  In this paper, we study the compactness of a boundary value problem for
hyperkaehler 4-manifolds. We show that under certain topological conditions and
the positive mean curvature condition on the boundary, a sequence of
hyperkaehler 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.
