Harmonic maps and representations of non-uniform lattices of {\rm PU}(m,1)

Vincent Koziarz, Julien Maubon

Type: Article

Publication Date: 2008-01-01

Citations: 38

DOI: https://doi.org/10.5802/aif.2359

Abstract

We study representations of lattices of PU (m,1) into PU (n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex hyperbolic n-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into PU (n,1) of non-uniform lattices in PU (1,1), and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.

Locations

French digital mathematics library (Numdam) - View - PDF
Annales de l’institut Fourier - View - PDF

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+	Bounded Cohomology, Boundary Maps, and Rigidity of Representations into Homeo+(S1) and SU(1, n)	2002	Alessandra Iozzi
+ PDF Chat	Some properties and applications of harmonic mappings	1978	J. H. Sampson
+ PDF Chat	Flat $G$-bundles with canonical metrics	1988	Kevin Corlette
+	Harmonic Mappings of Riemannian Manifolds	1964	James Eells J. H. Sampson
+	Harmonic maps into singular spaces andp-adic superrigidity for lattices in groups of rank one	1992	M. Gromov Richard Schoen