Gromov's compactness theorem for pseudo-holomorphic curves
type: Item
instance of: scholarly article
title: Gromov's compactness theorem for pseudo-holomorphic curves
zbMATH Open document ID: 0870.53002
author: Christoph Hummel
published in: Progress in Mathematics
publication date: 1997-05-26T00:00:00Z
review text: Let \(M\) be an almost complex manifold. A pseudo-holomorphic curve in \(M\) is a smooth map \(f:S\to M\), where \(S\) is a Riemann surface, with the property that at each point \(p\in S\), the differential \(Tf_p\) is complex linear, that is, satisfies the relation \(Tf\circ j=J\circ Tf\), where \(j\) (respectively \(J\)) denotes the complex structure on \(S\) (respectively on \(M\)). In his paper [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] \textit{M. Gromov} developed the study of pseudo-holomorphic curves in symplectic manifolds. In particular, he formulated existence and compactness results for families of such curves. The details of Gromov's theory were later on worked out by several authors, and the chronology of the subject is outlined in the introduction of the book under review. In this book, the author continues along the same line of thought and he studies pseudo-holomorphic curves in general almost complex manifolds. The outline of the book is as follows: Chapter 1 contains preliminaries and background results on Riemannian manifolds, almost complex manifolds, symplectic manifolds, Riemann surfaces, pseudo-holomorphic maps and hyperbolic geometry. Chapter 2 contains proofs of Gromov's monotonicity lemma and Gromov's Schwarz lemma for pseudo-holomorphic curves. The proofs use isoperimetric inequalities. Chapter 3 contains estimates of higher order derivatives of pseudo-holomorphic maps, with applications to a proof of a theorem of Gromov on the removal of singularities of pseudo-holomorphic maps. The author proves, as another application, that the derivatives of a locally uniformly convergent sequence of pseudo-holomorphic maps also converge locally uniformly, and he calls this result a ``generalized Weierstrass theorem''. Chapter 4 contains material about pairs of pants decompositions and the thin-thick decomposition for complete hyperbolic surfaces of finite area. In Chapter 5, the author formulates Gromov's compactness theorem for pseudo-holomor…
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 30F20
zbMATH DE Number: 1012989
zbMATH Keywords: pseudo-holomorphic curves
zbMATH Keywords: symplectic manifolds
zbMATH Keywords: almost complex manifolds
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: pseudo-holomorphic maps
zbMATH Keywords: Schwarz lemma
zbMATH Keywords: J-holomorphic maps
zbMATH Keywords: hyperbolic surfaces
zbMATH Keywords: cusp curves
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1012989