Remarks on eternal classes in symplectic cohomology

Abstract

This paper studies special classes in the symplectic cohomology of a semipositive and convex-at-infinity symplectic manifold $W$. The classes under consideration lie in the image of every continuation map (for this reason, we call them eternal classes as they are never born and never die). Non-eternal classes in symplectic cohomology can be used to define spectral invariants for contact isotopies of the ideal boundary $Y$ of $W$. It is shown that the spectral invariants of non-eternal classes behave sub-additively with respect to the pair-of-pants product. This is used to define a spectral pseudo-metric on the universal cover of the group of contactomorphisms. We also give criteria for existence and non-existence of eternal classes. First, a compact monotone Lagrangian with odd Euler characteristic and minimal Maslov number at least $2$ implies the existence of non-zero eternal classes (e.g., $T^{*}\mathrm{RP}^{2n}$ has non-zero eternal classes). Second, no non-zero eternal classes exist if every compact set in $W$ is smoothly displaceable (e.g., $T^{*}T^{n}$ has no non-zero eternal classes).