Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products
type: Item
instance of: scholarly article
title: Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products
zbMATH Open document ID: 1254.32033
DOI: 10.1515/CRELLE.2011.142
author: Ngaiming Mok
author: Sui Chung Ng
publication date: 2012-09-12T00:00:00Z
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 6081358
zbMATH Keywords: algebraic correspondence
zbMATH Keywords: modular correspondence
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: bounded symmetric domain
MaRDI profile type: Publication
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cites work: Realization of Hermitian symmetric spaces as generalized half-planes
cites work: Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains
cites work: Local characterization of holomorphic automorphisms of Siegel domains
published in: Journal für die Reine und Angewandte Mathematik
review text: The main result of the article under review is the following general extension theorem: Let \(\Omega\) be an irreducible bounded symmetric domain in \(\mathbb C^n\), \(n\geq 2\), and \(d\mu_\Omega\) the volume form of the Bergman metric on \(\Omega\). Let \(\Omega_\alpha=\Omega\) denote the \(\alpha\)-th direct factor of \(\Omega^d\), \(1\leq\alpha\leq d\), and \(\pi_\alpha:\Omega^d\rightarrow\Omega_\alpha\) the natural surjection. Let \(d_1, d_2\in \mathbb N_+\) and \(f=(f_1,\dots,f_{d_2}):(\Omega,d_1d\mu_\Omega;0)\rightarrow (\Omega^{d_2},\pi_1^*d\mu_\Omega+\cdots+\pi_{d_2}^*d\mu_\Omega;0)\) the germ of a measure preserving holomorphic map such that each \(f_\alpha\), \(1\leq\alpha\leq d_2\), is of maximal rank at some point. Then \(d_1=d_2\) and \(f\) extends to a totally geodesic holomorphic embedding \(f:\Omega\rightarrow \Omega^{d_2}\).NEWLINENEWLINE The proof is based on methods of analytic continuation, in particular on a local characterization of automorphisms of \(\Omega\). When rank \(\Omega=1\) a theorem of \textit{T. Alexander} [Math. Ann. 209, 249--256 (1974; Zbl 0272.32006)] is applied which says that a nonconstant holomorphic map \(\varphi:U\rightarrow \mathbb C^n\) on a neighborhood \(U\) of \(b\in \partial B^n\) with \(\varphi(U\cap \partial B^n)\subset \partial B^n\) can be extended to an automorphism of the unit ball \(B^n\). For rank \(\Omega\geq 2\) the authors prove and apply an analogous theorem for \(\Omega\). Alternatively they deduce the total geodesy from a local characterization of automorphisms of \(\Omega\) by \textit{A. E. Tumanov} and \textit{G. M. Khenkin} [Sov. Math., Dokl. 26, 702--705 (1982); translation from Dokl. Akad. Nauk SSSR 267, 796--799 (1982; Zbl 0529.32014)].NEWLINENEWLINE The extension theorem cited at the beginning yields an affirmative answer to a question raised by \textit{L. Clozel} and \textit{E. Ullmo} [J. Reine Angew. Math. 558, 47--83 (2003; Zbl 1042.11027)] whether measure preserving algebraic correspondences …
reviewed by: Eberhard Oeljeklaus
description: scientific article; zbMATH DE number 6081358