Extension and convergence theorems of pseudoholomorphic maps
type: Item
instance of: scholarly article
title: Extension and convergence theorems of pseudoholomorphic maps
zbMATH Open document ID: 1190.32022
author: Fathi Haggui
author: Adel Khalfallah
published in: Osaka Journal of Mathematics
publication date: 2009-12-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1256564208
review text: The authors present two main purposes for their research paper: The first purpose is to generalize the Adachi Theorem (cf. Theorem 2.10) and the second purpose is to generalize the Noguchi Theorem for almost complex manifolds for which they also offer other variants. Concerning the first purpose, the authors generalize the Picard theorem for pseudoholomorphic maps. The big Picard Theorem states that any holomorphic map \(f\) from the punctured disc \(\Delta^*\) into \(\mathbb C\backslash \{0,1\}\) can be extended to a holomorphic map \(f : \Delta \rightarrow \mathbb P_1 (\mathbb C)\). This result has been extended to higher dimensional settings [see \textit{P. Kierman} Trans. Am. Math. Soc. 172(1972), 347--355 (1973; Zbl 0255.32014), \textit{S. Kobayashi} Hyperbolic manifolds and holomorphic mappings. (1970; Zbl 0207.37902), \textit{M. H. Kwack} Ann. Math. (2) 90, 9--22 (1969; Zbl 0179.12103)] by introducing the notion of hyperbolic embeddedness and \textit{Y. Adachi} [Kodai Math. J. 18, No.~3, 408--424 (1995; Zbl 0852.32012), cf. Theorem 2.2] proved an extension theorem for holomorphic mappings. Since Adachi's proof is heavily based on the Riemann extension theorem and winding arguments, the authors are forced to substantial modifications in order to apply it to the almost complex case. In section 2.1, they investigate the extension of pseudoholomorphic maps in different situations. Their proof of Theorem 2.10 in section 2.2 is based on the so-called Gromov monotonicity lemma, Corollary 2.9 and Proposition 2.5 (results obtained in section 2.1). In section 2.2 (cf. Theorem 2.16), they give a generalization of a result proved by \textit{J. Noguchi} in the complex case [cf. Invent. Math. 93, No.~1, 15--34 (1988; Zbl 0651.32012), Lemma 2.4, p.~21]. In section 2.3, they prove an interesting result (cf. Theorem 2.17) which deals with the study of the extension of pseudoholomorphic maps into hyperbolically embedded almost complex manifolds. Concerning the second purpose o…
reviewed by: Saeid Jafari
Mathematics Subject Classification ID: 32Q60
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32Q65
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32T35
Mathematics Subject Classification ID: 32A19
Mathematics Subject Classification ID: 32H25
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 5644239
zbMATH Keywords: almost complex manifold
zbMATH Keywords: holomorphic map
zbMATH Keywords: peudoholomorphic map
zbMATH Keywords: the big Picard theorem
zbMATH Keywords: Adachi theorem
zbMATH Keywords: Noguchi theorem
zbMATH Keywords: relative intrinsic psudodistance
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5644239