Abstract: Volume forms are very flexible. The only invariants of smooth manifolds under volume preserving diffeomorphisms are the total volume and, when it is noncompact, the finiteness of the volume at each end, proven by Greene--Shiohama. We prove a parametric version of the flexibility of volume forms with some extra conditions, as well as a more general framework of fiber bundles where the volumes forms are on the fibers.
Gromov's seminal non-squeezing theorem, distinguishing symplectic forms from volume forms, shed light on the rigidity aspect of symplectic forms. After that, vast theories of symplectic capacities have been developed to give necessary conditions for symplectic embeddings. Moser proved that on a compact manifold, a smooth deformation within a cohomology class would leave symplectic forms diffeomorphic. Gromov's h-principle provides smooth cohomological paths between symplectic forms on noncompact manifolds and we prove that the Moser's theorem still applies if the path is subject to a growth condition at the infinity. Then we display how to remove a ray from the manifold without changing the symplectic structure.
We then consider Hamiltonian $\mathbb{R}^n$-spaces, which are integrable systems. We are interested in singular points of the focus-focus type and we give a complete classification of the germ at a compact fiber of the momentum map with multiple such points by a tuple of formal power series.
