Cubic model \(CR\)-manifolds without the assumption of complete nondegeneracy
type: Item
instance of: scholarly article
title: Cubic model \(CR\)-manifolds without the assumption of complete nondegeneracy
zbMATH Open document ID: 1405.32061
DOI: 10.1134/S1061920818020024
published in: Russian Journal of Mathematical Physics
publication date: 2018-10-02T00:00:00Z
review text: 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 holomo…
reviewed by: Masoud Sabzevari
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 6946512
zbMATH Keywords: CR manifolds
zbMATH Keywords: nondegeneracy
zbMATH Keywords: cubic models
author: Valeriĭ K. Beloshapka
MaRDI profile type: Publication
cites work: On``type'' conditions for generic real submanifolds of \(\mathbb{C}^n\)
cites work: Universal models for real submanifolds
cites work: A cubic model of a real manifold.
cites work: The envelope of holomorphy of a model third-degree surface and the rigidity phenomenon
cites work: Q4230686
cites work: Infinitesimal CR Automorphisms of Real Hypersurfaces
cites work: CR automorphisms of real analytic manifolds in complex space
cites work: Q3791444
cites work: Einige Bemerkungen über polynomiale Vektorfelder, Jordanalgebren und die Automorphismen von Siegelschen Gebieten. (Some remarks on polynomial vector fields)
cites work: Q3801872
cites work: Automorphisms of degenerate hypersurfaces in \(\mathbb{C}^2\) and a dimension conjecture
full work available at URL: s1061920818020024
OpenAlex ID: W2805246902
Recommended article: CR-manifolds of finite Bloom-Graham type: the model surface method
Recommended article: The dimension conjecture: solution and future prospects
Recommended article: Polynomial model \textit{CR}-manifolds with the rigidity condition
Recommended article: Holomorphic classification of four-dimensional surfaces in $ \mathbb C^3$
Recommended article: Vitushkin's germ theorem for Engel-type CR manifolds
Recommended article: Classification of homogeneous CR-manifolds in dimension 4
Recommended article: Q3807539
Recommended article: On equivalence problem for 2-nondegenerate CR geometries with simple models
Recommended article: NORMAL SYSTEMS OF COORDINATES ON MANIFOLDS OF CHERN-MOSER TYPE
Recommended article: Chern classes of spherical CR manifolds
description: scientific article; zbMATH DE number 6946512