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From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
Kai Cieliebak, Ludwig-Maximilians-Universität, München, Germany, and Yakov Eliashberg, Stanford University, CA
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Colloquium Publications
2012; 364 pp; hardcover
Volume: 59
ISBN-10: 0-8218-8533-2
ISBN-13: 978-0-8218-8533-8
List Price: US$78
Member Price: US$62.40
Order Code: COLL/59
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See also:

\(J\)-holomorphic Curves and Symplectic Topology: Second Edition - Dusa McDuff and Dietmar Salamon

Introduction to the \(h\)-Principle - Y Eliashberg and N Mishachev

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.

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."

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