The real Schwarz Lemma and geometric applications
type: Item
instance of: scholarly article
title: The real Schwarz Lemma and geometric applications
zbMATH Open document ID: 1035.53038
DOI: 10.1007/BF02392826
author: G. Besson
author: Gilles Courtois
author: Sylvestre (Sylvain) Gallot
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: In the first result of the present article the authors give a Riemannian analogue of the classical Schwarz lemma (that holomorphic maps between complex manifolds with suitable curvature conditions decrease volume): Let \(X\), \(Y\) be complete Riemannian manifolds of the same dimension \(n \geq 3\), such that \(\text{Ricci}_{g_{Y}} \geq -(n-1)g_{Y}\) and \(K_{g_{X}} \leq -1\). Then in any homotopy class of maps \(f:Y \rightarrow X\) and for all \(\varepsilon > 0\), there exists a map \(F_{\varepsilon}\) such that \(| \text{Jac}(F_{\varepsilon})| \leq 1 + \varepsilon\) pointwise. If, moreover, \(X\) and \(Y\) are compact and homotopically equivalent with \(K_{g_{Y}} \leq -1\), there exists \(F_{0}\) of class \(C^{\infty}\) such that \(| \text{Jac}(F_0)| \leq 1\) pointwise; and if \(| \text{Jac}F_0(y))| = 1\) for some \(y\) then \(d_yF_{0}\) is an isometry. The proof is a well-readable version of proof ideas already used by the authors in their articles [Geom. Funct. Anal. 5, 731--799 (1995; Zbl 0851.53032) and Ergodic Theory Dyn. Syst. 16, 623--649 (1996; Zbl 0887.58030)]. The key observation of the present paper is that with the same proof ideas one can obtain a much more general result: In fact, the authors get such maps \(F_{\varepsilon}\) and \(F_{0}\) also if \(X\) and \(Y\) are not necessarily of the same dimension, and these maps depend only on a representation of the fundamental group of \(Y\) in the one of \(X\); more generally, the authors show that one can start with a representation \(\rho : \Gamma \rightarrow \Gamma'\) of discrete isomorphism groups \(\Gamma\) of \(Y\) and \(\Gamma'\) of \(X\) and gets \(\rho\)-equivariant maps \(F_{\varepsilon}\) resp. \(F_{0}\) still satisfying inequalities of the type \(| \text{Jac}F_{\varepsilon}| \leq (\frac{\delta (\Gamma)}{p-1}(1+\varepsilon))^{p}\), where \(p=\text{dim}(Y)\) and \(\delta (\Gamma)\) is the critical exponent of the Poincaré series of \(\Gamma\). As a corollary the authors give a rigidity result for…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 1541216
zbMATH Keywords: negative curvature
zbMATH Keywords: minimal volume
zbMATH Keywords: symmetric spaces
zbMATH Keywords: quasi-Fuchsian representation
zbMATH Keywords: minimal entropy
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: rigidity
reviewed by: Gerd Dethloff
MaRDI profile type: Publication
cites work: Entropy and rigidity of locally symmetric spaces of strictly negative curvature
cites work: Minimal entropy and Mostow's rigidity theorems
cites work: On the cross-ratio at the boundary of \(CAT(-1)\)-spaces
cites work: Hausdorff dimension of quasi-circles
cites work: Archimedean superrigidity and hyperbolic geometry
cites work: Harmonic mappings of Kähler manifolds to locally symmetric spaces
cites work: Conformally natural extension of homeomorphisms of the circle
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Sur les groupes hyperboliques d'après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov)
cites work: Local rigidity of discrete groups acting on complex hyperbolic space
cites work: Sobolev and isoperimetric inequalities for riemannian submanifolds
cites work: Harmonic mappings and Kähler manifolds
cites work: Q3992565
cites work: Compactification of minimal submanifolds of hyperbolic space
cites work: Dimension conforme et sphère à l'infini des variétés à courbure négative
cites work: Q4222330
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
cites work: The density at infinity of a discrete group of hyperbolic motions
cites work: Dimension and rigidity of quasi-Fuchsian representations
cites work: The ergodic theory of discrete isometry groups on manifolds of variable negative curvature
full work available at URL: https://doi.org/10.1007/bf02392826
OpenAlex ID: W100527680
Recommended article: Q4217460
Recommended article: A geometric extension of Schwarz's Lemma and applications
Recommended article: Q3586026
Recommended article: Hyperbolic geometric versions of Schwarz’s lemma
Recommended article: General Schwarz Lemmata and their applications
Recommended article: Q4606839
Recommended article: Q4703197
Recommended article: Geometric versions of Schwarz's lemma and symmetrization
Recommended article: A multidimensional real Schwarz Lemma for the Hilbert metric
Recommended article: Geometric versions of Schwarz’s lemma for quasiregular mappings
description: scientific article; zbMATH DE number 1541216