25，第252页]，和基本存在定理[25，卡普。V]。本文的重点是对所有n=> 2都成立的结果，而且在这种情况下的方法是真实的变量方法而不是复变量方法。关于n维拟正则映射的理论有三种方法，每种方法都基于不同的证明方法。第一种方法，由于俞。G. Reshetnyak 1966-69，是分析的性质，并利用工具，从偏微分方程理论和微分几何。第二种方法是由O. Martio，S. Rickman和J. Viis~ il~ i在1969-72年。第二种方法是基于几何函数论的工具，特别是A。Beurling and LV Ahlfors [1，p. 81].在本文中，实际上使用曲线族F的模M（F）比F的极值长度（等于I/M（F））更方便。B的第三种方法是最近的。Bojarski和T. Iwaniec [7]是基于极大函数和其他思想从真实的分析。
The theory of quasiconformal and quasiregular mappings in R', n _---2, has resulted from efforts to generalize the theory of conformal mappings and analytic functions of one complex variable (cf. the survey article [50]). The systematic study of n-dimensional quasiconformai mappings, ie one-to-one quasiregular mappings, was started in 1960 by FW Gehring and J. V/iis~ il/i. The study of nonhomeomorphic n-dimensional quasiregular mappings was started by Yu. G. Reshetnyak in 1966. The usual methods of the classical complex analysis such as those based on geometric or algebraic special properties of complex numbers (power series, infinite or finite sums and products, complex integration) are not applicable to the study of quasiregular mappings in R", n> 3. The case n= 2 is somewhat exceptional because in this case one can apply some powerful results, which have no counterpart in the higher dimensional case n _-> 3. Among such results are Riemann's mapping theorem [25, p. 14], Stoilow's decomposition theorem [25, p. 252], and the fundamental existence theorem [25, Kap. V]. In this paper the stress is on results which hold for all n=> 2 and the methods in this case are rather real variable than complex variable m~ thods. There are three approaches to the theory of n-dimensional quasiregular mappings, each based on different methods of proof. The first approach, due to Yu. G. Reshetnyak 1966-69, is analytic in character and makes use of tools from the PDE theory and differential geometry. The second approach, geometric in character, was developed by O. Martio, S. Rickman, and J. Viis~ il~ i in 1969-72. The second approach is based on tools from the geometric function theory, in particular, on the extremal length method of A. Beurling and LV Ahlfors [1, p. 81]. In the present context it is actually more convenient to use the modulus M (F) of a curve family F rather than the extremal length of F which is equal to I/M (F). The third, very recent, approach of B. Bojarski and T. Iwaniec [7] is based on maximal functions and other ideas from real analysis.