A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories

TL;DR

This paper establishes a local-to-global inequality for spectral invariants in symplectic geometry, extending previous locality results to more general Hamiltonians and manifolds, using a novel energy bound for Floer trajectories.

Contribution

It introduces the first local-to-global spectral invariant inequality applicable to non-contact type Hamiltonian supports and irrational symplectic manifolds, with a new Floer trajectory energy estimate.

Findings

First examples of local-to-global inequalities beyond contact type boundaries

Extension to irrational symplectic manifolds

New energy lower bound for Floer trajectories crossing tubular neighborhoods

Abstract

We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a ``large enough" tubular neighborhood on semipositive symplectic manifolds. In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.