On the supports in the Humilière completion and $\gamma$-coisotropic sets

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.

63 pages, 9 figures. Some minor errors and typos corrected. A previous version had an appendix joint with V. Humilière, which is now included in a joint paper with M.-C. Arnaud and V. Humilière, arXiv:2404.00804 (to appear in Journal de l'Ecole polytechnique)

References in corpus (4)

• Exact Lagrangian submanifolds in simply-connected cotangent bundles
• Sheaf Quantization of Lagrangians and Floer cohomology
• Higher Dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique)
• Regular Lagrangians are smooth Lagrangians