Lecture Notes in Mathematics Virtual Series on Symplectic Geometry
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Holomorphic Curves in Low Dimensions

From Symplectic Ruled Surfaces to Planar Contact Manifolds

Authors: Wendl, Chris

  • Provides an up-to-date perspective on certain foundational results in 4-dimensional symplectic topology
  • Includes the first exposition aimed at graduate students on the classification of uniruled symplectic 4-manifolds
  • Illustrates the connection between McDuff's classic results on rational/ruled surfaces and more recent developments involving symplectic fillings of contact 3-manifolds and the Weinstein conjecture
  • Offers a concise survey of the essential analytical results in the theory of punctured holomorphic curves
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eBook 41,64 €
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  • ISBN 978-3-319-91371-1
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Softcover 51,99 €
price for Spain (gross)
  • ISBN 978-3-319-91369-8
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About this book

This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three.

The first half of the book focuses on McDuff's characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. The proof presented here uses the language of Lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Emphasizing applications rather than technical results, the analytical survey mostly refers to other sources for proofs, while aiming to provide precise statements that are widely applicable, plus some informal discussion of the analytical ideas behind them. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as classifying the symplectic fillings of planar contact manifolds.

This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details.

 

This book is also part of the Virtual Series on Symplectic Geometry

http://www.springer.com/series/16019

About the authors

Chris Wendl is known among symplectic topologists for contributions to the study of symplectic fillability of contact manifolds, and for transversality results in the theory of pseudoholomorphic curves. He is currently Professor of Differential Geometry and Global Analysis at the Humboldt University in Berlin, and is also the author of two other forthcoming books on holomorphic curves and symplectic field theory.

Table of contents (9 chapters)

Table of contents (9 chapters)
  • Introduction

    Pages 1-20

    Wendl, Chris

  • Background on Closed Pseudoholomorphic Curves

    Pages 21-66

    Wendl, Chris

  • Blowups and Lefschetz Fibrations

    Pages 67-98

    Wendl, Chris

  • Compactness

    Pages 99-115

    Wendl, Chris

  • Exceptional Spheres

    Pages 117-122

    Wendl, Chris

Buy this book

eBook 41,64 €
price for Spain (gross)
  • ISBN 978-3-319-91371-1
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 51,99 €
price for Spain (gross)
  • ISBN 978-3-319-91369-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Holomorphic Curves in Low Dimensions
Book Subtitle
From Symplectic Ruled Surfaces to Planar Contact Manifolds
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2216
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-91371-1
DOI
10.1007/978-3-319-91371-1
Softcover ISBN
978-3-319-91369-8
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XIII, 294
Number of Illustrations
2 b/w illustrations, 31 illustrations in colour
Topics