Many closed symplectic manifolds have infinite Hofer-Zehnder capacity

TL;DR

This paper constructs numerous closed symplectic manifolds with autonomous Hamiltonians lacking nonconstant periodic orbits, expanding known examples and suggesting such properties are widespread among symplectic four-manifolds.

Contribution

The authors provide new explicit examples of closed symplectic manifolds with autonomous Hamiltonians having no nonconstant periodic orbits, using symplectic sum techniques.

Findings

Examples include K3 surfaces and related manifolds.

Construction involves symplectic sums and perturbations.

Conjecture that many symplectic four-manifolds admit such forms.

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.