Harmonic maps to Hadamard spaces and a universal higher Teichmüller space
J. Maxwell Riestenberg, Peter Smillie
TL;DR
This work develops a stability-driven framework to locate harmonic maps near coarse Lipschitz targets from a complete manifold with $\mathrm{Ric}\ge -K$ to a proper Hadamard space. It proves that stable coarse Lipschitz maps are within a bounded distance of a (unique) harmonic map, using Ptolemy-type estimates and higher-rank Morse theory to circumvent strict negative curvature assumptions. The authors generalize Schoen–Li–Wang to higher rank via $\Theta$-quasi-isometric embeddings with positive or finitely non-transverse boundary data, yielding a robust correspondence between boundary maps and harmonic extensions. They then define a universal Hitchin component $\mathrm{UHit}_d$ as the space of $\Pi$-quasi-isometric embeddings with positive boundary maps, and show it coincides with the space of positive quasisymmetric boundary maps $\mathbb{RP}^1\to \mathrm{Flag}(\mathbb{R}^d)$, realizing a universal higher Teichmüller space via harmonic map theory with a concrete asymptotic-Dirichlet description.
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.