  We exhibit many examples of closed symplectic manifolds on which there is an
autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits
(the only previous explicit example in the literature was the torus T^2n (n\geq
2) with an irrational symplectic structure). The underlying smooth manifolds of
our examples include, for instance: the K3 surface and also infinitely many
smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many
minimal four-manifolds having any given finitely-presented group as their
fundamental group; and simply connected minimal four-manifolds realizing all
but finitely many points in the first quadrant of the geography plane below the
line corresponding to signature 3. The examples are constructed by performing
symplectic sums along suitable tori and then perturbing the symplectic form in
such a way that hypersurfaces near the "neck" in the symplectic sum have no
closed characteristics. We conjecture that any closed symplectic four-manifold
with b^+>1 admits symplectic forms with a similar property.
