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IMRN International Mathematics Research Notices 2004, No. 18

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 2n-dimensional 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
d 0, iX
1. (1.1)
The main result of this paper is about the global dynamics of the Reeb vector field on
three-dimensional contact manifolds. More precisely, we will prove an existence result
for the so-called "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


Source: Abbas, Casim - Department of Mathematics, Michigan State University


Collections: Mathematics