Analytic discs attached to a generating CR-manifold. (English) Zbl 0856.32012

Černe, Miran

Ark. Mat. 33, No. 2, 217-248 (1995).

The author gives a complete finite-dimensional parametrization of perturbations by analytic discs with their boundaries kept in a generating CR-fibration \(\{M (\xi) \}_{\xi \in \partial D}\), \(M (\xi) \subset \mathbb{C}^{m + n}\) over the unit circle \(\partial D\), under the condition that the partial indices of the original path are all greater than or equal to \(-1\). Here CR-bundle means that each fiber is a real vector subspace of CR-dimension \(m\), and “generating” means that each fiber is a generating subspace of \(\mathbb{C}^{m + n}\). This is a generalization of results by Globevnik and other people (including the author himself) in case when the boundaries are kept in a maximal real submanifold of \(\mathbb{C}^n\), which corresponds to \(m = 0\) in the above. This problem has applications such as the determination of the polynomial hull or the extension of CR-functions. Some geometric properties of the perturbations are discussed, e.g. the dimension of the subspaces of cotangent vectors of the perturbations conormal to the original fibers is shown to be constant along the circle.

Reviewer: A.Kaneko (Komaba/Meguro-ku)

MSC:

32V05	CR structures, CR operators, and generalizations
32V99	CR manifolds
32D10	Envelopes of holomorphy
32C25	Analytic subsets and submanifolds

Keywords:

CR-manifold; analytic discs; CR-fibration

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