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

Work of Song Dai

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the n-Fuchsian fibers. More precisely, we show that the energy density of the 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 a certain product Riemannian (or pseudo-Riemannian) manifold. In particular we generalize a theorem of R. Schoen. Our proof is based on establishing an algebraic inequality generalizing a theorem of L. Ness on the nilpotent orbits to general orbits, which has its own interests in the geometric invariant theory. This is a joint work with Qiongling Li.

Proc. London Math. Soc. (3) 2022;124:427–477.

https://doi.org/10.1112/plms.12431