Spectral invariants in Lagrangian Floer theory

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 $L^{\prime }$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L^{\prime }$. 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. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.

34 pages, 7 figures. v2: normalization of the spectral invariants. An explicit computation added (section 4.3). Published in Journal of Modern Dynamics

References in corpus (1)

• Spectral invariants and length minimizing property of Hamiltonian paths