Spaces of harmonic surfaces in non-positive curvature

Abstract

Let $\mathfrak{M}(\Sigma)$ be an open and connected subset of the space of hyperbolic metrics on a closed orientable surface, and $\mathfrak{M}(M)$ 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 $\mathfrak{M}(M)$, which are often satisfied when the metrics in $\mathfrak{M}(M)$ have non-positive curvature. Under these conditions, the data of a homotopy class of maps from $\Sigma$ to $M$ gives $\mathfrak{M}(\Sigma)\times \mathfrak{M}(M)$ the structure of a space of harmonic maps. Using transversality theory for Banach manifolds, we prove that the set of somewhere injective harmonic maps is open, dense, and connected in the moduli space. We also prove some results concerning the distribution of harmonic immersions and embeddings in the moduli space.