Harmonic maps and representations of non-uniform lattices of PU(m,1)

v2: the case of lattices of PU(1,1) has been rewritten and is now treated in full generality + other minor modificationsWe 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