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For a negatively curved manifold M and a continuous map: M from a closed surface, we study complex submanifolds of Teichm\"uller space S () such that the harmonic maps \hX: X M for X\ in the homotopy class of 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 X: 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 ₁ (M).
Ognjen Tošić (Sun,) studied this question.