本篇論文主要分成兩個部分。第一部分，包括了第一節到第十二節，主要是在討論在Rn空間中一些環域的性質和結果。在介紹完定義和符號後，第二節中，我們導出了一個引理，並用它在第三節推導出一個很有用的不等式。在第六第七節中，我們就應用到了這個不等式，並用它來探討某種環域上函數，Extremal Function，的性質。接下來的三節，我們將焦點繼續放在這種函數上，討論了它的解析性質，包括它是可解析的，並且在環域中任意緊緻子集上，梯度向量的長度是有限的。第十一節和十二節的主要內容是在討論環域的模函數，Modulus。我們討論了它的連續性，並在十二節中給出了一個它的上下界。
論文的第二部分包括了第十三節到第二十九節。在所需要的環域的性質都推導出來後，接下來，就是利用這些推導出來的結果來討論Rn空間中的擬保角變換。首先是定義。在有界環域R中，滿足不等式：(1/K)modR≦modR’≦(K)modR的任何拓樸變換，我們稱之為K擬保角變換。不等式中的R’是R在映射後的像。第十四節，我們推導出了K擬保角變換的一個很重要的性質，Compactness Property。第十五，十六和二十節，我們討論了擬保角變換的解析性質。特別是ACL（Absolutly Continuous on Lines）性質，和殆遍的可微性。在第十八節裡，我們給出了另一個等價的K擬保角變換的解析定義，並在第十九節中討論若將以上不等式改成只有其中的一半時所需要做的條件改變。第二十一和二十二節，我們又給出了第三個擬保角變換的等價定義，並證明了一個定理。為了第二十五節證明Distortion Theorem的需要，在第二十三節，我們又回來討論兩種特殊的環域，Grotzsch環和 Taichmuler環，並在二十四節裡導出一個不等式。利用二十五節的結果，我們即可在第二十六節推導出了另一個定理Keobe Viertelsatz，和在第二十七節推導出另一個在正規族（Normal Family）裡的重要定理。在第二十八節，我們證明了１擬保角變換事實上就是Moebius 變換。最後，在第二十九節，我們利用二十八節的結果簡化了保角變換的Liouville定理的條件，不再要求定理中擬保角變換的可微性。

This paper consists of two parts. In the first part, from section 1 to section 12, we are talking about rings. We begin by the definition of conformal capacity and a lemma in section 2. Using the lemma, we then establish an inequality in section 3. We next apply this inequality in sections 6-7 to show that the extremal function μ for a ring R exists whenever both components of boundary R are nondegenerate. In section 8-10 we then show that this extremal function μ is real analytic provided ∣▽μ∣is bounded away from 0 and ∞ a.e. on each compact set in R. In section 11 we prove a continuity property for the moduli of rings and in section 12 we use extremal lengths to obtain a pair of bounds for this modulus.
In the second part, from section 13 to section 29, we use the results on rings developed in the first part to study quasiconformal mappings. A topological mapping of a domain D is said to be K-quasiconformal if it satisfies the two sided modulus condition
(1/K)modR ≦ modR’ ≦ (K)modR
for all bounded rings R with its closure belongs to D. Here R’ is the image of R under the mapping. In section 14 we establish an important compactness property for the K-quasiconformal mappings of a domain. Next in section 15-17 and in section 20 we study some analytic properties of quasiconformal mappings. In particular we show that they are absolutely continuous on lines and that they are differentiable with nonzero Jacobians a.e. We formulate an equivalent analytic definition for K-quasiconformal mappings in section 18 and discuss the possibility of replacing the two sided modulus condition by a one sided one in section19. Sections 21-22 are devoted to a third definition for quasiconformality and a theorem on exceptional sets.
In section 23 we introduce some background about Grotzsch and Teichmuler rings, and then establish an inequality in section 24. After that, we then can prove a general distortiontheorem in section 25, which leads to an analogue of the Koebe Viertelsatz in section 26 and to an important theorem on normal family in section 27. In section 28 we identify the 1-quasiconformal mappings and show that these are just the Moebius transformations. Finally, we use this result to establish a very general form of Liouville’s theorem on the conformal mappings, which requires no differentiability hypotheses, in section 29.

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