Images of real hypersurfaces under holomorphic mappings
This paper is devoted to the mapping problem for real analytic hypersurfaces in complex vector spaces. It is a continuation of the authors' investigations in J. Differ. Geom. 31, No. 2, 473-499 (1990; Zbl 0702.32014) and also of these one with \textit{S. R. Bell} in Duke Math. J. 56, No. 3, 503-530 (1988; Zbl 0655.32015). Given a germ of real analytic hypersurface \(M\) through 0 in \(\mathbb{C}^{n+I}\) and a germ \(H\) at 0 of a holomorphic mapping of \(\mathbb{C}^{n+I}\) into itself, the problem, whether there exists a real analytic hypersurface \(M'\) in \(\mathbb{C}^{n+I}\) such that \(H(M)\subset M'\), is completely solved for the case when \(M\) is essentially finite. The case when \(H(M)\) is a smooth real hypersurface is characterized for finite \(H\). Assuming that \(H(M)=M'\) is a smooth submanifold, it is proved that \(M'\) is real analytic. An example is given that the smoothness assumption for \(M\) cannot be weakened. Images of hypersurfaces under nonfinite \(H\) are studied too. There are many interesting details in the proposed proofs, as well as different comments and statements of open problems.
essentially finite analytic hypersurface
scientific article; zbMATH DE number 147457
scientific article