Free boundary Hamiltonian stationary Lagrangian discs in ${\mathbb{C}}^2$

Abstract

Let $\Omega \subset {\mathbb{C}}^2$ 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 ${\mathbb{C}}^2$ is a $\Omega$-free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1(0)$-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary data, then $u(D^2)$ must be a Lagrangian equatorial plane disc. We also present examples of $\Omega$-free boundary Hamiltonain stationary discs, demonstrating the optimality of our assumptions.

arXiv comment13 pages