CR hypersurfaces with a contracting automorphism
Let \(M\) be a Levi nondegenerate real-analytyic germ of CR hypersurface through the origin in \(\mathbb C^{n+1}\).A real analytic germ of CR automorphism \(f\in\mathrm{Aut}(M,0)\) of \(M\) fixing \(0\) is \textit{weakly contracting} if there exists a contact form \(\theta\) and a dual vector field \(T\) such that \(|\theta(\mathrm df_0T)|<1\) (the definition is independent from \(\theta\) and \(T\)). The main result of the paper is the following: Assume that \(M\) does not contain any germ of complex analytic variety through the origin, and let \(f\in\mathrm{Aut}(M,0)\) be a weakly contracting automorphism. Then: {1.} \(f\) is linearizable, i.e. there exists a germ \(\psi\) of biholomorphism of \(\mathbb{C}^{n+1}\) fixing the origin such that \(\psi^{-1}\circ f\circ\psi\) is linear; {2.} \(M\) is biholomorphically equivalent to a hypersurface of the form \[ \mathrm{Im}\,z_0=P(z_1,\dots,z_n,\bar z_1,\dots,\bar z_n), \] where \(P\) is a \textit{weighted homogeneous polynomial}, i.e. there exist integers \(m,m_1,\dots,m_n\) such that for all \(\lambda>0\): \[ P(\lambda^{m_1}z_1,\dots,\lambda^{m_n}z_n,\lambda^{m_1}\bar z_1,\dots,\lambda^{m_n}\bar z_n) =\lambda^mP(z_1,\dots,z_n,\bar z_1,\dots,\bar z_n). \] The result generalizes previous work in the pseudoconvex case by \textit{R. Schoen} [Geom. Funct. Anal. 5, No. 2, 464--481 (1995; Zbl 0835.53015)]. Applications are given to the geometry of bounded pseudoconvex domains with non-compact automorphism group.
scientific article; zbMATH DE number 5322596