Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group

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dc.creatorOh, Yong-Geun
dc.date2002-06-10
dc.date2002-07-25
dc.date.accessioned2026-07-07T04:49:00Z
dc.date.available2026-07-07T04:49:00Z
dc.descriptionIn this paper, we first 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 diffeomorphisms on arbitrary compact symplectic manifold (M,omega). To each given time dependent Hamiltonian function H and quantum cohomology class 0 not equal a element of QH^*(M), we associate an invariant rho(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 \widetilde Omega_0(M) of the contractible loop space Omega_0(M). We call them the Novikov cycles. We then use the spectral invariants to construct a new invariant norm on the Hamiltonian diffeomorphism group and a partial order on the set of time-dependent Hamiltonian functions of arbitrary compact symplectic manifolds. As some applications, we obtain a new lower bound of the Hofer norm of non-degenerate Hamiltonian diffeomorphisms in terms of the area of certain pseudo-holomorphic curves and prove the semi-global C^1-flatness of the Hofer norm.
dc.description75 pages, the section 8 in the previous version is taken out and incoporated into a stronger existence theorem proven in a separate paper math.SG/0207214
dc.identifierhttps://arxiv.org/abs/math/0206092
dc.identifierhttp://arxiv.org/abs/math/0206092
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/64266
dc.subjectSymplectic Geometry
dc.subjectQuantum Algebra
dc.subject53D35, 53D40, 53D45; 70H05, 70H25
dc.titleMini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group
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