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