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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 1(343), Pages 3–44 (Mi umn474)  

This article is cited in 31 scientific papers (total in 31 papers)

Real submanifolds in complex space: polynomial models, automorphisms, and classification problems

V. K. Beloshapka

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This is a survey of results on the local theory of real submanifolds of a complex space. Most of the results included here were obtained in Vitushkin's seminar at Moscow State University over the last fifteen years. The most important achievement is a technique for computing automorphisms, invariants, and classifications of real submanifolds, which includes as a main step the construction of a “good” model surface (an analogue of an osculating paraboloid in classical differential geometry).


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English version:
Russian Mathematical Surveys, 2002, 57:1, 1–41

Bibliographic databases:

UDC: 517.55+514.76
MSC: Primary 32V40; Secondary 32V05, 32C05, 32H02, 32M05, 32B10, 32S25
Received: 03.12.2001

Citation: V. K. Beloshapka, “Real submanifolds in complex space: polynomial models, automorphisms, and classification problems”, Uspekhi Mat. Nauk, 57:1(343) (2002), 3–44; Russian Math. Surveys, 57:1 (2002), 1–41

Citation in format AMSBIB
\by V.~K.~Beloshapka
\paper Real submanifolds in complex space: polynomial models, automorphisms, and classification problems
\jour Uspekhi Mat. Nauk
\yr 2002
\vol 57
\issue 1(343)
\pages 3--44
\jour Russian Math. Surveys
\yr 2002
\vol 57
\issue 1
\pages 1--41

