Abstract:
We discuss a complete solution of the problem on the local description and classification of holomorphically homogeneous real hypersurfaces in $ \mathbb{C}^3$. Large families of such manifolds have been studied by several groups of mathematicians using various approaches in the last 25 years. The final results in this problem have been obtained by the present author with the use of the classification of abstract five-dimensional real Lie algebras and the technique of their holomorphic representations in complex 3-space.
The complete list of pairwise nonequivalent holomorphically homogeneous hypersurfaces in our classification contains forty-seven types of such manifolds, including standalone hypersurfaces as well as one- and two-parameter families of hypersurfaces.
Key words and phrases:
homogeneous manifold, real hypersurface, normal form, holomorphic transformation, vector field, Lie algebra.
Citation:
A. V. Loboda, “Holomorphically homogeneous real hypersurfaces in $ \mathbb{C}^3$”, Tr. Mosk. Mat. Obs., 81, no. 2, MCCME, M., 2020, 205–280; Trans. Moscow Math. Soc., 81:2 (2020), 169–228
This publication is cited in the following 9 articles:
I. I. Zavolokin, “Odnorodnye $\mathrm{CR}$-mnogoobraziya v $\mathbb{C}^4$”, Matem. zametki, 117:3 (2025), 388–401
Joël Merker, Paweł Nurowski, “Homogeneous CR and Para-CR Structures in Dimensions 5 and 3”, J Geom Anal, 34:1 (2024)
Vladislav Krutskikh, Alexandr Loboda, “APPLICATION OF SYSTEMS ANALYSIS AND COMPUTER ALGORITHMS IN STUDYING ORBITS OF 7-DIMENSIONAL LIE ALGEBRAS”, Mathematical Physics and Computer Simulation, 27:3 (2024), 38
A. V. Atanov, A. V. Loboda, “O nevyrozhdennykh orbitakh $7$-mernykh algebr Li, soderzhaschikh $3$-mernyi abelev ideal”, Trudy Voronezhskoi zimnei matematicheskoi shkoly S. G. Kreina — 2024, SMFN, 70, no. 4, Rossiiskii universitet druzhby narodov, M., 2024, 517–532
A. V. Loboda, R. S. Akopyan, V. V. Krutskikh, “O 7-mernykh algebrakh golomorfnykh vektornykh polei v $\Bbb C^4$, imeyuschikh 5-mernyi abelev ideal”, Dalnevost. matem. zhurn., 23:1 (2023), 55–80
A. V. Loboda, “O 7-mernykh algebrakh Li, dopuskayuschikh Levi-nevyrozhdennye orbity v $\mathbb{C}^4$”, Tr. MMO, 84, no. 2, MTsNMO, M., 2023, 205–230
A. V. Atanov, “Orbits of decomposable $7$-dimensional Lie algebras with $\mathfrak{sl}(2)$ subalgebra”, Ufa Math. J., 14:1 (2022), 1–19
A. V. Loboda, V. K. Kaverina, “On degeneracy of orbits of nilpotent Lie algebras”, Ufa Math. J., 14:1 (2022), 52–76
Doubrov B., Merker J., The D., “Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in C-3”, Int. Math. Res. Notices, 2021, rnab147
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