On the unique continuation problem for CR mappings into nonminimal hypersurfaces
Let \(M\) and \(M'\) be germs at 0 of \(C^\infty\) real hypersurfaces of \(\mathbb{C}^{n+1}\) and \(\mathbb{C}^{n'+1}\) respectively. As such we can assume \(M\) is given by \(Im w= \Phi(Z,\overline Z), Rew)\) for a \(C^\infty\) function \(\Phi: \mathbb{C}^{n+1}\to\mathbb{R}\) in coordinates \((Z,w)\) for \(\mathbb{C}^n\times\mathbb{C}\), and \(M'\) is similarly represented by \(\Phi':\mathbb{C}^{n'+1}\to\mathbb{R}\) in coordinates \((Z', w')\) for \(\mathbb{C}^{n'}\times \mathbb{C}\). Let \(H=(f_1, \dots, f_{n'},g)\) be a CR function from \(M\) to \(M'\). The author is interested in the question of whether infinite order of vanishing of \(H\) at 0 \((H\) ``flat'' at 0) forces \(H\) to be identically 0. The author obtains the following results. If \(M'\) contains a holomorphic hypersurface (Hence \((Z',w')\) can be choosen so that \(\Sigma'= \{w'= 0\}\subset M'.)\), \(H\) extends holomorphically to one side of \(M\), and \(H(M) \not\subset\Sigma'\) (Hence \(g\not\equiv 0.)\), then \(g\) is not flat at 0. (Hence \(H\) is not flat.) The author points out that any CR map \(f:M\to\mathbb{C}^{n'}\) yields a CR map \(H=(f,0):M\to M'\) which can be flat but not identically 0, hence the necessity of the hypothesis \(g\not\equiv 0\). The author continues the investigation without the hypothesis that \(H\) extends holomorphically to one side of \(M\). In case \(M'\) is Levi flat he shows there is a nonconstant CR mapping \(H:M\to M'\) which is flat at the origin. Further he provides an example of non-Levi flat \(M\) and \(M'\) in \(\mathbb{C}^2\) for which such a CR mapping exists. Finally in case \(M\) and \(M'\) are real analytic the author obtains sufficient condition for \(H\) flat to imply \(H\) constant. There conditions involve \(M\) being ``sufficiently far from being Levi-flat''.
scientific article; zbMATH DE number 1160126
scientific article