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

Let (E, ) 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, ) 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) (t 1) decays in an exponential way. In this paper, we shall study how the exponential rate is improved.