Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles

The moduli space of stable Higgs bundles of degree $0$ is equipped with the hyperkähler metric, called the Hitchin metric. On the locus where the spectral curves are smooth, there is the hyperkähler metric called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. We prove the exponentially rapid decay of the difference between the Hitchin metric and the semi-flat metric along the ray $(E,t\theta )$ as $t\to \mathrm{\infty }$.