Spaces of harmonic surfaces in non-positive curvature

TL;DR

This paper studies the space of harmonic maps from surfaces to manifolds with non-positive curvature, proving openness, density, and connectedness of certain harmonic maps within the moduli space.

Contribution

It establishes the topological and geometric properties of harmonic maps in the moduli space under non-positive curvature conditions, using transversality theory.

Findings

The set of somewhere injective harmonic maps is open, dense, and connected.

Results on the distribution of harmonic immersions and embeddings.

Conditions on metrics ensuring harmonic map structures.

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.