On Hitchin's equations for cyclic G-Higgs bundles

TL;DR

This paper offers a Lie-theoretic approach to Hitchin's equations for cyclic G-Higgs bundles, proving key conjectures about energy density and curvature for Coxeter cyclic cases across all groups except certain exceptional types.

Contribution

It introduces a Lie-theoretic framework for analyzing Hitchin's equations and proves conjectures on energy density monotonicity and negative curvature for Coxeter cyclic G-Higgs bundles.

Findings

Proved Dai-Li's conjecture on energy density monotonicity for Coxeter cyclic G-Higgs bundles.

Confirmed Dai-Li's negative curvature conjecture for Coxeter cyclic G-Higgs bundles, excluding E7 and E8 types.

Abstract

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 $\mathrm{E}_8.$