Journal of Modern Dynamics (JMD)

Spectral invariants in Lagrangian Floer theory

Pages: 249 - 286, Issue 2, April 2008      doi:10.3934/jmd.2008.2.249

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Rémi Leclercq - Department of Mathematics and Statistics, Université de Montréal, CP 6128 Succ. Centre Ville, Montréal, QC H3C 3J7, Canada (email)

Abstract: Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian submanifold $L'$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L'$. We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case and carry strictly more information than the classical ones. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.

Keywords:  Spectral invariants, spectral sequence, Lagrangian submanifolds, Lagrangian Floer homology, Hofer's distance.
Mathematics Subject Classification:  Primary: 57R17; Secondary: 53D12, 53D40, 55T10.

Received: June 2007;      Revised: September 2007;      Available Online: January 2008.