Floer homology, symplectic and complex hyperbolicities

On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, we define and study a notion of symplectic hyperbolicity and a symplectic capacity measuring it. On the other side, the usual notions of complex hyperbolicity can be straightforwardly generalized to the case of almost-complex manifolds by using pseudo-holomorphic curves. That’s why we study the links between these two notions of hyperbolicities when a manifold is provided with some compatible symplectic and almost-complex structures. We mainly explain how the non-symplectic hyperbolicity implies the existence of pseudoholomorphic curves, and so the non-complex hyperbolicity. Thanks to this analysis, we could both better understand the Floer cohomology and get new results on almost-complex hyperbolicity. We notably prove results of stability for non-complex hyperbolicity under deformation of the almost-complex structure among the set of the almost-complex structures compatible with a fixed non-hyperbolic symplectic structure, thus generalizing Bangert theorem that gave this same result in the special case of the standard torus. Symplectic manifolds are naturally provided with compatible almost-complex structure. Let us recall that a symplectic structure and an almost-complex structure are “compatible” if they define a Riemannian metric on the manifold; this metric is called almost-Kähler (and is a Kähler metric if the almost complex structure is integrable). This naturally raises the issue of links between the symplectic properties and the almost-complex properties of the manifold when it is provided with two compatible structures. This analysis is likely to provide a new and interesting approach to both these fields, and in fact it has already been done. When an almost-complex structure is given, you can define pseudo-holomorphic curves which generalize the notion of holomorphic curves in complex manifolds. Their introduction by Gromov [18] in symplectic geometry led to an explosion of research in this field, allowing to solve many problems and to define new symplectic invariants such as the Gromov invariant defined by counting pesudoholomorphic curves. This work fits in the framework of this analysis about links between symplectic and almostcomplex properties, and more precisely, we tackle it from the point of view of hyperbolicities notions.