Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems

Abstract

Let $I$ be an open interval containing zero, let $M$ be a real manifold, let ${\mathrm{Ṫ}}^{\ast }M$ be its cotangent bundle with the zero-section removed, and let $\Phi =\{{\phi }_t{\}}_{t\in I}$ be a homogeneous Hamiltonian isotopy of ${\mathrm{Ṫ}}^{\ast }M$ with ${\phi }_0=id$. Let $\Lambda \subset {\mathrm{Ṫ}}^{\ast }M\times {\mathrm{Ṫ}}^{\ast }M\times T^{\ast }I$ be the conic Lagrangian submanifold associated with $\Phi$. We prove the existence and unicity of a sheaf $K$ on $M\times M\times I$ whose microsupport is contained in the union of $\Lambda$ and the zero-section and whose restriction to $t=0$ is the constant sheaf on the diagonal of $M\times M$. We give applications of this result to problems of nondisplaceability in contact and symplectic topology. In particular we prove that some strong Morse inequalities are stable by Hamiltonian isotopies, and we also give results of nondisplaceability for nonnegative isotopies in the contact setting.