The spectral diameter of a Liouville domain

TL;DR

This paper establishes a criterion linking the spectral diameter of Hamiltonian diffeomorphisms on Liouville domains to the non-vanishing of symplectic cohomology, advancing understanding of symplectic topology.

Contribution

It proves that the spectral diameter of a Liouville domain is infinite if and only if its symplectic cohomology is non-zero, generalizing previous results.

Findings

Spectral diameter is infinite iff symplectic cohomology is non-zero.

Generalization of Monzner-Vichery-Zapolsky's result.

Applications to closed symplectic manifolds.

Abstract

The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming from spectral invariants in Hamiltonian Floer homology. This distance has found numerous applications in symplectic topology. However, its diameter is still unknown in general. In fact, for closed symplectic manifolds there is no unifying criterion for the diameter to be infinite. In this paper, we prove that for any Liouville domain this diameter is infinite if and only if its symplectic cohomology does not vanish. This generalizes a result of Monzner-Vichery-Zapolsky and has applications in the setting of closed symplectic manifolds.