Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

TL;DR

This paper compares spectral invariants in periodic orbits and Lagrangian Floer homology for certain symplectic manifolds, establishing relationships and properties like subadditivity, with implications for Hofer's distance.

Contribution

It provides a comparison of spectral invariants in Floer theory for both periodic orbits and Lagrangian submanifolds under specific conditions, introducing a new product and proving its properties.

Findings

Spectral invariants in periodic and Lagrangian Floer theories are comparable under certain conditions.

A new product on Floer homology is defined and shown to satisfy subadditivity of invariants.

Corollaries relate spectral invariants to Hofer's distance in symplectic geometry.

Abstract

We compare spectral invariants in periodical orbits and Lagrangian Floer homology case, for closed symplectic manifold $P$ and its closed Lagrangian submanifolds $L$, when $\omega|_{\pi_2(P,L)}=0$, and $\mu|_{\pi_2(P,L)}=0$. From this result, we derive a corollary considering comparison of Hofer's distance in periodic orbits and Lagrangian case. We also define a product $HF_*(H)\otimes HF_*(L,\phi^1_H(L))\to HF_*(L,\phi^1_H(L))$ and prove subadditivity of invariants with respect to this product.