CR maps and point Lie transformations
The motivation of the paper under review lies in the following well-known finite jet determination result which follows from the work of \textit{S. S. Chern} and \textit{J. K. Moser} [Acta Math. 133, 219-271 (1974; Zbl 0302.32015)]. Real-analytic CR automorphisms of Levi-nondegenerate real-analytic hypersurfaces in complex space are determined by their 2-jets at any given point. This kind of uniqueness result in CR geometry has been generalized to much wider classes of real-analytic hypersurfaces (and even CR manifolds of arbitrary codimension) by \textit{M. S. Baouendi}, \textit{L. P. Rothschild} and the reviewer [J. Geom. Anal. 12, No. 4, 543-580 (2002)]. In the paper under review, the author proposes a new approach in deriving Chern-Moser's uniqueness result by studying Lie symmetries of certain holomorphic second-order partial differential equations. Indeed, by a fundamental observation due to \textit{S. Webster} [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)], Segre varieties are invariant under biholomorphic mappings and hence such mappings can be seen as Lie symmetries of associated second-order partial differential equation (PDE) systems. The finite jet determination result of Chern-Moser can then be seen as a consequence of a 2-jet determination property for a certain class of PDE systems (Theorem 1.1). The well-known above-mentioned uniqueness result seems however the only application to CR geometry that the author can offer in the paper under review.
Levi-nondegenerate real-analytic hypersurfaces
scientific article; zbMATH DE number 1943226
scientific article