The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology--two important ingredients of the mirror symmetry conjecture.

This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds.

The book is suitable for graduate students and researchers interested in symplectic geometry and its applications, especially in the theory of Gromov-Witten invariants.

Readership

Graduate students and research mathematicians interested in symplectic geometry and its applications, especially in the theory of Gromov-Witten invariants.

Reviews

"The book offers a systematic treatment of one of the important new fields in mathematics and should be available in every library."

-- EMS Newsletter

"Written by two top specialists on this topic, it will be useful to all those interested in this fascinating domain. Last but not least the graphical conditions offered by the American Mathematical Society are excellent."

-- Zentralblatt Math

Table of Contents

• Introduction
• $J$-holomorphic curves
• Moduli spaces and transversality
• Compactness
• Stable maps
• Moduli spaces of stable maps
• Gromov-Witten invariants
• Hamiltonian perturbations
• Applications in symplectic topology
• Gluing
• Quantum cohomology
• Floer cohomology
• Fredholm theory
• Elliptic regularity
• The Riemann-Roch theorem
• Stable curves of genus zero
• Singularities and intersections (written with Laurent Lazzarini)
• Bibliography
• List of symbols
• Index