Constant energy families of harmonic maps

Abstract

For a negatively curved manifold $M$ and a continuous map $\psi :\Sigma \to M$ from a closed surface $\Sigma$, we study complex submanifolds of Teichmüller space $\mathcal{S}\subset \mathcal{T}(\Sigma )$ such that the harmonic maps $\{h_X:X\to M\text{ for }X\in \mathcal{S}\}$ in the homotopy class of $\psi$ all have equal energy. When $M$ is real analytic with negative Hermitian sectional curvature, we show that for any such $\mathcal{S}$, there exists a closed Riemann surface $Y$, such that any $h_X$ for $X\in \mathcal{S}$ factors as a holomorphic map ${\varphi }_X:X\to Y$ followed by a fixed harmonic map $h:Y\to M$. 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 ${\pi }_1(M)$.

39 pages, 0 figures