On the dimension of a group acting transitively on hypersurfaces in \(\mathbb{C}^3\)
Let \(M\) be a real-analytic CR-hypersurface in \({\mathbb C}^3\) with non degenerate Levi form. Assume that \(M\) is nonspherical (i.e. not locally isomorphic to a quadric). Let \(\xi\in M\). It is known that the group of germs of holomorphic automorphisms of \({\mathbb C}^3\) mapping \(M\) into \(M\) and stabilizing \(\xi\) is a finite-dimensional real Lie group which is linear in appropriately chosen local coordinates. The article under review establishes sharp upper bounds for the dimension of this group. It is of real dimension three if \(M\) is isomorphic to \(\{\Im(z_3)=(z_1\bar z_2 + z_2\bar z_1) + |z_1|^2 \}\) and at most two otherwise. As a consequence one obtains \(\dim_{\mathbb R}Aut(M)\leq 8\) for a homogeneous such \(M\).
scientific article; zbMATH DE number 1407616