John P. D’Angelo - Rational Sphere Maps

More information about this series at http://www.springer.com/series/4848

This book is published under the imprint Birkhuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AGThis book is published under the imprint Birkhuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG

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The unit circle in the complex number system and its self-mappings have played a major role in the history of mathematics. Below we give many striking examples. The central theme throughout this book will be to understand higher dimensional analogues, where things are more subtle and ideas from many fields of mathematics make their appearance.

In one dimension, if is holomorphic (complex analytic) in a neighborhood of the closure of the unit disk , and maps the circle to itself, then is a finite Blaschke product. One can draw the same conclusion assuming only that is a proper holomorphic mapping from to itself. In particular such functions are rational. Our primary topic will be the study of holomorphic rational maps sending the unit sphere in the source complex Euclidean space to the unit sphere in some target space . We call such mappings rational sphere maps. We use the terms monomial sphere map and polynomial sphere map with obvious meaning; even these mappings exhibit remarkably interesting and complicated behavior as the source and target dimensions rise.

In this book, a rational sphere map is complex analytic where it is defined. In other words, depends on the variables but not on the variables. In Chap. we briefly discuss some differences between holomorphic polynomial sphere maps and real polynomial sphere maps. In particular, in complex dimension at least , the only non-constant holomorphic polynomial maps sending the unit sphere to itself are linear, whereas there are real polynomial sphere maps of every degree. I considered the title Complex Analytic Rational Sphere Maps to prevent possible confusion, but the shorter title seems more appealing.

In some sense, this book is a research monograph, as it develops in a systematic fashion most of the research on rational sphere maps done in the last forty years. It differs however from many monographs in several ways, which we now describe.

First of all, scattered throughout the book are a large number of computational examples; the author feels that merging the abstract and concrete enhances both. Many times in his work on this subject, a theorem resulted from trying to cast a collection of examples into one framework. Some readers will stare at these formulas, observe subtle patterns, and pose their own open questions. Other readers may find the formulas distracting. I hope that I have achieved the right balance. Chaps. includes recent code by Lichtblau [1].