2000/09/01 by Peter Ebenfelt, Ebenfelt, Peter
Mathematics · #32H40 #32V10 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32H40 #msc:32V10
paper · pdf · doi:10.48550/arxiv.math/0009010
19pp; LaTeX
arxiv created 2000/09/01 · arxiv updated 2009/11/30
Let M be a connected real-analytic hypersurface in two dimensional complex space, \mathbb C2, containing a connected complex hypersurface E, and let f be a smooth CR mapping sending M into another real-analytic hypersurface M' in \mathbb C2. In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E.