Spherical Tube Hypersurfaces
We examine Levi non-degenerate tube hypersurfaces in complex linear space which are "spherical," that is, locally CR-equivalent to the real hyperquadric. Spherical hypersurfaces are characterized by the condition of the vanishing of the CR-curvature form,
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For further volumes: http://www.springer.com/series/304
2020
•
Alexander Isaev
Spherical Tube Hypersurfaces
123
Prof. Alexander Isaev Australian National University Mathematical Sciences Institute 0200 Canberra Aust Capital Terr Australia [email protected]
ISBN 978-3-642-19782-6 e-ISBN 978-3-642-19783-3 DOI 10.1007/978-3-642-19783-3 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011925542 Mathematics Subject Classification (2011): 32-XX c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In this book we consider (connected) smooth real hypersurfaces in the complex vector space Cn+1 with n ≥ 1. Specifically, we are interested in tube hypersurfaces, i.e. real hypersurfaces of the form
Γ + iV, where Γ is a hypersurface in a totally real (n + 1)-dimensional linear subspace V ⊂ Cn+1 . From now on we fix the subspace V and choose coordinates z0 , . . . , zn in Cn+1 such that V = {Im z j = 0, j = 0, . . . , n}. Everywhere below V is identified with Rn+1 by means of the coordinates x j := Re z j , j = 0, . . . , n. Tube hypersurfaces arise, for instance, as the boundaries of tube domains, that is, domains of the form D + iRn+1, where D is a domain in Rn+1 . We refer to the hypersurface Γ and domain D as the bases of the above tubes. The study of tube domains is a classical subject in several complex variables and complex geometry, which goes back to the beginning of the 20th century. Indeed, already Siegel found it convenient to realize certain symmetric domains as tubes. For example (see Section 5.3 for details), the familiar unit ball in Cn+1 is biholomorphically equivalent to the tube domain with the base given by the inequality x0 >
n
∑ x2α .
(0.1)
α =1
Note that the boundary of the tube domain with base (0.1) is the tube hypersurface whose base is defined by the equation x0 =
n
∑ x2α .
(0.2)
α =1
This tube hypersurface is equivalent to the (2n + 1)-dimensional sphere in Cn+1 with one point removed.
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