For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{ C}^{ n}$ 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 ${ \mathbb{ C}}^{ n}$. 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.
Categories: math.DG math.AP