Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume

TL;DR

This paper studies the limits of sequences of Hamiltonian stationary Lagrangian submanifolds in complex Euclidean space, showing convergence properties and extension results under bounded curvature and volume conditions.

Contribution

It establishes convergence of such submanifolds to varifolds and provides a method to extend them across certain singular sets, advancing understanding of their compactification.

Findings

Sequences converge to points or Hamiltonian stationary varifolds

Limit varifolds are Hamiltonian stationary

Extension theorem across codimension ≥ 2 sets

Abstract

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.