Asymptotics of Hitchin's Metric on the Hitchin Section

We consider Hitchin's hyperkähler metric g on the moduli space M of degree zero SL(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in M, g converges to an explicit "semiflat" metric g^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 B⊂M.