40 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
Definition 1.10. We call a smooth vector field X on a closed orientable three-
manifold M Reeb-like if there exists a volume form n and a one-form ,\ satisfying
ixn = d>-
ix>.> 0.The interesting result is nowTheorem 1.11 (Hofer). Every Reeb-like vector field X on S^3 has a periodic orbit.Before we continue let us give a few exercises related to the notion "Reeb-like".Exercise 1.12. l. Show that a Reeb-like vector field X on a closed orientable
smooth manifold M generates a volume preserving flow.- Show that a vector field is Reeb-like iff there exists a one-form ,\ such that
d,\ is point-wise of maximal rank and ixd>-= 0 and ix,\ > 0.
3. If Xis Reeb-like so is f X, where f is a smooth map with values in JR\ {O}.
- Show that the set of Ree b-like vector fields on S^3 constitutes an open subset
of the space of all smooth vector fields preserving some volume form (de-
pending on the vector field). The space of vector fields is equipped with the
C^00 -topology. - Show that the vector field generating the Hopf fibration is Reeb-like.
- Show that a vector field X on a closed orientable smooth three-manifold M
is Reeb-like iff there exists a Riemannian metric g, a volume form n, and a
smooth map f : M ~ (0, oo) such that
f · ixSl = d(ixg).
- Construct a non-singular volume preserving vector field on S^3 , which is not
Reeb-like.
8. Assume that H : JR^4 ~ JR is a smooth Hamiltonian and S = H-^1 (E) is a
regular level surface which is starshaped. Show that XHIS is Reeb-like.Theorem 1.11 proves a special case of the so-called Weinstein conjecture. Let
us give a reformulation of the Weinstein conjecture in dimension three.Conjecture 1.13 (Weinstein conjecture in dimension three). Every Reeb-like vec-
tor field on a closed three manifold has a periodic orbit.This is a difficult conjecture and it is not proved in generality yet. We come
back to it later.
In order to see that the Weinstein conjecture formulated for regular energy
surfaces in four-dimensional symplectic manifolds is equivalent to the reformulation
we give an exercise.Exercise 1.14. l. Assume that X is a Reeb-like vector field on the closed
three-manifold M. Identify M with {O} x M in W =JR x M. Construct a
symplectic form w on W such that the Hamiltonian vector field associatedto H(s, m) = s satifies XH(O, m) = f(m)X(m) for all m E M and some
smooth map f : M ~ JR \ { 0}, so that there exists ,\ on { 0} x M satisfying
d).. = wl( {O} x M) and >-(m)(XH(m)) =/= 0 for all m E M.
- Assume that the hypotheses of the Weinstein conjecture hold and W is
four-dimensional. Show that XH on I; is Reeb-like.