Constant energy families of harmonic maps

TL;DR

This paper investigates special families of harmonic maps from surfaces to negatively curved manifolds, revealing a factorization property and addressing questions in geometric analysis and group theory.

Contribution

It establishes a factorization theorem for harmonic maps with equal energy families, answering a question by Toledo and Gromov.

Findings

Existence of a fixed Riemann surface Y for harmonic map families

Harmonic maps factor through a holomorphic map to Y

Applications to harmonic maps from projective varieties and mapping class groups

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 Teichm\"uller 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 $\phi_X:X\to Y$ 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 $\pi_1(M)$.