Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

TL;DR

This paper investigates the asymptotic behavior of harmonic metrics on stable Higgs bundles over Riemann surfaces as the Higgs field is scaled, revealing decoupling phenomena and convergence to limiting configurations.

Contribution

It demonstrates the asymptotic decoupling of the Hitchin equation for generically regular semisimple Higgs fields and establishes convergence results for rank two bundles, introducing a rule for parabolic weights.

Findings

Hitchin equation becomes asymptotically decoupled for regular semisimple Higgs fields.

Sequence of harmonic metrics converges to a limiting configuration in rank two cases.

Results extend to Higgs bundles with Hermitian-Einstein metrics on curves.

Abstract

Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,t\theta)$ such that $\det(h_t)=h_{\det(E)}$. We study the behaviour of $h_t$ when $t$ goes to $\infty$. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence $(E,\overline{\partial}_E,\theta,h_t)$ in the case where the rank of $E$ is two. We introduce a rule to determine the parabolic weights of a "limiting configuration", and we show the convergence of the sequence to the limiting configuration in an appropriate sense. The results can be appropriately generalized in the context of Higgs bundles with a Hermitian-Einstein metric on curves.