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

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

2002-06-10

arxiv.org/abs/math/0206092v4

Mathematics

Symplectic Geometry

75 pages, the section 8 in the previous version is taken out and incoporated into a stronger existence theorem proven in a sep

Scientific paper

In 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.

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