This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.
Table of Contents
Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.- 3 Proofs of the symplectic rigidity results.- 4 What is in the book… and what is not.- 1: Basic symplectic geometry.- I An introduction to symplectic geometry.- 1 Linear symplectic geometry.- 2 Symplectic manifolds and vector bundles.- Appendix: the Maslov class M. Audin, A. Banyaga, F. Lalonde, L. Polterovich.- II Symplectic and almost complex manifolds.- 1 Almost complex structures.- 2 Hirzebruch surfaces.- 3 Coadjoint orbits (of U(n)).- 4 Symplectic reduction.- 5 Surgery.- Appendix: The canonical almost complex structure on the manifold of 1-jets of pseudo-holomorphic mappings between two almost complexmanifolds P. Gauduchon.- 2: Riemannian geometry and linear connections.- III Some relevant Riemannian geometry.- 1 Riemannian manifolds as metric spaces.- 2 The geodesic flow and its linearisation.- 3 Minimal manifolds.- 4 Two-dimensional Riemannian manifolds.- 5 An application to pseudo-holomorphic curves.- Appendix: the isoperimetric inequality M.-P. Muller.- IV Connexions linéaires, classes de Chern, théorème de Riemann-Roch.- 1 Connexions linéaires.- 2 Classes de Chern.- 3 Le théorème de Riemann-Roch.- Bibliographie.- 3: Pseudo-holomorphic curves and applications.- V Some properties of holomorphic curves in almost complex manifolds.- 1 The equation $$
\bar \partial f$$
in C.- 2 Regularity of holomorphic curves.- 3 Other local properties.- 4 Properties of the area of holomorphic curves.- 5 Gromov’s compactness theorem for holomorphic curves.- Appendix: Stokes’ theorem for forms with differentiable coefficients.- VI Singularities and positivity of intersections of J-holomorphic curves.- 1 Elementary properties.- 2 Positivity of intersections.- 3 Local deformations.- 4 Perturbing away singularities.- Appendix: The smoothness of the dependence on Gang Liu.- VII Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves.- 1 Introduction.- 2 A review of some classical Schwarz lemmas.- 3 Isoperimetric inequalities for J-curves.- 4 The Schwarz and monotonicity lemmas.- 5 Continuous Lipschitz extension across a puncture.- 6 Higher derivatives.- VIII Compactness.- 1 Riemann surfaces with nodes.- 2 Cusp-curves.- 3 Proof of the compactness theorem 2.2.1.- 4 Convergence of parametrised curves.- IX Exemples de courbes pseudo-holomorphes en géométrie riemannienne.- 1 Immersions isométriques elliptiques.- 2 Courbure de Gauss prescrite.- 3 Autres exemples et constructions.- Appendice: convergence d’applications pseudo-holomorphes.- Bibliographie.- X Symplectic rigidity: Lagrangian submanifolds.- 1 Lagrangian constructions.- 2 Symplectic area and Maslov classesrigidity in split manifolds.- 3 Soft and hard Lagrangian obstructions to Lagrangian embeddings in Cn.- 4 Rigidity in cotangent bundles and applications to mechanics.- 5 Pseudo-holomorphic curves: proof of the main rigidity theorem.- Appendix: Exotic structures on R2n.- Authors’ addresses.