50 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
This prompts a certain number of conjectures.Conjecture 1.41. On a homotopy three sphere a tight contact structure admits
an unknotted periodic orbit xr with sl(xr) = -1.
Of course, this conjecture is implied by the Poincare conjecture and theorem
1.40. However it might be that just the knowlegde that the manifold is a homotopy
sphere allows to construct a periodic orbit.
Conjecture 1.42. A dynamically convex homotopy three sphere is 83.
This conjecture seems easier than the previous one. As it turns out it sufficesto find an unknotted periodic orbit with self-linking number -1.
Conjecture 1.43. A tight homotopy three sphere is 53.
A solution of the first conjecture implies a solution of the second conjecture
(not obvious). The third conjecture is presumably the hardest. Some specialists
in symplectic topology strongly believe that every orientable closed three manifold
admits a tight contact structure. If that would be true, a solution of conjecture
1.43 would imply the Poincare conjecture. Therefore let us stateConjecture 1.44. Every closed orientable three-manifold admits a tight contact
structure.
Theorem 1.39 is extremely sharp. On 81 x 82 there exists a contact form A such
that 'PE. = 0 and for every contractible periodic orbit (x, T) with minimal period T
we have
I(x, T) c [1, oo)
and X admits periodic orbits ( x, T) with minimal period T and sl ( xr) = -1, x r
unknotted. In Theorem 1.39 we have I(x, T) C (1, oo) for the contractible periodic
orbits with minimal periods T.
Exercise 1.45. Construct such a A on 81 x 82. Assume that 81 x 53 c 81 x IR.^3
with coordinates (B, p,x,y). Consider the restriction A of the 1-form
1
pd()+2
[xdy - ydx]to 81 x 82.
From a dynamical point of view it is interesting to understand t ight Reeb flows
on 83. By a classification result due to Eliashberg this boils down to understanding
the Reeb vector fields associated to f Ao, where f : 83 --+ (0, oo) is a smooth function.
The Reeb flows obtained this way are quite general. Using Eliashberg's results one
can prove the following:
Proposition 1.46. Let g be a Riemannian metric on 82 and let T 182 be the unit
sphere bundle equipped with the induced vector field X generating the geodesic flow.
Then X is a Reeb vector field. Moreover there exists a double covering map 'Y :
53 --+ T 182 = RP^3 , which pulls X back to the Reeb vector field Y associated to the
form f>.o, where f: 83 --+ (0, oo) has the symmetry f(z) = f(-z) for z E 53.
So Reeb flows associated to forms f Ao are more general than the doubly-covered
geodesic flows on T 182. They even include the flows associated to Finsler metrics.
Starting with Ao = 1 · Ao and deforming 1 to an arbitrary positive function f
there is no reason that Xf>-.o admits a global disk-like surface of section. However