Gromov's Compactness Theorem for Pseudo-Holomorphic Curves

ISBN-13: 9783764357351 / Angielski / Twarda / 1997 / 135 str.

Christoph Hummel

Opis

Mikhail Gromov introduced pseudo-holomorphic curves into symplectic geometry in 1985. This book aims to present in detail the original proof for Gromov's compactness theorum for pseudo-holomorphic curves. 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 thick-thin decomposition is described. This text should be accessible to students with a basic knowledge of differentiable manifolds and covering spaces.

Recenzje

 
  "...the book provides a self-contained and for the most part excellent elaboration of Gromov's proof of the compactness theorem."   
  -- Mathematical Reviews 

Spis treści

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.