I will review some known results and several open questions concerning the L^2 cohomology of various non-compact moduli spaces carrying a hyperkähler metric. Particular attention will be given to the moduli space M of Higgs bundles over a Riemann surface, which arises as the space of solutions of a certain system of nonlinear PDEs, called self-duality equations. Although many aspects of M as a hyperkähler manifold are by now well-established, it is still an open problem to describe the geometry of this non-compact manifold near its ends. In this talk, I shall present some preliminary results in that direction.

(Joint work with Rafe Mazzeo, Hartmut Weiß, and Frederik Witt).