Berlin-Hamburg-Seminar am 4.11.2019

Marco Mazzucchelli (Lyon) Spectral characterizations of Besse and Zoll Reeb flows

A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all periodic with the same minimal period. This class of manifolds has been thoroughly studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors. It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and only if its unit-speed periodic geodesics all have the same minimal period.
In this talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds on the work of Lusternik and Schnirelmann. I will then present a stronger version of this statement valid for general Reeb flows on closed contact 3-manifolds: the closed orbits of any such Reeb flow admit a common period if and only if every orbit of the flow is closed. Time permitting, I will also summarize some related results for Reeb flows on higher dimensional contact spheres and for geodesic flows on simply connected compact rank-one symmetric spaces.
The talk is based on joint works with Suhr, Cristofaro Gardiner, and Ginzburg-Gürel.

Cedric de Groote (Leipzig) On the orderability up to conjugation of certain open contact manifolds

Eliashberg and Polterovich introduced in 2000 a notion of orderability for the group of contact isotopies of a contact manifold, which provides insights into the geometry of that group. Later, this same notion "up to conjugation" was used by Borman, Eliashberg and Murphy in their proof of the flexibility of overtwisted contact manifolds of all dimensions. I will review some of the history of that problem, and then present a new result on the orderability up to conjugation of certain contact annuli. This involves restating the problem as a contact non-squeezing result, which is then shown using a version of contact homology.