Work of Song Dai
( joint with Qiongling Li )
Let Σ be a closed Riemann surface of genus g>1. Let G be a reductive Lie group. Consider the moduli space of the polystable G-Higgs bundles over Σ. From the non-Abelian Hodge theory, a Higgs bundle in the moduli space corresponds to an equivariant harmonic maps from the universal cover of Σ to the symmetric space of G.
In this paper, we made some new observations under the setting of cyclic Higgs bundles. We introduced a new maximum principle for certain elliptic systems. By using this tool, we obtain some domination results. Firstly, we improved the domination result in the Hitchin section to the monotonicity result along a natural ray in the whole space. Secondly, in certain cases we obtained a lower bound of the extrinsic curvature of the harmonic map. Thirdly, we observed in certain cases, the energy density of the Higgs bundles in the Hitchin section is maximal in their Hitchin fibers. This observation gives an evidence to the conjecture of the maximality of the Hitchin representations.