Topics Related to Complex Analysis
First, there is the remarkable theorem of Liouville (which we shall prove later): except for isometries, dilations, and inversions (to be defined below), there are no smooth conformai mappings in dimension three or greater. Thus one seeks a natural generalization of conformai mappings to higher dimensions.
Second, many properties that are commonly proved for conformai mappings (in dimension two) are really properties of quasiconformal mappings. Mathematical facts, particularly geometric ones, are often most clearly understood when the fewest hypotheses are used.
Third, taking the first remark into account, any basic geometric property that distinguishes among dimensions—particularly among Euclidean spaces—is of a priori interest.
Fourth, the theory is used in the study of differentials on, and moduli for, Riemann surfaces—work pioneered in Teichmüller .
KeywordsEigenvalue Problem Conformal Mapping Quasiconformal Mapping Jordan Curve Rectangular Parallelepiped
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