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

Koziarz, Vincent; Maubon, Julien

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 507-558.

Nous étudions les représentations des réseaux de $\mathrm{PU}(m,1)$ dans $\mathrm{PU}(n,1)$. Nous montrons que si la représentation est réductive et si $m$ est supérieur ou égal à 2, il existe une application équivariante harmonique d’énergie finie de l’espace hyperbolique complexe de dimension $m$ dans l’espace hyperbolique complexe de dimension $n$. Ceci nous permet de donner une preuve géométrique de résultats de rigidité obtenus par M. Burger et A. Iozzi. Nous définissons aussi un nouvel invariant associé aux représentations dans $\mathrm{PU}(n,1)$ des groupes fondamentaux des surfaces orientables de type topologique fini et de caractéristique d’Euler négative. Nous montrons que cet invariant est borné par une constante dépendant uniquement de la caractéristique d’Euler de la surface et nous donnons une caractérisation complète des représentations d’invariant maximal, généralisant ainsi les résultats de D. Toledo sur les surfaces compactes.

We study representations of lattices of $\mathrm{PU}(m,1)$ into $\mathrm{PU}(n,1)$. 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 $\mathrm{PU}(n,1)$ of non-uniform lattices in $\mathrm{PU}(1,1)$, 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.

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