On Symplectic Invariants Associated to Zoll Manifolds.

[摘要] In this thesis, we provide a partial classification for M. Audin’s polarized symplectic manifolds, which are smooth symplectic manifolds endowed with a Morse-Bott function having only two critical values—a minimum, which is attained on a Lagrangian submanifold, and a maximum, which is attained on a symplectic submanifold of codimension 2. We provide examples via Lerman’s symplectic cut construction in which the Lagrangian minima are the Zoll manifolds, i.e. Riemannian manifolds all of whose geodesics are simply closed and of the same period. Given a polarized symplectic manifold with some additional assumptions on the Morse-Bott function, we prove that the Lagrangian minimum must be Zoll and obtain a local equivalence of such manifolds on a neighborhood of the Lagrangian. We then extend the equivalence out towards the symplectic maximum using gradient flows. The main tools we use are arguments in symplectic geometry and Morse(-Bott) theory. In low dimensions, we make use of a result due to McDuff and Lalonde to classify the 4-dimensional polarized symplectic manifolds up to symplectomorphism.

[发布机构] University of Michigan

[效力级别] Polarized Symplectic Manifolds

[关键词] Zoll Manifolds;Polarized Symplectic Manifolds;Mathematics;Science;Mathematics