  We prove that every Reeb flow on a closed connected three-manifold has either
two or infinitely many simple periodic orbits, assuming that the associated
contact structure has torsion first Chern class. As a special case, we prove a
conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth
and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely
many simple periodic orbits on any regular compact connected energy level that
is transverse to the radial vector field. Other corollaries settle some old
problems about Finsler metrics: we show that every Finsler metric on $S^2$ has
either two or infinitely many prime closed geodesics; and we show that a
Finsler metric on $S^2$ with at least one closed geodesic that is not
irrationally elliptic must have infinitely many prime closed geodesics. The
novelty of our work is that we do not make any nondegeneracy hypotheses.
