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Spherical Tube Hypersurfaces

  • Book
  • © 2011

Overview

Authors:
  1. Alexander Isaev

    1. Mathematical Sciences Institute, Australian National University, Canberra, Australia

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  • This is a research monograph which is quite unique in a number of ways
  • However, it is hard to state the main features of the book briefly for non-experts
  • As a result, I am afraid I cannot come up with simple selling points that would be understood by the general reader and even by mathematicians who are not experts in the area of several complex variables
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2020)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xii
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  2. Invariants of CR-Hypersurfaces

    • Alexander Isaev
    Pages 1-33

  3. Rigid Hypersurfaces

    • Alexander Isaev
    Pages 35-40

  4. Tube Hypersurfaces

    • Alexander Isaev
    Pages 41-53

  5. General Methods for Solving Defining Systems

    • Alexander Isaev
    Pages 55-82

  6. Strongly Pseudoconvex Spherical Tube Hypersurfaces

    • Alexander Isaev
    Pages 83-96

  7. (n − 1,1)-Spherical Tube Hypersurfaces

    • Alexander Isaev
    Pages 97-121

  8. (n − 2,2)-Spherical Tube Hypersurfaces

    • Alexander Isaev
    Pages 123-184

  9. Number of Affine Equivalence Classes of (k, n − k) -Spherical Tube Hypersurfaces for k ≤ n − 2

    • Alexander Isaev
    Pages 185-194

  10. Further Results

    • Alexander Isaev
    Pages 195-212

  11. Back Matter

    Pages 213-220
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Keywords

About this book

We consider 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, so such hypersurfaces are flat from the CR-geometric viewpoint. On the other hand, such hypersurfaces are of interest from the point of view of affine geometry. Thus our treatment of spherical tube hypersurfaces in this book is two-fold: CR-geometric and affine-geometric. Spherical tube hypersurfaces turn out to possess remarkable properties. For example, every such hypersurface is real-analytic and extends to a closed real-analytic spherical tube hypersurface in complex space. One of our main goals is to give an explicit affine classification of closed spherical tube hypersurfaces whenever possible. In this book we offer a comprehensive exposition of the theory of spherical tube hypersurfaces starting with the idea proposed in the pioneering work by P. Yang (1982) and ending with the new approach due to G. Fels and W. Kaup (2009).

Reviews

From the book reviews:

“The main goal and purpose of Isaev’s book is to explore the invariant theory of the special class of spherical tube hypersurfaces. … this book will be of interest and of value to everyone working on the equivalence problem for CR structures.” (Thomas Garrity, Bulletin of the American Mathematical Society, Vol. 51 (4), 2014)

Authors and Affiliations

  • Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Alexander Isaev

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