On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel)
type: Item
instance of: scholarly article
title: On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel)
zbMATH Open document ID: 0829.53030
DOI: 10.1007/BF01245189
author: Jürgen Moser
published in: Inventiones Mathematicae
publication date: 1996-01-22T00:00:00Z
full work available at URL: https://eudml.org/doc/144262
review text: In this paper, the non-compact pseudo-holomorphic curves on an almost complex torus \((T^{2n},J)\) are studied. These curves carry a complex structure and, therefore, represent Riemannian surfaces. The author is interested in the question of the persistence of these curves under perturbation of the almost complex structure. He considers a family of almost complex structures \[ J^\varepsilon = J^0 + \varepsilon J^1 (x, \varepsilon), \quad (J^\varepsilon)^2 = -\text{id} \] depending continuously on a parameter \(\varepsilon\). Assuming that \(J^0\) is constant, the question to solve is: Can a \(J^0\)-holomorphic curve, say \(f(\xi) = \alpha \xi + \overline {\alpha} \overline {\xi}\), be extended to a \(J^\varepsilon\)-holomorphic curve \(f^\varepsilon (\xi)\) with \(f^0 (\xi) = f(\xi)\) and similar topological structure? If such a continuation exists for every \(J^1\), the author speaks of persistence of \(f^0\). The main results of this paper are concerning foliations of holomorphic curves, including Bangert's theorem, and persistence of non-compact pseudo-holomorphic curves. Some examples are also exhibited. The paper includes an appendix of J. Pöschel who provides results which yield the proof of the main theorem.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 783995
zbMATH Keywords: persistence under perturbation
zbMATH Keywords: pseudo-holomorphic curves
zbMATH Keywords: almost complex structures
zbMATH Keywords: foliations of holomorphic curves
zbMATH Keywords: Bangert's theorem
Wikidata QID: Q110897020
MaRDI profile type: Publication
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OpenAlex ID: W2167075206
description: scientific article; zbMATH DE number 783995