عنوان

Gromov's Compactness Theorem for Pseudo-holomorphic Curves

(Number (ISBN

9783034889520

(Number (ISBN

9783034898423

Number

b406309

Title Proper

Gromov's Compactness Theorem for Pseudo-holomorphic Curves

General Material Designation

[Book]

First Statement of Responsibility

by Christoph Hummel.

Place of Publication, Distribution, etc.

Basel :

Name of Publisher, Distributor, etc.

Imprint: Birkhäuser,

Date of Publication, Distribution, etc.

1997.

Series Title

Progress in Mathematics ;

Volume Designation

151

Text of Note

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.

0

Text of Note

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.

International Standard Book Number

9783034898423

Title

Springer eBooks

Mathematics.

Hummel, Christoph.

SpringerLink (Online service)

Date of Transaction

20190301082800.0

Electronic name

[Book]

Y