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From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
 
Kai Cieliebak Ludwig-Maximilians-Universität, München, Germany
Yakov Eliashberg Stanford University, Stanford, CA
From Stein to Weinstein and Back
From Stein to Weinstein and Back
Hardcover ISBN:  978-0-8218-8533-8
Product Code:  COLL/59
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1582-2
Product Code:  COLL/59.E
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
Hardcover ISBN:  978-0-8218-8533-8
eBook: ISBN:  978-1-4704-1582-2
Product Code:  COLL/59.B
List Price: $188.00$143.50
MAA Member Price: $169.20$129.15
AMS Member Price: $150.40$114.80
From Stein to Weinstein and Back
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From Stein to Weinstein and Back
From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
Kai Cieliebak Ludwig-Maximilians-Universität, München, Germany
Yakov Eliashberg Stanford University, Stanford, CA
Hardcover ISBN:  978-0-8218-8533-8
Product Code:  COLL/59
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1582-2
Product Code:  COLL/59.E
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
Hardcover ISBN:  978-0-8218-8533-8
eBook ISBN:  978-1-4704-1582-2
Product Code:  COLL/59.B
List Price: $188.00$143.50
MAA Member Price: $169.20$129.15
AMS Member Price: $150.40$114.80
  • Book Details
     
     
    Colloquium Publications
    Volume: 592012; 364 pp
    MSC: Primary 32; 53;

    A beautiful and comprehensive introduction to this important field.

    Dusa McDuff, Barnard College, Columbia University

    This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a superb introduction to this area and also contains the authors' new results.

    Tomasz Mrowka, MIT

    This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from “Stein to Weinstein”) and its applications in the complex geometric world of Stein manifolds (the road “back”). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology.

    Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, \(h\)-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.

    Readership

    Graduate students and research mathematicians interested in functions in several complex variables and symplectic and contact topology.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. Introduction
    • Part 1. $J$-convexity
    • Chapter 2. $J$-convex functions and hypersurfaces
    • Chapter 3. Smoothing
    • Chapter 4. Shapes for $i$-convex hypersurfaces
    • Chapter 5. Some complex analysis
    • Part 2. Existence of Stein structures
    • Chapter 6. Symplectic and contact preliminaries
    • Chapter 7. The $h$-principles
    • Chapter 8. The existence theorem
    • Part 3. Morse–Smale theory for $J$-convex functions
    • Chapter 9. Recollections from Morse theory
    • Chapter 10. Modifications of $J$-convex Morse functions
    • Part 4. From Stein to Weinstein and back
    • Chapter 11. Weinstein structures
    • Chapter 12. Modifications of Weinstein structures
    • Chapter 13. Existence revisited
    • Chapter 14. Deformations of flexible Weinstein structures
    • Chapter 15. Deformations of Stein structures
    • Part 5. Stein manifolds and symplectic topology
    • Chapter 16. Stein manifolds of complex dimension two
    • Chapter 17. Exotic Stein structures
    • Appendix A. Some algebraic topology
    • Appendix B. Obstructions to formal Legendrian isotopies
    • Appendix C. Biographical notes on the main characters
  • Reviews
     
     
    • This book is a remarkable mix of classical topics and research done by the authors over the last twenty years. It is both a textbook and a research monograph. ... [M]ore classical material...is presented from the perspective of their applications in the book. It is fascinating and refreshing to see these classical tools in action, combined and applied to create something new. The exposition is very beautiful; whenever possible, the geometry of the situation is fully brought out, and formulas and computations are used only to check a geometric intuition.

      DMV
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 592012; 364 pp
MSC: Primary 32; 53;

A beautiful and comprehensive introduction to this important field.

Dusa McDuff, Barnard College, Columbia University

This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a superb introduction to this area and also contains the authors' new results.

Tomasz Mrowka, MIT

This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from “Stein to Weinstein”) and its applications in the complex geometric world of Stein manifolds (the road “back”). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology.

Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, \(h\)-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.

Readership

Graduate students and research mathematicians interested in functions in several complex variables and symplectic and contact topology.

  • Chapters
  • Chapter 1. Introduction
  • Part 1. $J$-convexity
  • Chapter 2. $J$-convex functions and hypersurfaces
  • Chapter 3. Smoothing
  • Chapter 4. Shapes for $i$-convex hypersurfaces
  • Chapter 5. Some complex analysis
  • Part 2. Existence of Stein structures
  • Chapter 6. Symplectic and contact preliminaries
  • Chapter 7. The $h$-principles
  • Chapter 8. The existence theorem
  • Part 3. Morse–Smale theory for $J$-convex functions
  • Chapter 9. Recollections from Morse theory
  • Chapter 10. Modifications of $J$-convex Morse functions
  • Part 4. From Stein to Weinstein and back
  • Chapter 11. Weinstein structures
  • Chapter 12. Modifications of Weinstein structures
  • Chapter 13. Existence revisited
  • Chapter 14. Deformations of flexible Weinstein structures
  • Chapter 15. Deformations of Stein structures
  • Part 5. Stein manifolds and symplectic topology
  • Chapter 16. Stein manifolds of complex dimension two
  • Chapter 17. Exotic Stein structures
  • Appendix A. Some algebraic topology
  • Appendix B. Obstructions to formal Legendrian isotopies
  • Appendix C. Biographical notes on the main characters
  • This book is a remarkable mix of classical topics and research done by the authors over the last twenty years. It is both a textbook and a research monograph. ... [M]ore classical material...is presented from the perspective of their applications in the book. It is fascinating and refreshing to see these classical tools in action, combined and applied to create something new. The exposition is very beautiful; whenever possible, the geometry of the situation is fully brought out, and formulas and computations are used only to check a geometric intuition.

    DMV
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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