Hyper-Kähler metrics from periodic monopoles

Relative moduli spaces of periodic monopoles provide novel examples of asymptotically locally flat hyper-K\"ahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four dimensional, this construction yields interesting examples of metrics with a self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described.