 
Summary: IMRN International Mathematics Research Notices
2004, No. 18
The Chord Problem and a New Method of Filling
by Pseudoholomorphic Curves
Casim Abbas
1 Introduction
Let M be a closed (2n + 1)dimensional manifold with contact form , that is, is a one
form on M such that (d)n
is a volume form. The contact structure associated to is
the 2ndimensional vector bundle = ker M, which is a symplectic vector bundle
with symplectic structure d. There is a distinguished vector field associated to a
contact form, the Reeb vector field X, which is defined by the formulas
iX
d 0, iX
1. (1.1)
The main result of this paper is about the global dynamics of the Reeb vector field on
threedimensional contact manifolds. More precisely, we will prove an existence result
for the socalled "characteristic chords." These are trajectories x of the Reeb vector field,
which hit a given Legendrian submanifold L at two different times t = 0 and T > 0. We
also ask for x(0) = x(T), otherwise, the chord would actually be a periodic orbit. Recall
