Extension and convergence theorems of pseudoholomorphic maps
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 of the paper, in section 3, the authors generalize Noguchi's Theorem [loc. cit.] for almost complex manifolds (cf. Theorem 3.1) and they offer other variants (cf. Theorems 3.2, 3.3 and 3.7). In section 4, by introducing the notion of relative pseudodistance, they are able to prove that when \((\mathbb C, J)\) is a pseudoholomorphic curve in an almost complex manifold \((S, J)\) of real dimension 4, then the pseudodistance w.r.t. \(S\backslash \mathbb C\) is a distance on \(S\) iff \((S\backslash \mathbb C, J)\) is hyperbolically embedded in (S, J). In the last section, the authors prove the finiteness of the automorphism group of \(S\backslash \mathbb C\), where \((S, J)\) is a compact almost complex manifold of real dimension 4 and \((\mathbb C, J)\) is a pseudoholomorphic curve in \((S, J)\) such that \((S\backslash \mathbb C, J)\) is hyperbolically embedded in \((S, J)\).
scientific article; zbMATH DE number 5644239
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