Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Gamma be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p >= 2, Sp(2, R) or SO( p, 2) with p >= 3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO* (8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kahler manifolds and Higgs bundles we study representations of the lattice Gamma into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU( p, 2) with p >= 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) x U(2)), on which it acts cocompactly.