## On Equivalence of Two Constructions of Invariants of Lagrangian Submanifolds

Venue: | Pacific J. Math |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Milinković_onequivalence,

author = {Darko Milinković},

title = {On Equivalence of Two Constructions of Invariants of Lagrangian Submanifolds},

journal = {Pacific J. Math},

year = {},

pages = {371--415}

}

### OpenURL

### Abstract

We give the construction ofsymplectic invariants which incorporates both the “infinite dimensional ” invariants constructed by Oh in 1997 and the “finite dimensional ” ones constructed by Viterbo in 1992. 1. Introduction. Let M be a compact smooth manifold. Its cotangent bundle T ∗M carries a natural symplectic structure associated to a Liouville form θ = pdq. For a given compactly supported Hamiltonian function H: T ∗M → R and a closed submanifold N ⊂ M Oh [30, 27] defined a symplectic invariants of

### Citations

203 |
Morse theory for lagrangian intersections
- Floer
- 1988
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Citation Context ...N : E) is onto. Consequently, for (J, S, H) ∈ (J c ω(E) ×S (E,Q) ×H(E))reg M (J,H,S)(x, y) is a smooth finite dimensional manifold. Similarly, we have the parameterized version of Proposition 12 (see =-=[9, 11]-=-): Proposition 13. Let N and Q be fixed as in Proposition 12, and (J α ,S α ,H α ), (J β ,S β ,H β ) ∈ (J c ω(E) ×S(E,Q) ×H(E))reg. Then there exists a dense subset (J c ω(E) × S (E,Q) × H(E))reg in a... |

133 |
Morse theory for periodic solutions of Hamiltonian systems and the Maslov index
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- 1992
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Citation Context ...∗(H, S : N) canonical grading. The existence of such grading is established in [10] and similar computations to ours are given for the case S ≡ 0, m = 0 in [30] and for the periodic orbits problem in =-=[6, 36]-=-. 4.1. The Maslov index. Maslov index for paths of Lagrangian subspaces has been studied by several authors (see [1, 3, 34, 33]). We will follow the notation and terminology of [34] and [33]. Denote b... |

89 |
On the topological properties of symplectic maps
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- 1990
(Show Context)
Citation Context ... According to Lemma 39 and Definition 40 (61) and (62) σ(a, 0, ˜ S : N) =c(a, LS : N) σ(a, ˜ H ⊕ 0,Q: N) =ρ(a, LS : N). Now, (59) follows from (60), (61) and (62). □ 7. A note on Hofer’s geometry. In =-=[16]-=- Hofer introduced a biinvariant metric on a group Dc ω(P ) of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold P . For H ∈ C∞ c (P × [0, 1]) define the oscillation of Ht by osc... |

82 |
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- 1986
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Citation Context ... f ◦ u attains its maximum at z. Ifzis an interior point in D it contradicts the maximum principle for subharmonic functions. If z ∈ ∂D then d dt |t=1((f ◦ u)(tz))=0 which contradicts Hopf lemma (see =-=[32]-=-). □ 3.2. The structure ofthe space oftrajectories. In this section we prove the following analogue of well-known Floer’s theorem (see [11, 15, 35]). Proposition 7. If U := (u, v) is a solution of Equ... |

78 |
Functors and computations in Floer homology with applcations
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- 1999
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Citation Context ...ing. The purpose of this paper is to give the details of this construction. Another way of interpolating Floer and Morse homologies for generating functions, in the case M = N was given by Viterbo in =-=[39, 37]-=-. The dependence of the above invariants on the subset N ⊂ M, in particular the continuity with respect to the C1-topology of submanifolds is an interesting question, which was further studied by Kast... |

76 |
Coherent orientations for periodic orbit problems in symplectic geometry
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- 1993
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Citation Context ...n. In order to define Floer homology for arbitrary coefficients we need the orientation of manifolds M (J,H,S) and M (J αβ ,H αβ ,S αβ ). Contrary to the case of holomorphic spheres or cylinders (see =-=[14]-=-, [24]), manifolds of holomorphic discs with Lagrangian boundary conditions need not to be orientable in general. However, in case of cotangent bundle such manifold are orientable under the boundary c... |

