LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 51there is still something like a system of surfaces of section. We will only give the
result and refer the reader to the forth-coming article [56]. Figure 5 shows some
generalisation.
Next we introduce the main concept in dealing with more general flows.
We begin with an important definition.
Definition 1.47. A finite energy foliation for (M, .A, J) is by definition a 2-dim-
ensional smooth foliation F of JR x M such that the following holds:
• There exists a universal constant C > 0 such that for every leaf F E F there
exists an embedded finite energy curve (S, r, ii) for (M, .A, J) satisfying
F = ii(S \ r)
and E(ii)::; C. (All punctures are assumed to be not removable).- For every a E JR the set Fa = T a(F) belongs to F , where T a(b, m) = (a+
b, m). In particular either Fa= For Fan F = 0.
Let us consider an explicit example. Take M = S^3 being the round sphere in
C^2. Take the standard contact form
1
.Ao= "2[q · dp-p · dq]
and on fo complex multiplication by i. The images of the planes
C x {O}, where c EC\ {O}
Figure 5. The figure shows the trace of the projection o f a finite energy
foliation on a two-dimensional plane. Here we have three spanning orbits E1
and E 2 which are elliptic and one hyperbolic one denoted by H. Moreover
the foliation contains planes and cylinders. The dashed lines are the traces of
the stable and unstable manifold of the hyperbolic orbit H. We assume the
non-generic situation that they precisely match up creating several invariant
sets. The dotted lines are periodic orbits for the Reeb vector field. The fat
lines represent rigid pieces of the finite energy foliation, n a m e ly two cylinders
and two planes. The three-sphere is viewed as IR^3 U { oo}.