CR hypersurfaces with a contracting automorphism
type: Item
instance of: scholarly article
title: CR hypersurfaces with a contracting automorphism
zbMATH Open document ID: 1187.32030
author: Kang-Tae Kim
author: Sung-Yeon Kim
published in: The Journal of Geometric Analysis
publication date: 2008-09-08T00:00:00Z
review text: 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.
reviewed by: Andrea Altomani
Mathematics Subject Classification ID: 32V30
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 5322596
zbMATH Keywords: holomorphic mapping
zbMATH Keywords: hyperquadric
zbMATH Keywords: Levi-nondegenerate
zbMATH Keywords: signature
zbMATH Keywords: projective space
zbMATH Keywords: CR hypersurface
zbMATH Keywords: contraction
zbMATH Keywords: linearization
MaRDI profile type: Publication
full work available at URL: s12220-008-9033-z
OpenAlex ID: W2010425867
cites work: Q4744562
cites work: Domains in \({\mathbb{C}}^{n+1}\) with noncompact automorphism group
cites work: Q3770811
cites work: CHARACTERIZATION OF MODELS IN C2 BY THEIR AUTOMORPHISM GROUPS
cites work: Boundary invariants of pseudoconvex domains
cites work: Real hypersurfaces, orders of contact, and applications
cites work: Analytic sets extending the graphs of holomorphic mappings
cites work: Characterization of convex domains with noncompact automorphism group
cites work: Complete Localization of Domains with Noncompact Automorphism Groups
cites work: Complex scaling and domains with non-compact automorphism group
cites work: Normal analytic polyhedra in \(\mathbb{C}^2\) with a noncompact automorphism group
cites work: Dynamics of local automorphisms of embedded CR-manifolds.
cites work: Analytic polyhedra in ℂ2with a non-compact automorphism group
cites work: Q4000467
cites work: Sur une caractérisation de la boule parmi les domaines de \(\mathbb{C}^n\) par son groupe d'automorphismes
cites work: On the conformal and CR automorphism groups
cites work: Smooth real hypersurfaces in \(\mathbb{C}^n\) with a non-compact isotropy group of CR transformations
cites work: A note on uniform extendability of automorphisms
cites work: Characterization of the unit ball in \(\mathbb{C}^n\) by its automorphism group
DOI: 10.1007/S12220-008-9033-Z
description: scientific article; zbMATH DE number 5322596