scholarly journals Spectral invariants for monotone Lagrangians

2018 ◽  
Vol 10 (03) ◽  
pp. 627-700 ◽  
Author(s):  
Rémi Leclercq ◽  
Frol Zapolsky
Keyword(s):  
Generating Functions ◽  
Floer Homology ◽  
Spectral Invariants ◽  
Seminal Paper ◽  
Floer Theory ◽  
Cotangent Bundles

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

The arnold conjecture

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Index Theory ◽  
Intersection Problem ◽  
Arnold Conjecture ◽  
Conley Index Theory ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.

scholarly journals Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1161-1174 ◽  
Author(s):  
Jovana Djuretic ◽  
Jelena Katic ◽  
Darko Milinkovic
Keyword(s):  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Spectral Invariants ◽  
Floer Theory

We compare spectral invariants in periodic orbits and Lagrangian Floer homology case, for a closed symplectic manifold P and its closed Lagrangian submanifolds L, when ?|?2(P,L)=0, and ?|?2(P,L)=0. We define a product HF*(H)?HF*(H:L) ? HF*(H:L) and prove subadditivity of invariants with respect to this product.

scholarly journals Spectral invariants in Lagrangian Floer homology of open subset

2017 ◽  
Vol 53 ◽  
pp. 220-267
Author(s):  
Jelena Katić ◽  
Darko Milinković ◽  
Jovana Nikolić
Keyword(s):  
Open Subset ◽  
Floer Homology ◽  
Spectral Invariants

scholarly journals Spectral invariants in Lagrangian Floer theory

2008 ◽  
Vol 2 (2) ◽  
pp. 249-286 ◽  
Author(s):  
Rémi Leclercq ◽  
Keyword(s):  
Spectral Invariants ◽  
Floer Theory

scholarly journals Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle

2017 ◽  
Vol 102 (116) ◽  
pp. 17-47
Author(s):  
Jovana Duretic
Keyword(s):  
Morse Function ◽  
Triangle Inequality ◽  
Floer Homology ◽  
Zero Section ◽  
Spectral Invariants ◽  
Conormal Bundle ◽  
Morse Homology

We give a construction of the Piunikhin-Salamon-Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.

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