1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE l. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 43

There is a priori the possibility that a Reeb-like vector field is simultaneously

tight and overtwisted. This is ruled out by the following exercise.

Exercise 1.20. If Xis Reeb-like and there are volume forms n 1 and one-forms >.. 1

such that ixD 1 = d>.. 1 and ix >..i > 0 for j = 1, 2, then either the >..i are both tight

or both overtwisted. ·

The main result on the three-dimensional Weinstein conjecture is

Theorem 1.21. {Hofer) Let X be a Reeb-like vector field on the closed three-

manif old M. Then the Weinstein conjecture holds if either M is diff eomorphic

to 83 or if n 2 ( M) =I- 0 or if X is overtwisted.

1.2. Holomorphic curves and dynamics

After seeing all these results, but not the proofs, one might be interested to see what

makes the difference between a Reeb vector field and an ordinary vector field. The

answer to that question is surprising and so is t he starting point of that discussion.

The following is also true in higher dimension but since we are interested in

three-dimensional dynamics we only consider the relevant case.

We begin with a fresh look at the complex vector space (C^2 , i). We define a

diffeomorphism

which we turn into a biholomorphic m ap by equipping the target space with a
complex structure Jo which is defined by

Jo o T<I> = T<I> o i.

A straight forward calculation shows that

Jo(a, u)(h, k) = (->..o(u)(k), inok + hXo(u)),

where Ao = ~ [q. dp - p. dq] 183. Here 83 is viewed as the unit sphere in C^2 and

the coordinates in C^2 are given by z = q + ip with q,p E ~^2. Observe that >..o is

a contact form. Its associated contact structure fo is the bundle of complex lines
in T8^3 and the Reeb vector field is Xo. It generates the Hopf fibration. Moreover
no : T 83 ----> fo is the projection along X 0. As complex multiplication on fo we take
i.
Hence studying holomorphic curves in C^2 \ {O} is equivalent to the study of

J 0 -holomorphic curves in ~ x 83. We note that there are no closed holomorphic

curves in ~ x 83 as a consequence of the Liouville theorem on C^2.
In C^2 there is an admittedly nice set of holomorphic curves namely the affine
algebr~ic sets of (complex) dimension 1. We might wonder how the image under
of such a set, perhaps removing the point 0, looks like. The following theorem has
been proved in [41].