Spectral invariants of distance functions

TL;DR

This paper explores spectral invariants of Floer homology for distance functions to identify superheavy subsets in symplectic manifolds, extending known results and providing new lower bounds for Poisson bracket invariants.

Contribution

It introduces a method to determine superheavy subsets via spectral invariants and extends results to new classes of symplectic manifolds.

Findings

Convex open subsets in Euclidean space are superheavy when embedded in certain symplectic manifolds.

The $S^1$ bouquet in a Riemann surface with genus ≥ 1 is superheavy.

Results extend to properties of monotone closed symplectic manifolds and Poisson bracket bounds.

Abstract

Calculating the spectral invariant of Floer homology of the distance function, we can find some kind of superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their compliment is superheavy. In particular, the $S^1$ bouquet in a closed Riemann surface with genus $g\geq 1$ is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni's result about lower bounds of Poisson bracket invariant.