Inversion Theory and Conformal Mapping

ISBN-13: 9780821826362 / Angielski / Miękka / 2000 / 118 str.

David E. Blair

Opis

It is rarely taught in undergraduate or even graduate curricula that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. This fact is taught in most complex analysis courses. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof, due to Nevanlinna, in general dimension and a differential geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Carath'eodory with the remarkable result that any circle-preserving transformation is necessarily a M'obius transformation - not even the continuity of the transformation is assumed.