Some totally real embeddings of three-manifolds
type: Item
instance of: scholarly article
title: Some totally real embeddings of three-manifolds
zbMATH Open document ID: 0586.32027
DOI: 10.1007/BF01168610
author: Franc Forstnerič
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155124
review text: Let \(\Omega \subset {\mathbb{C}}^ 3\) be an open subset, \(h: \Omega \to {\mathbb{C}}^ a \)holomorphic function and \(\rho: \Omega\to{\mathbb{R}}^ a\) \(C^ k\)-function \((k\geq 1)\) such that \(dh\wedge d\bar h\wedge d\rho\) does not vanish on \(M=\{x\in \Omega | h(x)=\rho (x)=0\}.\) The main result is: There exists a small \(C^ 1\)-perturbation \(\tilde M\) of M such that \(\tilde M\) is a totally real submanifold of \({\mathbb{C}}^ 3.\) Moreover \(\tilde M\) has a rather explicit form. The result is a conseqence of a theorem of \textit{M.L. Gromov} [Math. USSR Izv. 7(1973), 329-343 (1974; Zbl 0281.58004)]. As an application, certain totally real embeddings in \({\mathbb{C}}^ 3\) of quotients of \(S^ 3\) are shown to exist. The case of \(S^ 3\) itself was treated by \textit{P. Ahern} and \textit{W. Rudin} [Proc. Am. Math. Soc. 94, 460-462 (1985; Zbl 0567.32006)].
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 3939711
zbMATH Keywords: totally real embeddings in \({\mathbb{C}}^ 3\)
reviewed by: Nicolae Mihalache
MaRDI profile type: Publication
cites work: CONVEX INTEGRATION OF DIFFERENTIAL RELATIONS. I
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description: scientific article; zbMATH DE number 3939711