Global surfaces of section for the restricted three body problem

In the restricted three body problem one studies the motion of a massless body which is attracted by two massive bodies according to Newton's law of gravitation. Due to preservation of energy if the massless body moves in the plane spanned by the two massive objects its dynamics is described by a Hamiltonian flow on a three dimensional energy hypersurface in four dimensional phase space. A global surface of section is a gadget which stores the dynamics on a three dimensional energy hypersurface in an area preserving disk map. In the talk I explain how global methods from the theory of holomorphic curves can be applied to prove existence of such a gadget. These methods contrast sharply with perturbative methods mainly used so far.