Holomorphic Curves in Symplectic Geometry

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 classes—rigidity 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.