Cubic model \(CR\)-manifolds without the assumption of complete nondegeneracy
In [Rend. Circ. Mat. Palermo 23, 185--220 (1907; JFM 38.0459.02)], \textit{H. Poincaré} initiated his celebrated approach of studying real hypersurfaces of the complex space \(\mathbb C^2\) by means of the associated model surface, namely the Heisenberg sphere. Several years later, \textit{S. S. Chern} and \textit{J. K. Moser} in their seminal work [Acta Math. 133 (1974), 219--271 (1975; Zbl 0302.32015)] notably developed this approach by associating appropriate models to nondegenerate real \textit{hypersurfaces} in \textit{arbitrary} complex spaces. Afterwards, the author [Math. Notes 70, No. 4, 457--470 (2001); translation from Mat. Zametki 70, No. 4, 503--519 (2001; Zbl 1033.32027)] established his universal model surfaces associated with \textit{totally nondegenerate} CR manifolds of \textit{arbitrary} CR dimensions and codimensions and designed an effective method to construct their defining equations. In this framework, many questions about automorphism groups, classification, invariants and others can be reduced to similar problems about the associated models. The main idea behind the paper under review is actually along the lines of the approach introduced by the author in the above-mentioned paper [loc. cit.]. More precisely, in the present paper the author endeavors to find appropriate model surfaces associated with (not necessarily totally) \textit{nondegenerate} and generic \textit{cubic} CR manifolds with Hörmander numbers \((2, 3)\) and with arbitrary multiplicities. His approach is based upon a certain weight analysis of the determining equations of nondegenerate submanifolds as described in Sections III and IV of the famous book [\textit{M. S. Baouendi} et al., Real submanifolds in complex space and their mappings. Princeton, NJ: Princeton University Press (1999; Zbl 0944.32040)]. Accordingly, he finds some interesting results concerning the introduced cubic models. Among them, there are some simple criteria for the finiteness of the type, the holomorphic nondegeneracy and the holomorphic homogeneity of the models. Moreover, the author proves that these cubic models are the most symmetric ones in the class of all nondegenerate cubic submanifolds. Although this paper is devoted to the specific case of cubic models (with Hörmander numbers \((2,3)\)), it can be regarded as a perfect starting point to study model surfaces associated with arbitrary nondegenerate CR manifolds.
scientific article; zbMATH DE number 6946512
scientific article