  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.
