Gromov's compactness theorem for pseudo-holomorphic curves
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-holomorphic curves and he presents Gromov's proof of that theorem in detail. In Chapter 6, the author describes an application of pseudo-holomorphic curves and of the compactness theorem in symplectic geometry, namely Gromov's squeezing theorem. This last chapter is more of a survey nature. The book is very well written, and can be used as a reference to Gromov's compactness theorem.
scientific article; zbMATH DE number 1012989