A compactness theorem for hyperkahler 4-manifolds with boundary

A hyperkähler triple on a compact 4-manifold with boundary is a triple of symplectic 2-forms that are pointwise orthonormal with respect to the wedge product. It defines a Riemannian metric of holonomy contained in SU(2) and its restriction to the boundary defines a framing. In this talk, I will show that a sequence of hyperkähler triples converges smoothly up to diffeomorphims if their restrictions to the boundary converge smoothly up to diffeomorphisms, under certain topological assumptions and the “positive mean curvature” condition of the boundary framings.