On cyclic Higgs bundles

TL;DR

This paper develops a maximum principle for elliptic systems and applies it to analyze cyclic Higgs bundles, providing bounds on associated minimal immersions and characterizing maximal representations in specific Lie groups.

Contribution

It introduces a maximum principle for elliptic systems and applies it to derive geometric bounds and classification results for cyclic Higgs bundles and related representations.

Findings

Domination results on pullback metrics of minimal immersions

Bounds on extrinsic curvature of the immersion images

Complete classification of maximal representations in certain components

Abstract

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of $f$. As an application, we give a complete picture for maximal $Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the Hitchin components.