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    This publication is cited in the following articles:
    1. Kruglikov B., “Blow-Ups and Infinitesimal Automorphisms of Cr-Manifolds”, Math. Z.  crossref  mathscinet  isi
    2. Caglioti E., Chernov N., Lebowitz J.L., “Stability of solutions of hydrodynamic equations describing the scaling limit of a massive piston in an ideal gas”, Nonlinearity, 17:3 (2004), 897–923  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. V. K. Beloshapka, “Universal Models For Real Submanifolds”, Math. Notes, 75:4 (2004), 475–488  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. N. G. Kruzhilin, P. A. Soldatkin, “Affine and Holomorphic Equivalence of Tube Domains in $\mathbb C^2$”, Math. Notes, 75:5 (2004), 623–634  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. K. Beloshapka, “Symmetries of Real Hypersurfaces in Complex 3-Space”, Math. Notes, 78:2 (2005), 156–163  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. R. V. Gammel', I. G. Kossovskii, “The Envelope of Holomorphy of a Model Third-Degree Surface and the Rigidity Phenomenon”, Proc. Steklov Inst. Math., 253 (2006), 22–36  mathnet  crossref  mathscinet  elib
    7. I. G. Kossovskii, “On envelopes of holomorphy of model manifolds”, Izv. Math., 71:3 (2007), 545–571  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. V. K. Beloshapka, “Representation of the Group of Holomorphic Symmetries of a Real Germ in the Symmetry Group of Its Model Surface”, Math. Notes, 82:4 (2007), 461–463  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Beloshapka V.K., “Poincaré's program as an alternative to Klein's (centenary of the publication)”, Russ. J. Math. Phys., 14:4 (2007), 498–500  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Baouendi M.S., Rothschild L.P., Zaitsev D., “Deformation of generic submanifolds in a complex manifold”, Adv. Math., 214:1 (2007), 157–180  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    11. Beloshapka V.K., “Representation of the group of holomorphic symmetries of a real germ in the symmetry group of the model surface of the germ”, Russ. J. Math. Phys., 14:2 (2007), 213–215  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    12. Valerii K. Beloshapka, “Programma A. Puankare kak alternativa programme F. Kleina (k 100-letiyu publikatsii programmy)”, Zhurn. SFU. Ser. Matem. i fiz., 1:1 (2008), 63–67  mathnet
    13. Mamai I.B., “Model CR-manifolds with one-dimensional complex tangent”, Russ. J. Math. Phys., 16:1 (2009), 97–102  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Wu Qingyan, “On holomorphic automorphisms of a class of non-homogeneous rigid hypersurfaces in $\mathbb C^{N+1}$”, Chinese Annals of Mathematics Ser. B, 31:2 (2010), 201–210  crossref  mathscinet  zmath  isi  scopus  scopus
    15. Beloshapka V.K., Kossovskiy I.G., “Homogeneous Hypersurfaces in C-3, Associated with a Model CR-Cubic”, Journal of Geometric Analysis, 20:3 (2010), 538–564  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Beloshapka V.K., Kossovskiy I.G., “Classification of homogeneous CR-manifolds in dimension 4”, Journal of Mathematical Analysis and Applications, 374:2 (2011), 655–672  crossref  mathscinet  zmath  isi  scopus  scopus
    17. V. K. Beloshapka, “Model-surface method: An infinite-dimensional version”, Proc. Steklov Inst. Math., 279 (2012), 14–24  mathnet  crossref  mathscinet  isi  elib
    18. I. B. Mamai, “Moduli spaces of model surfaces with one-dimensional complex tangent”, Izv. Math., 77:2 (2013), 354–377  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. Alexander Isaev, Dmitri Zaitsev, “Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms”, J Geom Anal, 2013  crossref  mathscinet  isi  scopus  scopus
    20. J. Merker, M. Sabzevari, “The Cartan equivalence problem for Levi-non-degenerate real hypersurfaces $M^3\subset\mathbb C^2$”, Izv. Math., 78:6 (2014), 1158–1194  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. Kossovskiy I., Zaitsev D., “Convergent Normal Form and Canonical Connection For Hypersurfaces of Finite Type in C-2”, 281, 2015, 670–705  crossref  mathscinet  zmath  isi  scopus  scopus
    22. Merker J., “Rationality in Differential Algebraic Geometry”, Complex Geometry and Dynamics, Abel Symposia, eds. Fornaess J., Irgens M., Wold E., Springer, 2015, 157–209  crossref  mathscinet  zmath  isi
    23. Kossovskiy I., Shafikov R., “Analytic Differential Equations and Spherical Real Hypersurfaces”, 102, no. 1, 2016, 67–126  mathscinet  zmath  isi
    24. Isaev A., Kruglikov B., “A short proof of the Dimension Conjecture for real hypersurfaces in $\mathbb {C}^2$”, Proc. Amer. Math. Soc., 144:10 (2016), 4395–4399  crossref  mathscinet  zmath  isi  elib  scopus
    25. Sabzevari M., Hashemi A., Alizadeh B.M., Merker J., “Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN”, Filomat, 30:6 (2016), 1387–1411  crossref  mathscinet  zmath  isi  elib  scopus
    26. Isaev A., Kruglikov B., “On the Symmetry Algebras of 5-Dimensional Cr-Manifolds”, Adv. Math., 322 (2017), 530–564  crossref  mathscinet  zmath  isi  scopus  scopus
    27. Kolar M., Kossovskiy I., Zaitsev D., “Normal Forms in Cauchy-Riemann Geometry”, Analysis and Geometry in Several Complex Variables, Contemporary Mathematics, 681, eds. Berhanu S., Mir N., Straube E., Amer Mathematical Soc, 2017, 153–177  crossref  mathscinet  zmath  isi  scopus  scopus
    28. Kossovskiy I., Zaitsev D., “Convergent Normal Form For Real Hypersurfaces At a Generic Levi-Degeneracy”, J. Reine Angew. Math., 749 (2019), 201–225  crossref  mathscinet  zmath  isi  scopus
    29. M. A. Stepanova, “Ob avtomorfizmakh CR-podmnogoobrazii kompleksnogo gilbertova prostranstva”, Sib. elektron. matem. izv., 17 (2020), 126–140  mathnet  crossref
    30. Santi A., “Homogeneous Models For Levi Degenerate Cr Manifolds”, Kyoto J. Math., 60:1 (2020), 291–334  crossref  isi
    31. M. A. Stepanova, “Holomorphically homogeneous CR-manifolds and their model surfaces”, Izv. Math., 85:3 (2021), 529–535  mathnet  crossref  crossref  isi  elib
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