Moduli Spaces of Noncompact Special Lagrangian Submanifolds

The purpose of this thesis is to study the deformations of a 3-dimensional asymptotically cylindrical special Lagrangian submanifold. We first show that, using the Implicit Function Theorem for Banach Spaces, the moduli (or deformation) space of a 3-dimensional asymptotically cylindrical special Lagrangian submanifold with a given decay rate and fixed boundary at infinity is a smooth, finite-dimensional manifold for almost all decay rates. More importantly, we calculate, using the theory of elliptic operators on asymptotically cylindrical manifolds, the explicit decay rates of the asymptotically cylindrical special Lagrangian submanifolds for which the above result holds. Next, we show how to extend these arguments to include a moving boundary at infinity, that is, we show that the moduli space of a 3-dimensional asymptotically cylindrical special Lagrangian submanifold with a given decay rate and moving boundary at infinity is a smooth, finite-dimensional manifold for almost all decay rates. Finally, we show using algebraic topology that the moduli space of 3-dimensional asymptotically cylindrical special Lagrangian cycles can be immersed as a Lagrangian submanifold of the moduli space of the 2-dimensional compact special Lagrangian cycles at infinity.