Regular Lagrangians are smooth Lagrangians

TL;DR

This paper proves that in cotangent bundles, elements with smooth or compact $ extgamma$-supports in the Lagrangian completion are actually smooth Lagrangians and connected, respectively.

Contribution

It establishes that $ extgamma$-completions of smooth Lagrangians are themselves smooth when their support is smooth, and connected when support is compact.

Findings

Elements with smooth $ extgamma$-support are smooth Lagrangians.

Elements with compact $ extgamma$-support are connected.

Provides new insights into the structure of Lagrangian completions.

Abstract

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