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Mat. Zametki, 2004, Volume 75, Issue 4, Pages 507–522 (Mi mz49)  

This article is cited in 37 scientific papers (total in 37 papers)

Universal Models For Real Submanifolds

V. K. Beloshapka

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

Abstract: In previous papers by the present author, a machinery for calculating automorphisms, constructing invariants, and classifying real submanifolds of a complex manifold was developed. The main step in this machinery is the construction of a “nice” model surface. The nice model surface can be treated as an analog of the osculating paraboloid in classical differential geometry. Model surfaces suggested earlier possess a complete list of the desired properties only if some upper estimate for the codimension of the submanifold is satisfied. If this estimate fails, then the surfaces lose the universality property (that is, the ability to touch any germ in an appropriate way), which restricts their applicability. In the present paper, we get rid of this restriction: for an arbitrary type $(n,K)$ (where $n$ is the dimension of the complex tangent plane, and $K$ is the real codimension), we construct a nice model surface. In particular, we solve the problem of constructing a nondegenerate germ of a real analytic submanifold of a complex manifold of arbitrary given type $(n,K)$ with the richest possible group of holomorphic automorphisms in the given class.


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English version:
Mathematical Notes, 2004, 75:4, 475–488

Bibliographic databases:

UDC: 514.742
Received: 03.06.2003
Revised: 15.07.2003

Citation: V. K. Beloshapka, “Universal Models For Real Submanifolds”, Mat. Zametki, 75:4 (2004), 507–522; Math. Notes, 75:4 (2004), 475–488

Citation in format AMSBIB
\by V.~K.~Beloshapka
\paper Universal Models For Real Submanifolds
\jour Mat. Zametki
\yr 2004
\vol 75
\issue 4
\pages 507--522
\jour Math. Notes
\yr 2004
\vol 75
\issue 4
\pages 475--488

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    This publication is cited in the following articles:
    1. Sabzevari M., “Convergent Normal Form For Five Dimensional Totally Nondegenerate Cr Manifolds in C-4”, J. Geom. Anal.  crossref  isi
    2. Sabzevari M., “Totally Nondegenerate Models and Standard Manifolds in Cr Dimension One”, Bull. Iran Math. Soc.  crossref  mathscinet  isi
    3. Sabzevari M., Spiro A., “On the Geometric Order of Totally Nondegenerate Cr Manifolds”, Math. Z.  crossref  mathscinet  isi
    4. 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
    5. 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
    6. Beloshapka V. K., “Moduli space of model real submanifolds”, Russ. J. Math. Phys., 13:3 (2006), 245–252  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    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, “A Counterexample to the Dimension Conjecture”, Math. Notes, 81:1 (2007), 117–120  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. 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
    10. Beloshapka V., Ezhov V., Schmalz G., “Canonical cartan connection and holomorphic invariants on Engel CR manifolds”, Russ. J. Math. Phys., 14:2 (2007), 121–133  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    11. 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  scopus
    12. 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
    13. Beloshapka V. K., “A CR-manifold in general position as an $e$-structure”, Russ. J. Math. Phys., 14:1 (2007), 1–7  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    14. V. K. Beloshapka, V. V. Ezhov, G. Schmalz, “Holomorphic classification of four-dimensional surfaces in $\mathbb C^3$”, Izv. Math., 72:3 (2008), 413–427  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    15. 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
    16. Katarzyna S.-S., Wierzba W., Radowicki S., “Wpływ stosowania tibolonu na obraz endometrium u kobiet po menopauzie [The influence of a tibolone therapy on endometrium in post-menopausal women]”, Ginekol. Pol., 79:11 (2008), 758–761  isi
    17. Beloshapka V. K., “Space of orbits of the automorphism group of a model surface of type $(1,2)$”, Russ. J. Math. Phys., 15:1 (2008), 140–143  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. 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  scopus
    19. Beloshapka V.K., Kossovskiy I.G., “Homogeneous Hypersurfaces in C-3, Associated with a Model CR-Cubic”, J Geom Anal, 20:3 (2010), 538–564  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. Beloshapka V.K., Kossovskiy I.G., “Classification of homogeneous CR-manifolds in dimension 4”, J Math Anal Appl, 374:2 (2011), 655–672  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    21. V. K. Beloshapka, “Model-surface method: An infinite-dimensional version”, Proc. Steklov Inst. Math., 279 (2012), 14–24  mathnet  crossref  mathscinet  isi  elib
    22. Merker J., Sabzevari M., “Explicit Expression of Cartan's Connection for Levi-Nondegenerate 3-Manifolds in Complex Surfaces, and Identification of the Heisenberg Sphere”, Cent. Eur. J. Math., 10:5 (2012), 1801–1835  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    23. 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
    24. 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
    25. Sabzevari M., Hashemi A., M-Alizadeh B., Merker J., “Applications of Differential Algebra For Computing Lie Algebras of Infinitesimal Cr-Automorphisms”, Sci. China-Math., 57:9 (2014), 1811–1834  crossref  mathscinet  zmath  isi  scopus  scopus
    26. Masoud S., “Moduli Spaces of Model Real Submanifolds: Two Alternative Approaches”, Sci. China-Math., 58:11 (2015), 2261–2278  crossref  mathscinet  zmath  isi  scopus  scopus
    27. Merker J., Sabzevari M., “Cartan Equivalence Problem for 5-Dimensional Bracket-Generating CR Manifolds in
      $$\mathbb {C}^4$$
      C 4”, J. Geom. Anal., 26:4 (2016), 3194–3251  crossref  mathscinet  zmath  isi  elib  scopus
    28. 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
    29. 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
    30. Beloshapka V.K., “Cubic Model Cr-Manifolds Without the Assumption of Complete Nondegeneracy”, Russ. J. Math. Phys., 25:2 (2018), 148–157  crossref  mathscinet  isi  scopus  scopus
    31. M. A. Stepanova, “A modification of the Bloom–Graham Theorem: the introduction of weights in the complex tangent space”, Trans. Moscow Math. Soc., 2018, 201–208  mathnet  crossref  elib
    32. Sabzevari M., “On the Maximum Conjecture”, Forum Math., 30:6 (2018), 1599–1608  crossref  mathscinet  zmath  isi  scopus
    33. Beloshapka V.K., “Polynomial Model Cr-Manifolds With the Rigidity Condition”, Russ. J. Math. Phys., 26:1 (2019), 1–8  crossref  mathscinet  isi  scopus
    34. Sabzevari M., “Biholomorphic Equivalence to Totally Nondegenerate Model Cr Manifolds”, Ann. Mat. Pura Appl., 198:4 (2019), 1121–1163  crossref  mathscinet  isi
    35. M. A. Stepanova, “Ob avtomorfizmakh CR-podmnogoobrazii kompleksnogo gilbertova prostranstva”, Sib. elektron. matem. izv., 17 (2020), 126–140  mathnet  crossref
    36. Beloshapka V.K., “Cr-Manifolds of Finite Bloom-Graham Type: the Model Surface Method”, Russ. J. Math. Phys., 27:2 (2020), 155–174  crossref  mathscinet  isi
    37. 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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