Holomorphic Curves in Symplectic Geometry

ISBN-13: 9783034896566 / Angielski / Miękka / 2012 / 331 str.

Michele Audin;Jacques Lafontaine

Opis

The school, the book This book is based on lectures given by the authors of the various chapters in a three week long CIMPA summer school, held in Sophia-Antipolis (near Nice) in July 1992. The first week was devoted to the basics of symplectic and Riemannian geometry (Banyaga, Audin, Lafontaine, Gauduchon), the second was the technical one (Pansu, Muller, Duval, Lalonde and Sikorav). The final week saw the conclusion ofthe school (mainly McDuffand Polterovich, with complementary lectures by Lafontaine, Audin and Sikorav). Globally, the chapters here reflect what happened there. Locally, we have tried to reorganise some ofthe material to make the book more coherent. Hence, for instance, the collective (Audin, Lalonde, Polterovich) chapter on Lagrangian submanifolds and the appendices added to some of the chapters. Duval was not able to write up his lectures, so that genuine complex analysis will not appear in the book, although it is a very current tool in symplectic and contact geometry (and conversely). Hamiltonian systems and variational methods were the subject of some of Sikorav's talks, which he also was not able to write up. On the other hand, F. Labourie, who could not be at the school, wrote a chapter on pseudo-holomorphic curves in Riemannian geometry.

Spis treści

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.