Gromovs Compactness Theorem for Pseudo-holomorphic Curves [Paperback]

This book presents 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.

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. Gromovs 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 hyperblS8