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A short proof of the dimension conjecture for real hypersurfaces in \(\mathbb {C}^2\). (English) Zbl 1356.32013

Let \(M\) be a 3-dimensional connected real-analytic CR-manifold of hypersurface type. For \(p\in M\), denote by \(\mathfrak{hol\,}(M,p)\) the Lie algebra of germs of real-analytic infinitesimal CR-automorphisms of \(M\) at \(p\). Recently, I. Kossovskiy and R. Shafikov [J. Differ. Geom. 102, No. 1, 67–126 (2016; Zbl 1342.53079)] proved the following theorem:
If \(M\) is not Levi-flat, then for any \(p\in M\) the condition \(\dim\mathfrak{hol\,}(M,p)>5\) implies that \(M\) is spherical at \(p\).
The article [loc. cit.] is quite long and its method is rather involved and based on considering second-order compex ODEs with meromorphic singularity. In the reviewed paper, the authors present a short proof of the above theorem by using known facts on Lie algebras and their actions.

MSC:

32V40 Real submanifolds in complex manifolds
32C05 Real-analytic manifolds, real-analytic spaces

Citations:

Zbl 1342.53079
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References:

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