The unbounded Lagrangian spectral norm and wrapped Floer cohomology

TL;DR

This paper explores the spectral metric in symplectic geometry, showing it is unbounded for certain Lagrangian submanifolds, and introduces wrapped Floer cohomology as a key tool for this analysis.

Contribution

It establishes a link between the boundedness of the spectral metric and the vanishing of wrapped Floer cohomology, demonstrating the unboundedness of the Lagrangian Hofer diameter.

Findings

Spectral metric is bounded iff wrapped Floer cohomology vanishes.

The Lagrangian Hofer diameter of the orbit space is infinite.

Wrapped Floer cohomology is used to define spectral invariants in Weinstein domains.

Abstract

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid $Ham$-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.