Harmonic maps for Hitchin representations

Abstract

Let (S,g_0) be a hyperbolic surface, ρ be a Hitchin representation for PSL(n,R), and f be the unique ρ-equivariant harmonic map from (S,g_0) to the corresponding symmetric space. We show its energy density satisfies e(f) ≥ 1 and equality holds at one point only if e(f) ≡ 1 and ρ is the base n-Fuchsian representation of (S,g_0). In particular, we show given a Hitchin representation ρ for PSL(n,R), every ρ-equivariant minimal immersion f from the hyperbolic plane H^2 into the corresponding symmetric space X is distance-increasing, i.e. f∗gX ≥ gH^2. Equality holds at one point only if it holds everywhere and ρ is an n-Fuchsian representation.