On the geometry of analytic discs attached to real manifolds
type: Item
instance of: scholarly article
title: On the geometry of analytic discs attached to real manifolds
zbMATH Open document ID: 0821.32014
author: Linda Preiss Rothschild
author: Mohamed Salah Baouendi
author: Jean-Marie Trepreau
published in: Journal of Differential Geometry
publication date: 1994-07-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32D15
zbMATH DE Number: 590835
zbMATH Keywords: analytic disc
zbMATH Keywords: Banach space
zbMATH Keywords: implicit function theorem
Wikidata QID: Q115179773
reviewed by: R. R. Simha
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454877
OpenAlex ID: W1482319489
DOI: 10.4310/JDG/1214454877
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description: scientific article; zbMATH DE number 590835