Eversive maps of bounded convex domains in \(\mathbb{C}^{n+1}\)
type: Item
instance of: scholarly article
title: Eversive maps of bounded convex domains in \(\mathbb{C}^{n+1}\)
zbMATH Open document ID: 1039.32019
DOI: 10.1007/BF02930659
author: Stefano Trapani
author: Andrea Spiro
published in: The Journal of Geometric Analysis
publication date: 2003-10-21T00:00:00Z
review text: An automorphism group \(G = \Aut(B^{n+1})\) of the unit ball \(B^{n+1}\subset {\mathbb C}^{n+1}\) has the following two important properties: it has the largest admissible dimension for an automorphism group of an Hermitian manifold and it acts freely and transitively on the bundle \(\pi : U(B^{n+1})\rightarrow B^{n+1}\) of all unitary frames of \(B^{n+1}\) with respect to the Poincaré-Bergmann metric. Each element \(g \in G\) extends up to boundary \(S^{n+1} = \partial B^{n+1}\) and it acts as a CR transformation on \(S^{2n+1}.\) Moreover, \(G\) has the largest admissible dimension for an automorphism group of a Levi non-degenerate hypersurface and it acts freely and transitively on the Chern-Moser bundle \(\pi': P_{\text{CM}}(S^{2n+1}) \rightarrow S^{2n+1},\) associated with the CR structure of \(S^{2n+1}.\) It follows that both manifolds \(U(B^{n+1})\) and \(P_{\text{CM}}(S^{2n+1})\) are diffeomorphic to \(G.\) Therefore, given a point \(u \in U(B^{n+1})\) and a point in \(p \in P_{\text{CM}}(S^{2n+1}),\) there exists a unique \(G\)-equivariant diffeomorphism \(E: U(B^{n+1}) \rightarrow P_{\text{CM}} (S^{2n+1})\) mapping \(u\) into \(p.\) The associated diffeomorphism determines a quite interesting duality between the invariants of the Hermitian geometry of \(B^{n+1}\) and the invariants of the CR geometry of the unit sphere \(S^{2n+1}.\) Such duality constitutes the perfect analog of the well-known relation between the geometry of the hyperbolic space \(H^{n+1}\) and the conformal geometry of unit sphere \(S^{n}.\) The main goal of this article is to prove that one diffeomorphism as above exists for any smoothly bounded, strongly convex domain \(D \subset {\mathbb C}^{n+1}.\) A duality principle, relating the geometry of the Kobayashi metric with the CR geometry of the boundaries of smoothly bounded, strongly convex domain \(D \subset {\mathbb C}^{n+1},\) is established. As a by-product, the authors are able to show the following new fact: any holomorphic Jacobi…
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32V15
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32F27
zbMATH DE Number: 1983426
zbMATH Keywords: automorphism group of the unit ball
zbMATH Keywords: Hermitian manifold
zbMATH Keywords: CR transformation
zbMATH Keywords: Levi non-degenerate hypersurface
zbMATH Keywords: Chern-Moser bundle
zbMATH Keywords: holomorphic Jacobi vector field
zbMATH Keywords: duality principle
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02930659
OpenAlex ID: W2091947357
description: scientific article; zbMATH DE number 1983426