Free Boundary Hamiltonian Stationary Lagrangian Discs in C 2.

Abstract

Let $\Omega \subset C^2$ be a smooth domain. We establish conditions under which a weakly conformal, branched $\Omega$ -free boundary Hamiltonian stationary Lagrangian immersion u of a disc in $C^2$ is a $\Omega$ -free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1\left(0\right)$ -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then $u\left(D^2\right)$ is a Lagrangian equatorial plane disc. Furthermore, we present examples of $\Omega$ -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.