Parabolic Higgs bundles and gravitational instantons

I will survey recent results on the asymptotic geometry of Hitchin’s hyperkähler metric on the moduli space of Higgs bundles. I will then show how to obtain gravitational instantons from 4-dimensional Hitchin systems. Finally I will explain ongoing work with Fredrickson, Mazzeo and Swoboda on a particular case of Boalch’s modularity conjecture.