Pseudoholomorphic 2-tori in \(\mathbb{T}^ 4\)
type: Item
instance of: scholarly article
title: Pseudoholomorphic 2-tori in \(\mathbb{T}^ 4\)
zbMATH Open document ID: 0866.53043
DOI: 10.1016/0926-2245(96)89144-3
published in: Differential Geometry and its Applications
publication date: 1997-07-20T00:00:00Z
review text: In 1985, \textit{M. Gromov} published the first work on pseudo-holomorphic (PH) curves [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] as a powerful tool to study the geometry of a manifold. Attention was also given to manifolds formed by PH curves: in particular, PH spheres. In the interesting paper under review, the author studies PH-2 tori, using, á la Gromov, a special Hilbert manifold consisting of a pair: (tamed almost complex structure on \(T^2\), PH torus corresponding to it), and its projection \(\pi\) to the manifold of tamed almost complex structures. A direct analogy to this method was used by the author et al. to study solutions to nonlinear elliptic equations [\textit{M. I. Vishnik} and \textit{S. B. Kuksin}, Math. USSR, Sb. 58, 223-243 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 2(6), 222-242 (1986; Zbl 0625.35029)]. In the first two sections, the author gives the basic definitions and setting. The main section is the third one. After defining PH tori, he shows that \(\pi\) is a Fredholm map of such index that the image and its complement both have nonempty interior. He infers therefore that a tamed almost complex 4-torus typically has an even finite number of PH 2-tori. He also analyses the case where this number is either zero or nonzero and proves that both are typical, that is to say, they both happen for open nonempty sets of almost complex structures.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 897470
zbMATH Keywords: pseudoholomorphic torus
zbMATH Keywords: Hilbert manifold
zbMATH Keywords: tamed almost complex structures
Wikidata QID: Q115362491
author: Sergei Kuksin
reviewed by: L. Del Riego
MaRDI profile type: Publication
full work available at URL: 0926-2245(96)89144-3
OpenAlex ID: W2015507952
cites work: Uniqueness in Cauchy problems for elliptic systems of equations
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cites work: Pseudo holomorphic curves in symplectic manifolds
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cites work: Elliptic methods in symplectic geometry
cites work: On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel)
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description: scientific article; zbMATH DE number 897470