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학술연구/단체지원/교육 등 연구자 활동을 지속하도록 DBpia가 지원하고 있어요.
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The author previously defined the spectral invariants, denoted by ρ(H;a), of a Hamiltonian function H as the mini-max value
of the action functional A^H over the Novikov Floer cycles in the Floer
homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H;a) states that the mini-max value
is a critical value of the action functional A^H. The main purpose of
the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, ω). We also prove
that the spectral invariant function a : H ↦ ρ(H;a) can be pushed
down to a continuous function defined on the universal (etale) covering space Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, !). For a certain generic homotopy, which we
call a Cerf homotopy H = {H^s}0≤s≤1 of Hamiltonians, the function
ρ^a ․H : s ↦ ρ(H^s;a) is piecewise smooth away from a countable subset
of [0, 1] for each non-zero quantum cohomology class a.
The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent
interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter
family of Hamiltonian functions, a general structure theorem and the
handle sliding lemma of Novikov Floer cycles over such a family and a
family version of new transversality statements involving the Floer chain
map, and many others. We call this chain level Floer theory as a whole
the Floer mini-max theory.
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참고문헌 (24)
참고문헌 신청
A. Floer / 1988 / Morse theory for Lagrangian intersections / J. Differential Geom. 28 (3) : 513 ~ 547
A. Floer / 1988 / The unregularized gradient flow of the symplectic action / Comm. Pure Appl. Math. 41 (6) : 775 ~ 813
A. Floer / 1989 / Symplectic fixed points and holomorphic spheres / Comm. Math. Phys. 120 (4) : 575 ~ 611
A. Floer / 1989 / Witten’s complex and infinite dimensional Morse theory / J. Differential Geom. 30 : 207 ~ 221
A. Weinstein / 1978 / Bifurcations and Hamilton’s principle / Math. Z. 159 (3) : 235 ~ 248
A. Floer / 1988 / The unregularized gradient flow of the symplectic action / Comm. Pure Appl. Math. 41 (6) : 775 ~ 813
A. Floer / 1989 / Symplectic fixed points and holomorphic spheres / Comm. Math. Phys. 120 (4) : 575 ~ 611
A. Floer / 1989 / Witten’s complex and infinite dimensional Morse theory / J. Differential Geom. 30 : 207 ~ 221
A. Weinstein / 1978 / Bifurcations and Hamilton’s principle / Math. Z. 159 (3) : 235 ~ 248
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