Abstract

A Zoll contact manifold is a contact manifold all of whose Reeb orbits are periodic and of the same minimal period. Starting from such a Zoll contact manifold, we show the existence of periodic Reeb orbits through a prescribed domain for every sufficiently close perturbation of the contact form. In particular, this proves the Weinstein conjecture for contact hypersurfaces which are sufficiently close to a Zoll contact hypersurface. The proof relies on a homotopy stretching argument, in which the crucial step is showing that certain moduli spaces of solutions to an interpolated Floer equation are non-empty. A special case of a Zoll contact manifold is the geodesic flow over a Zoll Riemannian manifold (i.e., all geodesics are closed with the same length). As an application, we find periodic magnetic geodesics for every sufficiently weak magnetic field on a Zoll Riemannian manifold.