Domination results in $n$-Fuchsian fibers in the moduli space of Higgs bundles

Abstract

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the n-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an n-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.(1) the energy density satisfies e(f ) < e(f τn○j );(2) the pullback metric satisfies g f < g f τn○j ;(3) the translation length spectrum satisfies l ρ < λ ⋅ l τn○j for some positive constant λ < 1; (4) the energy satisfies E(f ) < E(f τn○j );(5) the entropy satisfies h(ρ) > h(τ n ○ j) = 6 n 3 −n , unless P(ρ) = P(τ n ○ j).Remark 1.2. For P(ρ) = P(τ n ○ j), we mean ρ is conjugate to (τ n ○ j) ⋅ µ (n) for some unitaryin which case, it has the same harmonic map and the same translation length spectrum as τ n ○ j.Remark 1.3. In the case of SL(2, C), Theorem 1.1 were shown by Deroin and Tholozan [9]. Note that in this case, every Hitchin fiber is an 2-Fuchsian fiber.Remark 1.4. The second author in [22] shows a more refined domination result inside the nilpotent cone.Remark 1.5. Potrie and Sambarino [25] showed that for any Hitchin representation ρ ∶ π 1 (S) → SL(n, R), one has the entropy h(ρ) ≤ h(τ n ○ j) = 6 n 3 −n and the equality holds only if ρ is n-Fuchsian. We can see that the n-Fuchsian fibers possess an opposite behavior comparing to the Hitchin section in the Hitchin fibration.