Appears in collection : 2025 - T2 - WS1 - Higher rank geometric structures, Higgs bundles and physics

Let $X$ be a pinched Cartan-Hadamard manifold, and $Y$ a symmetric space of non-compact type. We define a notion of stability for coarse Lipschitz maps $f: X \to Y$, and show that every stable map from $X$ to $Y$ is at bounded distance from a unique harmonic map. As an application, we extend any positive quasi-symmetric map from $\mathbb{RP}^1$ to the flag variety of $\textrm{SL}_n(\mathbb{R})$ to a harmonic map from $\mathbb H^2$ to the symmetric space of $\textrm{SL}_n(\mathbb{R})$. This allows us to define a universal Hitchin component in the style suggested by Labourie and Fock-Goncharov. This is all joint work with Max Riestenberg.