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 Mat. Zametki, 1997, Volume 62, Issue 3, Pages 391–403 (Mi mz1621)

Sphericity of rigid hypersurfaces in $\mathbb C^2$

A. V. Loboda

Abstract: The sphericity of hypersurfaces in the space $\mathbb C^2_{z,w}$ (locally) representable by equations of the form $\operatorname{Im}v=F(z,\overline z)$ is discussed. Invoking the notion of Moser normal form, a necessary and sufficient condition for these surfaces to be spherical is constructed. It is a partial differential third-order equation for the function $\mu(z,\overline z)=F_{zz\overline z}/F_{z\overline z}$. The similarity between this equation and the sphericity criterion for tube hypersurfaces makes it possible to reduce the problem to the familiar description of spherical tubes. Reduction mappings are written out explicitly. As a particular case, a description of Reinhardt spherical surfaces defined by the equations $\operatorname{Im}w=\alpha(|z|^2)$ is given.

DOI: https://doi.org/10.4213/mzm1621

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English version:
Mathematical Notes, 1997, 62:3, 329–338

Bibliographic databases:

UDC: 514.764.274

Citation: A. V. Loboda, “Sphericity of rigid hypersurfaces in $\mathbb C^2$”, Mat. Zametki, 62:3 (1997), 391–403; Math. Notes, 62:3 (1997), 329–338

Citation in format AMSBIB
\Bibitem{Lob97}
\by A.~V.~Loboda
\paper Sphericity of rigid hypersurfaces in $\mathbb C^2$
\jour Mat. Zametki
\yr 1997
\vol 62
\issue 3
\pages 391--403
\mathnet{http://mi.mathnet.ru/mz1621}
\crossref{https://doi.org/10.4213/mzm1621}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1620070}
\zmath{https://zbmath.org/?q=an:0923.32017}
\transl
\jour Math. Notes
\yr 1997
\vol 62
\issue 3
\pages 329--338
\crossref{https://doi.org/10.1007/BF02360874}

• http://mi.mathnet.ru/eng/mz1621
• https://doi.org/10.4213/mzm1621
• http://mi.mathnet.ru/eng/mz/v62/i3/p391

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Isaev A.V., “Zero CR-curvature equations for rigid and tube hypersurfaces”, Complex Variables and Elliptic Equations, 54:3–4 (2009), 317–344
2. Isaev A., “Spherical Tube Hypersurfaces”, Spherical Tube Hypersurfaces, Lect. Notes Math., 2020, Springer-Verlag Berlin, 2011, 1–217
3. Ezhov V., Schmalz G., “The zero curvature equation for rigid CR-manifolds”, Complex Var. Elliptic Equ., 61:4 (2016), 443–447
4. Ebenfelt P., Son D.N., “Umbilical Points on Three Dimensional Strictly Pseudoconvex Cr Manifolds i: Manifolds With U(1)-Action”, Math. Ann., 368:1-2 (2017), 537–560
5. M. A. Stepanova, “Ob avtomorfizmakh CR-podmnogoobrazii kompleksnogo gilbertova prostranstva”, Sib. elektron. matem. izv., 17 (2020), 126–140
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