A compactness theorem for hyperkähler 4-manifolds with boundary

In this paper, we study the compactness of a boundary value problem for hyperkähler 4-manifolds. We show that under certain topological conditions and the positive mean curvature condition on the boundary, a sequence of hyperkähler triples converges smoothly up to diffeomorphisms if and only if their restrictions to the boundary converge smoothly up to diffeomorphisms. We also generalize this result to torsion-free hypersymplectic triples.