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Holomorphic Curves in Symplectic Geometry : Progress in Mathematics - Michele Audin

Holomorphic Curves in Symplectic Geometry

Progress in Mathematics

By: Michele Audin (Editor), Jacques Lafontaine (Editor)

Hardcover

Published: 1st February 1994
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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.

Introduction: Applications of pseudo-holomorphic curves to symplectic topology, J. Lafontaine and M. Audin: Examples of problems and results in symplectic topology; Pseudo-holomorphic curves in almost complex manifolds; Proofs of the symplectic rigidity results; What is in the book. . . and what is not...; Bibliography. Part 1: Basic symplectic geometry - An introduction to symplectic geometry, A. Banyaga: Linear symplectic geometry; Symplectic manifolds and vector bundles, Appendix: the Maslov class, M. Audin, A. Banyaga, F. Lalonde and L. Polterovich, Bibliography; Symplectic and almost complex manifolds, M. Audin: Almost complex structures, Hirzebruch surfaces, Coadjoint orbits (of U(n)), Symplectic reduction, Surgery?, Appendix: The canonical almost complex structure on the manifold of 1-jets of pseudo-holomorphic mappings between two almost complex manifolds, P. Gauduchon, Bibliography. Part 2: Riemannian geometry and linear connections - Some relevant Riemannian geometry J. Lafontaine: Riemannian manifolds as metric spaces, The geodesic flow and its linearisation, Minimal manifolds, Two-dimensional Riemannian Manifolds, An application to pseudo-holomorphic curves, Appendix: the isoperimetric inequality, M.-P. Muller, Bibliography; Connections lineaires, classes de Chern, theoreme de Riemann-Roch, P. Gauduchon: Connexions lineaires, Classes de Chern, Le theoreme de Riemann-Roch, Bibliographie. Part 3: Pseudo-holomorphic curves and applications - Some properties of holomorphic curves in almost complex manifolds, J, - C. Sikorav: The equation af = g in C, Regularity of holomorphic curves, Other local properties, Properties of the area of holomorphic curves, Gromov's compactness theorem for holomorphic curves, Appendix: Stokes' theorem for forms with differentiable coefficients, Bibliography; Singularities and positivity of intersections of J-holomorphic curves, D. McDuff: Elementary properties, Positivity of intersections, Local deformations, Perturbing away singularities, Appendix: The smoothness of the dependence on E - Gang Liu, Bibliography; Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves, M.-P. Muller: Introduction, A review of some classical Schwarz lemmas Isoperimetric inequalities for J-curves, The Schwarz and monotonicity lemmas, Continuous Lipschitz extension across a puncture, Higher derivatives, Bibliography; Compactness, P. Pansu: Riemann surfaces with nodes, Cusp-curves, Proof of the compactness theorem, Convergence of parametrised curves, Bibliography; Examples de courbes pseudo-holomorphes en geometrie riemannienne, F. Labourie: Immersions isometriques elliptiques, Courbure de Gauss prescrite, Autres exemples et constructions, Appendice: convergence d'applications pseudo-holomorphes, Bibliographie; Symplectic rigidity: Lagrangian submanifolds, M. Audin, F. Lalonde and L.Polterovich: Lagrangian constructions, Symplectic area and Maslov classes-rigidity in split manifolds, Soft

ISBN: 9783764329976
ISBN-10: 3764329971
Series: Progress in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 331
Published: 1st February 1994
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 0.66