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Some totally real embeddings of three-manifolds. (English) Zbl 0586.32027
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 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 P. Ahern and W. Rudin [Proc. Am. Math. Soc. 94, 460-462 (1985; Zbl 0567.32006)].
Reviewer: N.Mihalache

32V40 Real submanifolds in complex manifolds
57R40 Embeddings in differential topology
Full Text: DOI EuDML
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