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We exhibit many examples of closed symplectic manifolds on which there is anautonomous Hamiltonian whose associated flow has no nonconstant periodic orbits(the only previous explicit example in the literature was the torus T^2n (n\geq2) with an irrational symplectic structure). The underlying smooth manifolds ofour examples include, for instance: the K3 surface and also infinitely manysmooth manifolds homeomorphic but not diffeomorphic to it; infinitely manyminimal four-manifolds having any given finitely-presented group as theirfundamental group; and simply connected minimal four-manifolds realizing allbut finitely many points in the first quadrant of the geography plane below theline corresponding to signature 3. The examples are constructed by performingsymplectic sums along suitable tori and then perturbing the symplectic form insuch a way that hypersurfaces near the "neck" in the symplectic sum have noclosed characteristics. We conjecture that any closed symplectic four-manifoldwith b^+>1 admits symplectic forms with a similar property.

Many closed symplectic manifolds have infinite Hofer-Zehnder capacity