On the geometry of analytic discs attached to real manifolds
Let \(M\) be a (germ of) generic submanifold(s) at \(O \in \mathbb{C}^ n\), of real codimension \(\ell\); thus \(M = \{\zeta_ 1 = \cdots = \zeta_ \ell = 0\}\) where \(\partial \zeta_ 1 \wedge \cdots \wedge \partial \zeta_ \ell \neq 0\). An analytic disc attached to \(M\) at \(O\) is a continuous map \(A : \overline \Delta \to \mathbb{C}^ N\), where \(\overline \Delta = \Delta\cup S^ 1\) is the closed unit disc, \(A | \Delta\) is holomorphic, \(A(S^ 1) \subset M\), and \(A(1) = 0\). In this paper the authors assume that \(A \in C^{1, \alpha} (\overline \Delta)\), for a fixed \(\alpha \in (0,1)\), and study the set of all ``small'' discs attached to \(M\) at \(O\) (i.e. those which are close to the constant disc at \(O\) in \(C^{1, \alpha}\)-norm). They show that it is a smooth submanifold (of infinite dimension) in the Banach space of all discs at \(O\) in \(\mathbb{C}^ N\), and determine its tangent space. Using only the implicit function theorem for Banach manifolds, the authors are able to recover and improve earlier results of the third author [Math. Z. 211, No. 1, 93-103 (1992; Zbl 0759.32011)] and \textit{A. E. Tumanov} [Math. USSR, Sb. 64, No. 1, 129-140 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 1(5), 128-139 (1988; Zbl 0692.58005)] on holomorphic extensions of CR-functions and propagation of analyticity without resorting to micro-local analysis or explicitly solving Bishop's equation.
scientific article; zbMATH DE number 590835
scientific article