Pseudoholomorphic 2-tori in \(\mathbb{T}^ 4\)
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.
scientific article; zbMATH DE number 897470
scientific article