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Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Jovana Đuretić, Jelena Katić, Darko Milinković

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 $ω|_{π_2(P,L)}=0$, and $μ|_{π_2(P,L)}=0$. From this result, we derive a corollary considering comparison of Hofer's distance in periodic orbits and Lagrangian case. We also define a product $HF_*(H)\otimes HF_*(L,φ^1_H(L))\to HF_*(L,φ^1_H(L))$ and prove subadditivity of invariants with respect to this product.

Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Abstract

We compare spectral invariants in periodical orbits and Lagrangian Floer homology case, for closed symplectic manifold and its closed Lagrangian submanifolds , when , and . From this result, we derive a corollary considering comparison of Hofer's distance in periodic orbits and Lagrangian case. We also define a product and prove subadditivity of invariants with respect to this product.

Paper Structure

This paper contains 9 sections, 6 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Recalls and preliminaries
  3. Floer homology and $\operatorname{PSS}$ (iso)morphisms
  4. Spectral invariants and their comparison
  5. Action functionals
  6. Invariants
  7. Chimneys and comparison between invariants
  8. Proof of Theorem \ref{['thm:product']}
  9. Proof of Theorem \ref{['thm:product']}

Key Result

Theorem 1

Let $P$ be a closed symplectic manifold and $L\subset P$ its closed Lagrangian submanifold such that $\omega|_{\pi_2(P,L)}=0$, $\mu|_{\pi_2(P,L)}=0$. Let $H_j:P\times [0,1]\to \mathbb R$ be three (time dependent) Hamiltonians, for $j=1,2,3$. Then there exists a product which, in the case when $H_2=H_3$, turns the Lagrangian Floer homology $HF_*(H_2:L)$ into a module over Floer homology for period

Theorems & Definitions (9)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Proposition 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Remark 9