Geodesic
Sphericity of rigid hypersurfaces in $\mathbb C^2$
Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 391-403.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MZM_1997_62_3_a7,
     author = {A. V. Loboda},
     title = {Sphericity of rigid hypersurfaces in $\mathbb C^2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {391--403},
     publisher = {mathdoc},
     volume = {62},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a7/}
}
TY  - JOUR
AU  - A. V. Loboda
TI  - Sphericity of rigid hypersurfaces in $\mathbb C^2$
JO  - Matematičeskie zametki
PY  - 1997
SP  - 391
EP  - 403
VL  - 62
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a7/
LA  - ru
ID  - MZM_1997_62_3_a7
ER  - 
%0 Journal Article
%A A. V. Loboda
%T Sphericity of rigid hypersurfaces in $\mathbb C^2$
%J Matematičeskie zametki
%D 1997
%P 391-403
%V 62
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a7/
%G ru
%F MZM_1997_62_3_a7
A. V. Loboda. Sphericity of rigid hypersurfaces in $\mathbb C^2$. Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 391-403. http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a7/

[1] Stanton N. K., “A normal form for rigid hypersurfaces in $\mathbb C^2$”, Amer. J. Math., 113:5 (1991), 877–910 | DOI | MR | Zbl

[2] Dadok J., Yang P., “Automorphisms of tube domains and spherical hypersurfaces”, Amer. J. Math., 107:4 (1985), 999–1013 | DOI | MR | Zbl

[3] Loboda A. V., “On homogeneous hypersurfaces in $\mathbb C^2$”, International Conference “Complex Analysis: Geometrical Methods”, St.-Petersburg, 1994

[4] Chern S. S., Moser J. K., “Real hypersurfaces in complex manifolds”, Acta Math., 133:3 (1974), 219–271 | DOI | MR

[5] Isaev A. V., “Rigid spherical hypersurfaces”, Complex Variables. Theory Appl., 31 (1996), 141–163 | Zbl