Hyperbolic embeddedness and extension-convergence theorems of \(J\)-holomorphic curves
type: Item
instance of: scholarly article
title: Hyperbolic embeddedness and extension-convergence theorems of \(J\)-holomorphic curves
zbMATH Open document ID: 1185.32017
author: Fathi Haggui
author: Adel Khalfallah
published in: Mathematische Zeitschrift
publication date: 2009-06-16T00:00:00Z
review text: This paper deals with Kobayashi hyperbolicity for almost complex manifolds. The analogue of the Kobayashi semi-distance is defined using \(J\)-holomorphic curves and hyperbolicity means that it is a true distance. In the paper under review several results that are classical in the complex case are generalized to this setting. \S 2 deals with hyperbolic embeddings, which are defined as usual (see. Def. 2.1). Denote by \(\mathcal {O}_J(\Delta,M)\) the set of \(J\)-holomorphic discs in the almost complex manifolds \((M,J)\). As in the complex case, a relatively compact almost complex submanifold \(M\) of an almost complex manifold \((N,J)\) is hyperbolically embedded iff \(\mathcal{O}_J(\Delta, M)\) is relatively compact in \(\mathcal{O}_J(\Delta, N)\). Other equivalent conditions are given in terms of length functions (see Thm. 1) and of limit \(J\)-complex lines (see Def. 2.5 and Thm 2). Next in \S 3 the stability of hyperbolically embedded submanifolds is considered under a small perturbation of the almost complex structure. Let \((M,J)\) be a compact almost complex manifold and let \(D \subset M\) be a \(J\)-holomorphic hypersurface. If \((D,J)\) is hyperbolic and \(M\setminus D\) is hyperbolically embedded, then the same holds for almost complex structures close to \(J\) in the \(\mathcal{C}^\infty\) topology (Thm. 3). \S\S 4-5 deal with extension of \(J\)-holomorphic maps from the pointed disc. Results of \textit{P. Kiernan} [Value-Distrib. Theory, Proc. Tulane Univ. Progr. Value-Distrib. Theory complex Analysis related Topics differ. Geom., Part A, 97-107 (1974; Zbl 0292.32018)] and \textit{J. Noguchi} [Invent. Math. 93, No.1, 15--34 (1988; Zbl 0651.32012)] are generalized to the almost complex setting. The proofs are completely different.
reviewed by: A. Ghigi
Mathematics Subject Classification ID: 32Q65
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32Q60
zbMATH DE Number: 5565291
zbMATH Keywords: Pseudoholomorphic curves
zbMATH Keywords: almost complex manifolds
zbMATH Keywords: hyperbolic embedding
MaRDI profile type: Publication
full work available at URL: s00209-008-0378-6
OpenAlex ID: W2074679894
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DOI: 10.1007/S00209-008-0378-6
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description: scientific article; zbMATH DE number 5565291