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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

##### MSC:
 32V40 Real submanifolds in complex manifolds 57R40 Embeddings in differential topology
##### Keywords:
totally real embeddings in $${\mathbb{C}}^ 3$$
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##### References:
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