!AS/Park City Mathematics Series
Volume 7, 1999Holomorphic Curves and Dynamics in
Dimension Three
Helmut Hofer
LECTURE 1
Problems, Basic Concepts and Overview
In t his section we provide some background material, introduce some basic
concepts and survey the main results.1.1. Periodic orbits of smooth vector fields on three-manifolds
In 1950 H. Seifert, [80], raised the question if a nowhere vanishing smooth vector
field on S^3 has a periodic orbit. This question mutated into what is commonly
referred to as the Seifert conjecture:
Conjecture 1.1. (Seifert) An everywhere nowhere vanishing smooth vector field
on S^3 has a periodic orbit.
Only in 1974 P. Schweitzer was able to give a C^1 -counter example, i.e. a
nowhere vanishing C^1 -vector field without any periodic orbits, see [79]. The prob-
lem was finally resolved by K. K uperberg who provided a cw-counter example,
[62].
Theorem 1.2. (K. Kuperberg) There exists a real analytic counter example to the
Seifert conjecture. With other words there exists a real analytic nowhere vanishing
vector field on S^3 without any periodic orbits.
K. Kuperberg's counter example is not volume preserving. What happens if we
look for a volume preserving counter example? In this case G. Kuperberg provides
a partial answer, [63].
Theorem 1.3. (G. Kuperberg) Given a smooth volume form n on S^3 there exists
a nowhere vanishing C^1 -vector field X having no periodic orbit and having a flow
(^1) Courant Institute, 251 Mercer Street, New York, NY 10012.
WWW: http://www.math.nyu.edu/research/hofer.
E-mail address: hof er©cims. nyu. edu.
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@ 1999 American Mathematical Society