Constant energy families of harmonic maps

For a negatively curved manifold M and a continuous map ψ:Σ→ M from a closed surface Σ, we study complex submanifolds of Teichm\"uller space S(Σ) such that the harmonic maps :X→ M for X\ in the homotopy class of ψ all have equal energy. When M is real analytic with negative Hermitian sectional curvature, we show that for any such S, there exists a closed Riemann surface Y, such that any hX for X factors as a holomorphic map φX:X→ Y followed by a fixed harmonic map h:Y→ M. This answers a question posed by both Toledo and Gromov. As a first application, we show a factorization result for harmonic maps from normal projective varieties to M. As a second application, we study homomorphisms from finite index subgroups of mapping class groups to π₁(M).