On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions.
type: Item
instance of: scholarly article
title: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions.
zbMATH Open document ID: 1042.32008
author: Xiaojun Huang
published in: Journal of Differential Geometry
publication date: 2000-06-26T00:00:00Z
review text: Let \(\mathbb{B}_n= \{z\in \mathbb{C}^n:| z|< 1\}\) be the unit ball. A proper holomorphic map \(F: \mathbb{B}_n\to \mathbb{B}_N\) is a totally geodesic embedding if there are automorphisms \(\sigma\in \Aut(\mathbb{B}_n)\) and \(\Sigma\in\Aut(\mathbb{B}_N)\) such that \(\Sigma\circ F\circ\sigma(z)= (z,0)\), \(z\in \mathbb{B}_n\). The author shows that a nonconstant CR mapping \(f: M_1\to M_2\) of class \(C^2\) between open subsets \(M_1\subseteq S^{2n-1}\) and \(M_2\subseteq S^{2N-1}\) (\(n> 1\), \(N< 2n-1\)) is the restriction of some totally geodesic embedding \(F: \mathbb{B}_n\to \mathbb{B}_N\). This is related to work by \textit{S. M. Webster} [Pac. J. Math. 81, 267--272 (1979; Zbl 0379.32018); Indiana Univ. Math. J. 28, 405--416 (1979; Zbl 0387.53020)] as it is known (cf. op. cit.) that any open subset of \(M\subseteq M^{2n-1}\) is rigid in \(S^{2n+1}\) (i.e. any CR diffeomorphism of \(M\) into another CR submanifold \(M'\subset S^{2n+1}\) extends to a CR automorphism of \(S^{2n+1}\)). A question raised by S. M. Webster, i.e. whether a proper holomorphic immersion of \(\mathbb{B}_n\) into \(\mathbb{B}_N\), \(N= n+1\), extends to an immersion of the boundary (this is true when \(N= n\) by a result of \textit{H. Alexander} [Indiana Univ. Math. J. 26, 137--146 (1977; Zbl 0391.32015)]) has been partially (for polynomial proper maps) dealt with by \textit{J. P. D'Angelo} [Mich. Math. J. 35, 83--90 (1988; Zbl 0651.32014); Mich. Math. J. 38, 53--65 (1991; Zbl 0732.32014)]. The relationship among the author's results and the differential geometric aspects (the geometry of the ``second fundamental form'' of \(f: M_1\to M_2\) such as built from the pseudo-Hermitian structure of \(S^{2N+1}\), cf. \textit{S. Dragomir} [Am. J. Math. 117, 169--202 (1995; Zbl 0827.32020)] and \textit{E. Barletta} and \textit{S. Dragomir} [Kodai Math. J. 19, 62--68 (1996; Zbl 0861.32004)]) remains unclear.
reviewed by: Sorin Dragomir
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 32V15
zbMATH DE Number: 1465061
zbMATH Keywords: proper holomorphic map
zbMATH Keywords: linear embedding
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214425024
OpenAlex ID: W1593111489
Wikidata QID: Q115194273
DOI: 10.4310/JDG/1214425024
description: scientific article; zbMATH DE number 1465061