Gromov’s Compactness Theorem for Pseudo-holomorphic Curves
Springer Basel (Verlag)
978-3-0348-9842-3 (ISBN)
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.
"...the book provides a self-contained and for the most part excellent elaboration of Gromov's proof of the compactness theorem."
-- Mathematical Reviews
| Erscheint lt. Verlag | 15.10.2012 |
|---|---|
| Reihe/Serie | Progress in Mathematics |
| Zusatzinfo | VIII, 135 p. |
| Verlagsort | Basel |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 237 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| Schlagworte | Geometry • manifold • Proof • Symplectic Geometry • Theorem |
| ISBN-10 | 3-0348-9842-8 / 3034898428 |
| ISBN-13 | 978-3-0348-9842-3 / 9783034898423 |
| Zustand | Neuware |
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