Floer mini-max theory, the Cerf diagram, and the spectral invariants
| dc.creator | OH, Yong-Geun | |
| dc.date | 2004-06-23 | |
| dc.date | 2007-09-06 | |
| dc.date.accessioned | 2026-07-07T12:52:05Z | |
| dc.date.available | 2026-07-07T12:52:05Z | |
| dc.description | 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 $Å_H$ 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 $Å_H$. The main purpose of the present paper is to prove this axiom for {\it nondegenerate} Hamiltonian functions in {\it irrational} symplectic manifolds $(M,ω)$. We also prove that the spectral invariant function $ρ_a: H \mapsto ρ(H;a)$ can be pushed down to a {\it continuous} function defined on the universal ({\it étale}) covering space $\widetilde{Ham}(M,ω)$ of the group $Ham(M,ω)$ of Hamiltonian diffeomorphisms on general $(M,ω)$. For a certain generic homotopy, which we call a {\it Cerf homotopy} $\HH = \{H^s\}_{0 \leq s\leq 1}$ of Hamiltonians, the function $ρ_a \circ \HH: s \mapsto ρ(H^s;a)$ 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 {\it 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 {\it Floer mini-max theory}. | |
| dc.description | 74 pages; An incorrect statement in Theorem 3.7 corrected which results in partial rewriting of section 8 and 9. A new theorem, Theorem V added. A new reference [22] added | |
| dc.identifier | https://arxiv.org/abs/math/0406449 | |
| dc.identifier | http://arxiv.org/abs/math/0406449 | |
| dc.identifier | J. Korean Math. Soc. 46 (2009), 363--447 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/223192 | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Dynamical Systems | |
| dc.subject | 53D35, 53D40 | |
| dc.title | Floer mini-max theory, the Cerf diagram, and the spectral invariants | |
| dc.type | text |