Harmonic maps to Hadamard spaces and a universal higher Teichm\"{u}ller space

TL;DR

This paper establishes a criterion for harmonic maps from manifolds with Ricci curvature bounds to Hadamard spaces, generalizes a conjecture on quasi-isometric embeddings, and introduces a universal higher Teichmüller space.

Contribution

It introduces the stability criterion for harmonic maps, generalizes the Schoen-Li-Wang conjecture, and defines a universal Hitchin component extending Teichmüller space.

Findings

Harmonic maps are close to Lipschitz maps under stability.

Generalization of the Schoen-Li-Wang conjecture to higher rank spaces.

Definition of a universal higher Teichmüller space for PGL_d(R).

Abstract

We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove uniqueness of the harmonic map under additional assumptions on $X$ and $Y$. Using this criterion, we prove a significant generalization of the Schoen-Li-Wang conjecture on quasi-isometric embeddings between rank 1 symmetric spaces. In particular, under a natural generalization of the quasi-isometric condition, we remove the assumption that the target has rank 1. This allows us to define a universal Hitchin component for each $\mathrm{PGL}_d(\mathbb{R})$, generalizing universal Teichm\"uller space, and show that it can be described both as a space of quasi-symmetric positive maps from $\mathbb{RP}^1$ to the flag variety, and as a space of harmonic maps.