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AuthorHummel, Christoph. author
TitleGromov's Compactness Theorem for Pseudo-holomorphic Curves [electronic resource] / by Christoph Hummel
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1997
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Descript VIII, 135 p. online resource


Mikhail Gromov introduced pseudo-holomorphic curves into symplectic 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 thickthin 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


I Preliminaries -- 1. Riemannian manifolds -- 2. Almost complex and symplectic manifolds -- 3. J-holomorphic maps -- 4. Riemann surfaces and hyperbolic geometry -- 5. Annuli -- II Estimates for area and first derivatives -- 1. Gromov's Schwarz- and monotonicity lemma -- 2. Area of J-holomorphic maps -- 3. Isoperimetric inequalities for J-holomorphic maps -- 4. Proof of the Gromov-Schwarz lemma -- 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 -- 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 -- V The compactness theorem -- 1. Cusp curves -- 2. Proof of the compactness theorem -- 3. Bubbles -- 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 -- References on pseudo-holomorphic curves

Mathematics Geometry Mathematics Geometry


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