Floer homology, symplectic and complex hyperbolicities

TL;DR

This paper explores the relationship between symplectic and complex hyperbolicities using Floer cohomology, establishing new links and stability results for almost-complex structures on symplectic manifolds.

Contribution

It introduces a symplectic capacity related to hyperbolicity and demonstrates how non-symplectic hyperbolicity implies complex hyperbolicity, extending stability results to broader settings.

Findings

Non-symplectic hyperbolicity implies the existence of pseudo-holomorphic curves.

Stability of non-complex hyperbolicity under deformation of almost-complex structures.

Generalization of Bangert's theorem to broader symplectic contexts.

Abstract

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 pseudo-holomorphic 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.