Harmonic maps for Hitchin representations

Let $(S,g_0)$ be a hyperbolic surface, $\rho$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $\rho$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\equiv 1$ and $\rho$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $\rho$ for $PSL(n,\mathbb R)$, every $\rho$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is distance-increasing, i.e. $f^*(g_{X})\geq g_{\mathbb H^2}$. Equality holds at one point only if it holds everywhere and $\rho$ is an $n$-Fuchsian representation.Comment: 14 pages, comments are welcom

On the Uniqueness of Vortex Equations and Its Geometric Applications

We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map u : C → H^2 satisfying ∂u ≠ 0 with prescribed polynomial Hopf differential; there is a unique affine spherical immersion u : C → R^3 with prescribed polynomial Pick differential. We also show that the uniqueness fails for non-polynomial entire functions with finitely many zeros

Minimal surfaces for Hitchin representations

Given a reductive representation ρ:π_1(S) → G, there exists a ρ-equivariant harmonic map f from the universal cover of a fixed Riemann surface Σ to the symmetric space G/K associated to G. If the Hopf differential of f vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: the q_n and q_(n−1) cases. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system