On the supports in the Humili\`ere completion and $\gamma$-coisotropic sets

TL;DR

This paper explores the concept of $oldsymbol{ extgamma}$-support and $oldsymbol{ extgamma}$-coisotropic sets within the symplectic spectral metric framework, analyzing their properties and examples in the completion of Lagrangian submanifolds.

Contribution

It introduces the notions of $oldsymbol{ extgamma}$-support and $oldsymbol{ extgamma}$-coisotropic sets, establishing their properties and relation to symplectic geometry completions.

Findings

$oldsymbol{ extgamma}$-support must be $oldsymbol{ extgamma}$-coisotropic

Examples of Lagrangians with large $oldsymbol{ extgamma}$-support

Analysis of 'regular Lagrangians' with small $oldsymbol{ extgamma}$-support

Abstract

The symplectic spectral metric on the set of Lagrangian submanifolds or Hamiltonian maps can be used to define a completion of these spaces. For an element of such a completion, we define its $\gamma$-support. We also define the notion of $\gamma$-coisotropic set, and prove that a $\gamma$-support must be $\gamma$-coisotropic toghether with many properties of the $\gamma$-support and $\gamma$-coisotropic sets. We give examples of Lagrangians in the completion having large $\gamma$-support and we study those (called "regular Lagrangians") having small $\gamma$-support. We compare the notion of $\gamma$-coisotropy with other notions of isotropy.