Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds

By Yong-Geun Oh. 2004

153 citations recorded by the source corpus.

In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian paths on arbitrary, especially on nonexact and nonrational, compact symplectic manifold (M, ω). To each given time dependent Hamiltonian function H and quantum cohomology class 0 ≠ a ∈ QH*(M), we associate an invariant ρ(H; a) which varies continuously over H in the C 0-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is “dual” to the given quantum cohomology class a on the covering space \( \tilde \Omega _0 (M) \) of the contractible loop space Ω0(M). We call them the Novikov Floer cycles. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5706c54f91fbd456ea622ee33a7d8ad5ca8f15ca

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In Discrete Mathematics and Combinatorics

1. Michael Usher
2. V. Ginzburg
3. Basak Z. Gurel
4. P. Biran
5. O. Cornea
6. F. Zapolsky
7. R. Leclercq
8. E. Shelukhin

In Geometry and Topology

1. E. Shelukhin
2. A. Kislev
3. Michael Usher
4. L. Polterovich
5. V. Ginzburg
6. Basak Z. Gurel
7. O. Cornea
8. F. Zapolsky

In Mathematical Physics

1. E. Shelukhin
2. A. Kislev
3. Michael Usher
4. L. Polterovich
5. V. Ginzburg
6. F. Zapolsky
7. Basak Z. Gurel
8. O. Cornea

Citations in the corpus

• Persistent homology and Floer-Novikov theory (2015, methodology): “[FlH94], [Sc00], [EP03], [Oh05], [U13], [HLS13]), it is only very recently that they have been considered from a persistent-homological point of view.”
• Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, methodology): “The spectrality property (AS10) is proved in [Oh2] in the rational case and in [U1] for non-degenerate Hamiltonians.”
• 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): “The properties of the Lagrangian spectral invariants listed in Theorem 2 have previously appeared in other contexts, and consequently they are more or less standard, see [Vit92, Oh05a, MVZ12].”
• Persistence modules with operators in Morse and Floer theory (2017, background): “Spectral invariant can be defined in complete generality (without assumptions on π2(M)), and have been used extensively in symplectic topology in the last few decades starting with the foundational papers [42,51,57] (see [40,53] for recent developments and numerous further references), before persis”
• Symplectic Topology and Floer Homology (2015, methodology): “List of conventions We follow the conventions of [Oh05c, Oh09a, Oh10] for the definition of Hamiltonian vector fields and action functional and others appearing in the Hamiltonian Floer theory and in the construction of spectral invariants and EntovPolterovich’s Calabi quasimorphisms.”
• Rigidity and uniruling for Lagrangian submanifolds (2008, methodology): “We refer the reader to Oh [47; 45; 46; 41], Schwarz [49] and McDuff and Salamon [40] for the foundations of the theory of spectral invariants.”
• The action spectrum and $$C^0$$ symplectic topology (2018, background): “Spectral invariants satisfy the following properties whose proofs can be found in [46] as well as [37, 38, 54].”
• Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds (2009, background): “The invariants that have so far most productively been used in this spirit are the so-called spectral numbers, which were introduced in the symplectically aspherical case in [37] and were extended to general closed symplectic manifolds by Oh (see [27]).”
• THE SHARP ENERGY-CAPACITY INEQUALITY (2008, methodology): “We use the Oh-Schwarz spectral invariant ( ; 1) ([ 19 ], [27]) in the role of this , but we should emphasize that we do not in fact show that this is an action selector: indeed, perhaps the most basic property of an action selector, namely the condition that (H) always belong to the action spectrum ”
• Quantum Structures for Lagrangian Submanifolds (2007, background): “We refer the reader to [59, 53, 51, 52, 46, 45, 44] for the foundations of the theory of spectral invariants.”