The space of hyperk\"ahler metrics on a 4-manifold with boundary

Let X be a compact 4-manifold with boundary. We study the space of hyperk\"ahler triples on X, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperk\"ahler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperk\"ahler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of SU(2).