Asymptotics of certain families of Higgs bundles in the Hitchin component

TL;DR

This paper investigates the asymptotic behavior of certain Higgs bundles and their associated flat connections and harmonic maps within the Hitchin component, revealing new insights into their geometric and analytical properties.

Contribution

It provides a detailed asymptotic analysis of Higgs bundles with specific differentials in the Hitchin component, connecting the behavior of flat connections and harmonic maps.

Findings

Asymptotic estimates for flat connections as differentials scale

Analysis of harmonic maps to symmetric spaces in the Hitchin component

Connections to recent work by Katzarkov, Noll, Pandit, and Simpson

Abstract

Using Hitchin's parameterization of the Hitchin-Teichm\"uller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.