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Lecture 5. Some Outlook

In the previous sections we described some of the details of the proof of the

Weinstein conjecture. If one wants to study the Reeb flows in more detail the whole

theory has to be extended.

One crucial ingredient is a Fredholm theory for finite energy curves. The Fred-

holm theory is subtle since we are working on non-compact manifolds. The be-

haviour of the curves in a neigbhourhood of a puncture has to be understood.

Moreover the Fredholm index depends on certain topological quantities which have

to be identified. Also the intersection theory for finite energy surfaces has to be

developed. The following is an overview of the literature addressing these points.

• A comprehensive and self-contained description of the machineries in the

proof of the Weinstein conjecture in dimension three: [1].

• Behaviour near an interior puncture: [53], [54].
  Behaviour near a puncture at the boundary in the case of suitable boundary
  value problems and the relationship to Arnold's chord conjecture: [2].


• Local and global behaviour of finite energy surfaces and topological invari-
  ants: [52].


• Fredholm theory. There are different ways of approaching a Fredholm the-
  ory:
  Fredholm theory in the embedded case: [50].
  General Fredholm theory: [57].


• Applications to dynamical systems: [55], [49].
  Applications to topology:[51], [48].


• Using sufficiently rich families of holomorphic curves in suitable compact
  symplectic manifolds of dimension 4 one can construct by a deformation
  method finite energy foliations, see [56].

There is also a potentially rich interface between the Seiberg-Witten theory, holo-
morphic curves in symplectic four-manifolds and holomorphic curves in symplecti-
sations: [83, 82],[8].

Acknowlegments: Special thanks go to Casim Abbas, Yasha Eliashberg, Dietmar
Salamon, Kris Wysocki and Eduard Zehnder for many stimulating mathematical

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