Harmonic maps to Hadamard spaces and a universal higher Teichmüller space

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 Teichmüller 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.

53 pages, 2 figures. Added Corollary 69 and a sentence to Corollary 70. Added acknowledgements and references