21 research outputs found
Global surfaces of section for Reeb flows in dimension three and beyond
We survey some recent developments in the quest for global surfaces of
section for Reeb flows in dimension three using methods from Symplectic
Topology. We focus on applications to geometry, including existence of closed
geodesics and sharp systolic inequalities. Applications to topology and
celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written
for Proceedings of the ICM, Rio 2018; in v3 we made minor additional
corrections, updated references, added a reference to work of Lu on the
Conley Conjectur
Global properties of tight Reeb flows with applications to Finsler geodesic flows on
We show that if a Finsler metric on with reversibility has flag
curvatures satisfying , then closed geodesics
with specific contact-topological properties cannot exist, in particular there
are no closed geodesics with precisely one transverse self-intersection point.
This is a special case of a more general phenomenon, and other closed geodesics
with many self-intersections are also excluded. We provide examples of Randers
type, obtained by suitably modifying the metrics constructed by Katok
\cite{katok}, proving that this pinching condition is sharp. Our methods are
borrowed from the theory of pseudo-holomorphic curves in symplectizations.
Finally, we study global dynamical aspects of 3-dimensional energy levels
-close to .Comment: 27 page