Comparison of the Hitchin metric and the semi-flat metric in the rank two case

TL;DR

This paper investigates the rate at which the difference between the Hitchin and semi-flat metrics on rank two Higgs bundles decays exponentially along certain curves, aiming to improve the understanding of their asymptotic behavior.

Contribution

It provides an analysis of the exponential decay rate of the metric difference, enhancing previous results on the asymptotic comparison of the Hitchin and semi-flat metrics.

Findings

Exponential decay of metric difference along Higgs bundle curves

Improved decay rate estimates over previous results

Enhanced understanding of metric asymptotics in rank two case

Abstract

Let $(E,\theta)$ be a Higgs bundle of rank $2$ and degree $0$ on a compact Riemann surface $X$ whose spectral curve is smooth. The tangent space of the moduli space of Higgs bundles at $(E,\theta)$ is equipped with two natural metrics called the Hitchin metric and the semi-flat metric. It is known that the difference between two metrics along the curve $(E,t\theta)$ $(t\geq 1)$ decays in an exponential way. In this paper, we shall study how the exponential rate is improved.