Hamiltonian stationary maps with infinitely many singularities

TL;DR

This paper constructs a class of Hamiltonian stationary Lagrangian maps in complex two-space with infinitely many singularities, maintaining high regularity up to the boundary.

Contribution

It introduces a method to create Hamiltonian stationary Lagrangian maps with infinitely many singularities while preserving smooth boundary behavior.

Findings

Existence of maps with infinitely many singularities

Maps are of class C^k up to the boundary

Maps have smooth trace on the boundary

Abstract

For any $k\in \mathbb{N}$ we construct an Hamiltonian stationary Lagrangian map from a disc to $\mathbb{C}^2$ with infinitely many Schoen-Wolfson singularities which is of class $C^k$ up to the boundary and has smooth trace.