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

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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/}
}
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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/