Deformations of minimal Lagrangian submanifolds with boundary

Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$ can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near $L$ with boundary on $W$ is found to be finite dimensional and is parametrised over the space of harmonic 1-forms of $L$ satisfying Neumann boundary conditions. The second is that if $W^{\mathrm{\prime }}$ is a symplectic, codimension 2 submanifold sufficiently near $W$, then under suitable conditions, there exists a minimal Lagrangian submanifold $L^{\mathrm{\prime }}$ near $L$ with boundary on $W^{\mathrm{\prime }}$.