Towards the HZ- and multiplicity conjectures for dynamically convex Reeb flows

In this talk, based on a joint work with Erman Cineli and Basak Gurel, we discuss the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain. The first of our two main results asserts that such a flow has at least n prime closed Reeb orbits, settling a conjecture which is usually attributed to Ekeland. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks’ celebrated 2-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture.