Asymptotics of Hitchin's metric on the Hitchin section

TL;DR

This paper proves that Hitchin's hyperk"ahler metric on the moduli space of degree zero SL(2)-Higgs bundles converges exponentially to a semiflat metric along generic rays within the Hitchin section, confirming a long-standing conjecture.

Contribution

It establishes the exponential convergence of Hitchin's metric to the semiflat metric specifically on the tangent bundle of the Hitchin section, advancing understanding of the metric's asymptotic behavior.

Findings

Proves exponential convergence of Hitchin's metric to the semiflat metric.

Confirms the conjecture for the restriction to the Hitchin section.

Provides detailed analysis of the metric's asymptotics along generic rays.

Abstract

We consider Hitchin's hyperk\"ahler metric $g$ on the moduli space $\mathcal{M}$ of degree zero $\mathrm{SL}(2)$-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in $\mathcal{M}$, $g$ converges to an explicit "semiflat" metric $g^{\mathrm{sf}}$, with an exponential rate of convergence. We show that this is indeed the case for the restriction of $g$ to the tangent bundle of the Hitchin section $\mathcal{B} \subset \mathcal{M}$.