Lagrangian Submanifolds of Euclidean Space

We give an exposition of the result that there is no closed exact Lagrangian submanifold L of (C, ω0) where ω0 is the standard symplectic structure. We show that the assertion is equivalent to the statement that the perturbed Cauchy-Riemann equation ∂̄J0u = g for maps u from the unit disc D to C which map the boundary circle ∂D to L has no solution for some function g0. To do this, we follow [1] and consider the universal moduli spaceM = {(u, g) : ∂̄J0u = g} and show that if we assume L to be exact, the projection (u, g) 7→ g is surjective in suitable spaces. To obtain surjectivity, it is necessary to show that this projection is proper, a property which follows from Gromov’s theorem of compactness for pseudoholomorphic curves. We provide a proof of this compactness theorem, following arguments in [12], by obtaining a subsequence which converges modulo bubbling and removing the bubble point singularities. A proof is given in the case of interior singularities and we give suggestions for how to modify the method for singularities on the boundary.