Compressions and contractions of Hermitian symmetric spaces
type: Item
instance of: scholarly article
title: Compressions and contractions of Hermitian symmetric spaces
zbMATH Open document ID: 0918.32016
DOI: 10.1007/PL00004648
author: Jean-Louis Clerc
published in: Mathematische Zeitschrift
publication date: 1999-08-17T00:00:00Z
review text: Let \(D=G/K\) be a bounded symmetric domain equipped with the Bergman metric \(d\). We say that function \(f:D\to D\) is a contraction for the metric \(d\) if for all \(p\) and \(q\) in \(D\) \[ d\bigl(f(p), f(q)\bigr)\leq d(p,q). \] We say that \(f:D\to D\) is a strict contraction if there is a \(K<1\) so that \[ d \bigl(f(p),f(q)\bigr)\leq Kd(p,q). \] Let \(G_\mathbb{C}\) be the complexification of \(G\) and consider the so-called group of compressions of \(D\) \[ \Gamma= \bigl\{a \in G_\mathbb{C},g(D) \subset D\bigr\}. \] It is known that \(\Gamma=G\exp(C)\) where \(C\) is a convex, closed, proper cone. Let \(C^0\) be the interior of the cone \(C\) and denote by \(\Gamma^0\) the set \(G\exp C^0\). In this paper the author proves that every element \(\gamma\in\Gamma\) is a contraction for the Bergman metric of \(D\). Moreover, every \(\gamma\in\Gamma^0\) is a strict contraction for the Bergman metric of \(D\).
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 1229578
zbMATH Keywords: bounded symmetric domain
zbMATH Keywords: group of compressions
zbMATH Keywords: contraction
zbMATH Keywords: Bergman metric
reviewed by: L. Barchini
MaRDI profile type: Publication
full work available at URL: pl00004648
OpenAlex ID: W2047682637
description: scientific article; zbMATH DE number 1229578