Gromov's Compactness Theorem for Pseudo-Holomorphic Curves

By: Christoph Hummel

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Hardcover | 1 December 1996

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148 Pages

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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.

Industry Reviews

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

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Introduction	p. 1
Preliminaries
Riemannian manifolds	p. 5
Almost complex and symplectic manifolds	p. 7
J-holomorphic maps	p. 10
Riemann surfaces and hyperbolic geometry	p. 11
Annuli	p. 17
Estimates for area and first derivatives
Gromov's Schwarz- and monotonicity lemma	p. 21
Area of J-holomorphic maps	p. 22
Isoperimetric inequalities for J-holomorphic maps	p. 23
Proof of the Gromov-Schwarz lemma	p. 30
Higher order derivatives
I-jets of J-holomorphic maps	p. 35
Removal of singularities	p. 41
Converging sequences of J-holomorphic maps	p. 45
Variable almost complex structures	p. 46
Hyperbolic surfaces
Hexagons	p. 49
Building hyperbolic surfaces from pairs of pants	p. 55
Pairs of pants decomposition	p. 59
Thick-thin decomposition	p. 68
Compactness properties of hyperbolic structures	p. 70
The compactness theorem
Cusp curves	p. 79
Proof of the compactness theorem	p. 82
Bubbles	p. 91
The squeezing theorem
Discussion of the statement	p. 99
Proof modulo existence result for pseudo-holomorphic curves	p. 102
The analytical setup: A rough outline	p. 104
The required existence result	p. 110
The classical isoperimetric inequality	p. 115
The C[superscript k]-topology	p. 117
References on pseudo-holomorphic curves	p. 121
Bibliography	p. 125
Index	p. 129
Table of Contents provided by Blackwell. All Rights Reserved.

Gromov's Compactness Theorem for Pseudo-Holomorphic Curves

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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