Many closed symplectic manifolds have infinite Hofer–Zehnder capacity

Abstract:

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.