On Hitchin's equations for cyclic G-Higgs bundles

We develop a Lie-theoretic perspective on Hitchin's equations for cyclic
$G$-Higgs bundles, which we use to study analytic and geometric properties of
harmonic maps. Among other things, we prove Dai-Li's conjecture on the
monotonicity of the energy density in the case of Coxeter cyclic $G$-Higgs
bundles, for all $G$, and Dai-Li's negative curvature conjecture for Coxeter
cyclic $G$-Higgs bundles, for all $G$ except those of type $\mathrm{E}_7$ and
$\mathrm{E}_8.$