Regular Lagrangians are smooth Lagrangians

Abstract

We prove that for any element in the ${\hat{I}}^3$-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its ${\hat{I}}^3$-support is a smooth Lagrangian submanifold, then the element itself is a smooth Lagrangian. We also prove that if the ${\hat{I}}^3$-support of an element in the completion is compact, then it is connected.

16 pages, 2 figures. v2: Revised, to appear in J. Math. Soc. Japan