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Convergent Normal Form for Five Dimensional Totally Nondegenerate CR Manifolds in \(\pmb {{\mathbb {C}}^4}\)

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Abstract

Applying the equivariant moving frames method, we construct a convergent normal form for real-analytic 5-dimensional totally nondegenerate submanifolds of \({\mathbb {C}}^4\). We develop this construction by applying further normalizations, the possibility of which completely relies upon vanishing/non-vanishing of some specific coefficients of the normal form. This in turn divides the class of our CR manifolds into several biholomorphically inequivalent subclasses, each of them has its own specified normal form with no further possible normalization applicable on it. It also is shown that, biholomorphically, Beloshapka’s cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR manifolds, in question.

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Notes

  1. There is some technical reason for indicating a bit unnaturally the indices j as \(u_j\) and \(v^j\).

  2. Contact forms play no essential role in this study.

  3. Indeed, the reason is that the degree of indeterminancy of the system (49) is zero (cf. [23, Definition 11.2]).

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Acknowledgements

The author gratefully expresses his sincere thanks to Peter Olver and Francis Valiquette for their helpful comments and discussions during the preparation of this paper. He is also very grateful to Maria Stepanova who noted via a counterexample the existence of a certain error concerning the results of Branch 3-3-1 in the first version of this paper. Grateful thanks are also addressed to an anonymous referee for a careful reading and for insightful suggestions. The research of the author was supported in part by a grant from IPM, No. 98510420.

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Authors and Affiliations

  1. Department of Mathematics, Shahrekord University, Shahrekord, 88186-34141, Iran

    Masoud Sabzevari

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, Iran

    Masoud Sabzevari

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  1. Masoud Sabzevari
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Correspondence to Masoud Sabzevari.

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Dedicated to Professor Peter J. Olver.

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Sabzevari, M. Convergent Normal Form for Five Dimensional Totally Nondegenerate CR Manifolds in \(\pmb {{\mathbb {C}}^4}\). J Geom Anal 31, 7900–7946 (2021). https://doi.org/10.1007/s12220-020-00558-0

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