1,720,981 research outputs found
Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.Addendum
No full textThis is an addendum to the author's earlier paper ''Floer Cohomology of Lagrangian Intersection and Pseudo-Holomorphic Discs, I,'' Comm. Pure Appl. Math. 46, 1993, pp. 949-993. The main result of this addendum extends the definition of the Fleer cohomology of Lagrangian intersection to the case where the minimal Maslov number is equal to 2. (C) 1996 John Wiley & Sons, Inc.X1145sciescopu
Spectral invariants and the length minimizing property of Hamiltonian paths
No full textIn this paper we provide a criterion for the quasi-autonomous Hamiltonian path ("Hofer's geodesic") on arbitrary closed symplectic manifolds (M, omega) to be length minimizing in its homotopy class in terms of the spectral invariants rho(G; 1) that the author has recently constructed. As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has no contractible periodic orbits of period one and it has a maximum and a minimum that are generically under-twisted, and all of its critical points are non-degenerate in the Floer theoretic sense.X1110sci
Floer mini-max theory, the Cerf diagram, and the spectral invariants
No full textThe author previously defined the spectral invariants, denoted by rho(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 rho(H; a) states that the mini-max value is a critical value of the action functional AH. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, omega). We also prove that the spectral invariant function rho(a) : H (sic) rho(H; a) can be pushed down to a continuous function defined on the universal (etale) covering space (sic)(M,omega) of the group Ham(M,omega) of Hamiltonian diffeomorphisms on general (M, omega). For a certain generic homotopy, which we call a Cerf homotopy H = {H(s)}(0 <= s <= 1) of Hamiltonians, the function rho(a) circle H : s (sic) rho(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.X118sciescopuskc
Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group
No full textIn this article, we apply spectral invariants constructed in [O5] and [O6] to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds (M, omega). Using spectral invariants, we first construct an invariant norm called the spectral norm on. the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the epsilon-regularity theorem and the symplectic area of certain pseudoholomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths among all paths. In addition to the construction of spectral invariants, these applications rely heavily on the chain-level Floer theory and on some existence theorems with energy bounds of pseudoholomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this article in turn relies on some careful geometric analysis involving adiabatic degeneration and thick-thin decomposition of the Floer moduli spaces which has an independent interest of its own. We assume that (M, omega) is strongly semipositive throughout this article.X1127sciescopu
Fredholm theory of holomorphic discs under the perturbation of boundary conditions
No full textX1124sciescopu
Symplectic topology as the geometry of action functional. I - Relative Floer theory on the cotangent bundle
No full textX1158sciescopu
Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings
No full textIn this paper, we give a new simple proof of Chekanov's positivity theorem of the disjunction energy of compact Lagrangian submanifolds in tame symplectic manifolds. As a consequence, it also gives rise to a simple proof of nondegeneracy of Hofer's norm on the group of Hamiltonian diffeomorphisms on any tame symplectic manifolds.X1113sciescopu
Normalization of the Hamiltonian and the action spectrum
No full textIn this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold (M, w) canonically relate the action spectra of different. normalized Hamiltonians on arbitrary symplectic manifolds (M, w). The natural classes of normalized Hamiltonians consist of those whose mean value is zero for the closed manifold, and those, which are compactly supported in IntM for the open manifold, We also study the effect of the action spectrum under the pi(1) of Hamiltonian diffeomorphism group. This forms a foundational basis for our study of spectral invariants of the Hamiltonian diffeomorphism in [8].X117sciescopuskc
Higher jet evaluation transversality of J-holomorphic curves
No full textIn this paper, we establish general stratawise higher jet evaluation transversality of J-holomorphic curves for a generic choice of almost complex structures J (tame to a given symplectic manifold (M, omega)). Using this transversality result, we prove that there exists a subset J(omega)(ram) subset of J(omega) of second category such that for every J is an element of J(omega)(ram), the dimension of the moduli space of (somewhere injective) J-holomorphic curves with a given ramification profile goes down by 2n or 2(n - 1) depending on whether the ramification degree goes up by one or a new ramification point is created. We also derive that for each J is an element of J(omega)(ram) there are only a finite number of ramification profiles of J-holomorphic curves in a given homology class beta is an element of H(2)(M; Z) and provide an explicit upper bound on the number of ramification profiles in terms of c(1)(beta) and the genus g of the domain surface.X1124sciescopuskc
Symplectic Topology as the Geometry of Action Functional, II - pants product and cohomological invariants
No full textX1133sciescopu
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