Spectral invariants in Lagrangian Floer theory

By R. Leclercq. 2006

58 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fc6803afceeb6df3b1c52ecfb3c4d8d30d6e885

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Citations in the corpus

Coisotropic rigidity and $C^{0}$-symplectic geometry (2013, methodology): “They have been adapted to the compact case by Leclercq [12] for weakly exact Lagrangians and Leclercq–Zapolsky [13] for monotone Lagrangians.”
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, result): “consistently introduce spanning half-disks. The rest of the theory comes though. It is appropriate to mention that a related approach, although in a diﬀerent context, was pursued by R´emi Leclercq in [Lec]. He deﬁned invariants of Lagrangian submanifolds instead of Hamiltonian diﬀeomorphisms, but in”
Bounds on spectral norms and barcodes (2018, methodology): “They were first introduced in symplectic topology by Viterbo in [118] by means of generating functions, then in Floer theory by Schwarz [97], and Oh [84, 85], and in Lagrangian Floer theory by Leclercq [64] and Leclercq-Zapolsky [65] (see also [44, 77, 81]).”
Spectral invariants for monotone Lagrangians (2015, methodology): “roduced into symplectic topology by Viterbo [Vit92], and subsequently their theory was developed by Schwarz [Sch00], Oh [Oh05a] in the context of periodic orbit Floer homology, by Oh [Oh97], Leclercq [Lec08], Monzner–Vichery–Zapolsky [MVZ12] in the context of Lagrangian Floer homology, by Chaperon [”