Biolley, Anne-Laure (2003) Floer cohomology, symplectic and almost-complex hyperbolicities. PhD thesis CMAT, CMAT, EP/X p.132.

Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Biolley_web.pdf

Abstract

On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, I 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 pseudoholomorphic curves. That's why I study the links between these two notions of hyperbolicities when a manifold is provided with some compatible symplectic and almost-complex structures. I mainly explain how the non-symplectic hyperbolicity implies the existence of pseudoholomorphic curves, and so the non-complex hyperbolicity. Thanks to this analysis, I could both better understand the Floer cohomology and get new results on almost-complex hyperbolicity. I 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.
Moreover, I tackle the issue of complex hyperbolicity of foliations: through the introduction of an invariant tensor associated to the foliation, I study the existence of foliated holomorphic cylinder.

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