Ebook: Holomorphic Curves in Symplectic Geometry

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.

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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.
Content:
Front Matter....Pages i-xi
Introduction Applications of pseudo-holomorphic curves to symplectic topology....Pages 1-14
Front Matter....Pages 15-15
An introduction to symplectic geometry....Pages 17-40
Symplectic and almost complex manifolds....Pages 41-74
Front Matter....Pages 75-75
Some relevant Riemannian geometry....Pages 77-112
Connexions lin?aires, classes de Chern, th?or?me de Riemann-Roch....Pages 113-162
Front Matter....Pages 163-163
Some properties of holomorphic curves in almost complex manifolds....Pages 165-189
Singularities and positivity of intersections of J-holomorphic curves....Pages 191-215
Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves....Pages 217-231
Compactness....Pages 233-249
Exemples de courbes pseudo-holomorphes en g?om?trie riemannienne....Pages 251-269
Symplectic rigidity: Lagrangian submanifolds....Pages 271-321
Back Matter....Pages 323-331

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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.
Content:
Front Matter....Pages i-xi
Introduction Applications of pseudo-holomorphic curves to symplectic topology....Pages 1-14
Front Matter....Pages 15-15
An introduction to symplectic geometry....Pages 17-40
Symplectic and almost complex manifolds....Pages 41-74
Front Matter....Pages 75-75
Some relevant Riemannian geometry....Pages 77-112
Connexions lin?aires, classes de Chern, th?or?me de Riemann-Roch....Pages 113-162
Front Matter....Pages 163-163
Some properties of holomorphic curves in almost complex manifolds....Pages 165-189
Singularities and positivity of intersections of J-holomorphic curves....Pages 191-215
Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves....Pages 217-231
Compactness....Pages 233-249
Exemples de courbes pseudo-holomorphes en g?om?trie riemannienne....Pages 251-269
Symplectic rigidity: Lagrangian submanifolds....Pages 271-321
Back Matter....Pages 323-331
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