On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel)
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.
scientific article; zbMATH DE number 783995
scientific article