© 2018

Holomorphic Curves in Low Dimensions

From Symplectic Ruled Surfaces to Planar Contact Manifolds

  • 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


Part of the Lecture Notes in Mathematics book series (LNM, volume 2216)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Chris Wendl
    Pages 1-20
  3. Chris Wendl
    Pages 67-98
  4. Chris Wendl
    Pages 99-115
  5. Chris Wendl
    Pages 117-122
  6. Chris Wendl
    Pages 123-131
  7. Chris Wendl
    Pages 133-189
  8. Back Matter
    Pages 275-294

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


MSC (2010): 57R17, 32Q65 symplectic ruled surfaces rational and ruled symplectic 4-manifolds planar contact manifolds pseudoholomorphic curves Gromov-Witten invariants uniruled symplectic manifolds inimal symplectic 4-manifolds Lefschetz fibrations Lefschetz pencils immersed symplectic spheres symplectic fillings

Authors and affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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.

Bibliographic information


“The book is well-written, well-referenced … . Anyone interested in McDuff’s characterization of rational and ruled symplectic 4-manifolds or her theorem that says ‘uniruled => rational/ruled’ should find this book quite useful.” (David E. Hurtubise, zbMATH 1432.57055, 2020)