Spaces of harmonic surfaces in non-positive curvature

Let M(Σ) be an open and connected subset of the space of hyperbolic metrics on a closed orientable surface, and M(M) an open and connected subset of the space of metrics on an orientable manifold of dimension at least 3. We impose conditions on M and M(M), which are often satisfied when the metrics in M(M) have non-positive curvature. Under these conditions, the data of a homotopy class of maps from Σ to M allows us to view M(Σ) × M(M) as a space of harmonic maps of surfaces. Using transversality theory for Banach manifolds, we prove that the set of somewhere injective harmonic maps is open, dense, and connected in the space of harmonic maps. We also prove some results concerning the distribution of harmonic immersions and embeddings in the space of harmonic maps.