Mathematics

On the persistence of closed Reeb orbits

Ely Kerman (UIUC Math)

Abstract: Reeb flows on contact manifolds are conjectured to always have closed orbits (the Weinstein Conjecture) and in many settings nontrivial lower bounds for their numbers are also expected. One obstacle to detecting many distinct closed orbits is the difficulty, well known from the study of closed geodesics, of distinguishing between simple and multiply covered orbits. In this talk I will introduce a $C^0$-distance between Reeb flows in the spirit of the Hofer norm and will describe a Floer theoretic proof that certain clusters of simple closed Reeb orbits must persist over surprisingly long distances. Among other things, this allows one to reprove a classic result of Ekeland and Lasry concerning multiple closed characteristics on convex hypersurfaces pinched between spheres of radius 1 and $\sqrt{2}$.