Constant energy families of harmonic maps

TL;DR

The paper analyzes families of harmonic maps from a closed surface into a negatively curved manifold by studying when the energy functional $E_\psi$ is constant along complex submanifolds of Teichmüller space. It proves that such constant-energy families necessarily factor through a fixed holomorphic map to a curve: there exists a closed Riemann surface $Y$ and a holomorphic $p:\Sigma_g\to Y$ such that each harmonic map in the family factors as $h_Y\circ p$ with $h_Y:Y\to M$ harmonic, and $\psi$ itself factors as $\theta\circ p$ for some $\theta:Y\to M$. The core method combines pluriharmonicity analysis of the combined map on the universal curve, Schiffer varieties from Teichmüller theory, and non-abelian Hodge theory for moving Riemann surfaces. Applications include a factorization result for harmonic maps from normal projective varieties and a mapping class group theorem showing that strongly point-pushing homomorphisms factor through the fundamental group of a hyperbolic curve after passing to finite-index subgroups. The work thereby links harmonic map theory, Teichmüller theory, and algebraic geometry to obtain a robust holomorphic-curve factorization phenomenon with broad structural consequences.

Abstract

For a negatively curved manifold $M$ and a continuous map $ψ:Σ\to M$ from a closed surface $Σ$, we study complex submanifolds of Teichmüller space $\mathcal{S}\subset\mathcal{T}(Σ)$ such that the harmonic maps $\{h_X:X\to M\text{ for }X\in\mathcal{S}\}$ in the homotopy class of $ψ$ 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 $φ_X:X\to Y$ 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 $π_1(M)$.

Theorems & Definitions (52)

• Theorem 1
• Definition 1.1
• Theorem 2
• Theorem 3
• Definition 1.2
• Theorem 4
• Remark 2.1
• Definition 2.3
• Remark 2.4
• Lemma 2.5