Regular Lagrangians are smooth Lagrangians

Abstract

We prove that for any element in the 𝛾-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its 𝛾-support is a smooth Lagrangian submanifold, then the element itself is a smooth Lagrangian. We also prove that if the 𝛾-support of an element in the completion is compact, then it is connected.