Gromov’s compactness theorem for pseudo-holomorphi

'...the book provides a self-contained and for the most part excellent elaboration of Gromov's proof of the compactness theorem.' -- Mathematical Reviews Mikhail Gormov introduced pseudo-holomorphic curves into sympletic geometry in 1985. Since then, pseudo-holomorphic curves have taken on great importance in many fields. The aim of this book is to present the original proof of Gromov's compactness theorem for pseudo-holomorphic curves in detail. Local properties of pseudo-holomorphic curves are investigated and proved from a geometric viewpoint. Properties of particular interest are isoperimetric inequalities, a monotonicity formula, gradient bounds and the removal of singularities. A special chapter is devoted to relevant features of hyperbolic surfaces, where pairs of pants decomposition and thick-thin decomposition are described. The book is essentially self-contained and should also be accessible to students with a basic knowledge of differentiable manifolds and covering spaces. Contents Introduction Chapter I Preliminaries 1. Riemannian manifolds 2. Almost complex and sympletic manifolds 3. J -holomorphic maps 4. Riemann surfaces and hyperbolic geometry 5. Annuli Chapter II Estimates for area and first derivatives 1. Gromov's Schwarz- and monotonicty lemma 2. Area of J -holomorphic maps 3. Isoperimetric inequalities for J -holomorphic maps 4. Proof of the Gromov-Schwarz lemma Chapter III Higher order derivatives 1. 1-jets of J -holomorphic maps 2. Removal of singularities 3. Converging sequences of J -holomorphic maps 4. Variable almost complex structures Chapter IV Hyperbolic surfaces 1. Hexagons 2. Building hyperbolic surfaces from pairs of pants 3. Pairs of pants decomposition 4. Thick-thin decomposition 5. Compactness properties of hyperbolic structures Chapter V The compactness theorem 1. Cusp curves 2. Proof of the compactness theorem 3. Bubbles Chapter VI The squeezing theorem 1. Discussion of the statement 2. Proof modulo existence result for pseudo-holomorphic curves 3. The analytical setup: A rough outline 4. The required existence result Appendix A The classical isoperimetric inequality Appendix B The C k -topology References on pseudo-holomorphic curves Bibliography Index Series: Progress in Mathematics, Volume 151