Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank

TL;DR

This paper studies the asymptotic behavior of solutions to Hitchin's equations for large parameters on stable Higgs bundles over Riemann surfaces, showing exponential convergence to a decoupled harmonic metric under specific conditions.

Contribution

It proves exponential convergence of harmonic metrics for Higgs bundles induced by line bundles on spectral curve normalizations, extending understanding of large-scale solutions.

Findings

Sequence of harmonic metrics converges exponentially.

Uniform convergence for a family of Higgs bundles.

Results apply to Higgs bundles induced by line bundles.

Abstract

Let $X$ be a compact Riemann surface. Let $(E,\theta)$ be a stable Higgs bundle of degree $0$ on $X$. Let $h_{\det(E)}$ denote a flat metric of the determinant bundle $\det(E)$. For any $t>0$, there exists a unique harmonic metric $h_t$ of $(E,\theta)$ such that $\det(h_t)=h_{\det(E)}$. We prove that if the Higgs bundle is induced by a line bundle on the normalization of the spectral curve, then the sequence $h_t$ is convergent to the naturally defined decoupled harmonic metric at the speed of the exponential order. We also obtain a uniform convergence for such a family of Higgs bundles.