COMPACTIFICATION OF THE SPACE OF HAMILTONIAN STATIONARY LAGRANGIAN SUBMANIFOLDS WITH BOUNDED TOTAL EXTRINSIC CURVATURE AND VOLUME

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in C-n with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian n-varifold locally uniformly in C-k for any nonnegative integer k away from a finite set of points, and the limit is Hamiltonian stationary in C-n. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds L across a compact set N of Hausdorff codimension at least 2 that is locally noncollapsing in volumes matching its Hausdorff dimension, provided the mean curvature of L is in L-n and a condition on local volume of L near N is satisfied.