Overview

"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 super 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.|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."

Full Product Details

Author:   Kai Cieliebak ,  Yakov Eliashberg
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Edition:   New ed.
Weight:   0.803kg
ISBN:  

9780821885338

ISBN 10:   0821885332
Pages:   364
Publication Date:   30 January 2013
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Introduction Part I. 𝐽-convexity 𝐽-convex functions and hypersurfaces Smoothing Shapes for 𝑖-convex hypersurfaces Some complex analysis Part II. Existence of Stein structures Symplectic and contact preliminaries The ℎ-principles The existence theorem Part III. Morse–Smale theory for 𝐽-convex functions Recollections from Morse theory Modifications of 𝐽-convex Morse functions Part IV. From Stein to Weinstein and back Weinstein structures Modifications of Weinstein structures Existence revisited Deformations of flexible Weinstein structures Deformations of Stein structures Part V. Stein manifolds and symplectic topology Stein manifolds of complex dimension two Exotic Stein structures Some algebraic topology Obstructions to formal Legendrian isotopies Biographical notes on the main characters Bibliography Index

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

"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"

oThis 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. a [M]ore classical materialais 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.o -- DMV

Author Information

Kai Cieliebak, Ludwig-Maximilians-Universität, München, Germany Yakov Eliashberg, Stanford University, Stanford, CA, USA

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