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- 1995
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Citation Context ...ed by E(ψ) :=inf{l(φ H t ) | φ H 1 = ψ}. The non-degeneracy of the energy functional, i.e., E(ψ) =0iffψ =id has been proved by Hofer [16] (see also [17]) in the case P = C n and by Lalonde and McDuff =-=[20]-=- in general. In the case P = C n Bialy and Polterovich [2] proved that (63) c(µ, Γψ) − c(1, Γψ) ≤ E(ψ) where c(µ, Γψ) − c(1, Γψ) is Viterbo’s norm (see [38]). Moreover, they proved that Viterbo’s and ... |

65 |
Morse-type index theory for flows and periodic solutions for Hamiltonian equations
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- 1984
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Citation Context ...∗(H, S : N) canonical grading. The existence of such grading is established in [10] and similar computations to ours are given for the case S ≡ 0, m = 0 in [30] and for the periodic orbits problem in =-=[6, 36]-=-. 4.1. The Maslov index. Maslov index for paths of Lagrangian subspaces has been studied by several authors (see [1, 3, 34, 33]). We will follow the notation and terminology of [34] and [33]. Denote b... |

61 |
On the hypotheses of Rabinowitz’ periodic orbit theorems
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- 1979
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Citation Context ...ssential ingredient of the proof is the version of maximum principle which states that a J-holomorphic curve cannot touch certain kind of hypersurfaces. 3.1. Contact type hypersurfaces. Definition 2 (=-=[40]-=-). A smooth hypersurface ∆ in a symplectic manifold (V,ω) is said to be of a contact type if there exists a vector field X defined in a neighborhood U of ∆ and transversal to ∆ such that d(X⌋ω) =ω in ... |

58 |
Morse theory, the Conley index and Floer homology
- Salamon
- 1990
(Show Context)
Citation Context ...n d dt |t=1((f ◦ u)(tz))=0 which contradicts Hopf lemma (see [32]). □ 3.2. The structure ofthe space oftrajectories. In this section we prove the following analogue of well-known Floer’s theorem (see =-=[11, 15, 35]-=-). Proposition 7. If U := (u, v) is a solution of Equation (10) which satisfies the condition (11), then there exist the limits x α (t) = lim τ→−∞ U(τ,t)ON EQUIVALENCE OF TWO CONSTRUCTIONS... 379 and... |

56 | On the Maslov-type index
- Cappell, Lee, et al.
- 1994
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Citation Context ... given for the case S ≡ 0, m = 0 in [30] and for the periodic orbits problem in [6, 36]. 4.1. The Maslov index. Maslov index for paths of Lagrangian subspaces has been studied by several authors (see =-=[1, 3, 34, 33]-=-). We will follow the notation and terminology of [34] and [33]. Denote by Λ(k) the Lagrangian Grassmanian, i.e., the manifold of Lagrangian subspaces in C k . The Maslov index assigns to every pair o... |

49 |
Characteristic class entering in quantization conditions
- Arnold
- 1967
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Citation Context ...NS... 373 functions S : E → R such that S = Q outside K and ∞∑ (3) εk‖S − Q‖Ck < ∞ k=0 for some sequence εk of positive real numbers. Similarly, let H(E) denote the set of smooth functions H : T ∗E × =-=[0, 1]-=- → R such that outside K H(x, ξ) =H1 ⊕ H2(x, ξ) :=H1(x)+H2(ξ) for some compactly supported functions H1 : T ∗M → R and H2 : Cm → R and ∞∑ (4) εk‖H‖C k < ∞. k=0 Equipped with norms (3) and (4) the spac... |

44 | The Maslov index for paths
- Robbin, Salamon
- 1993
(Show Context)
Citation Context ... given for the case S ≡ 0, m = 0 in [30] and for the periodic orbits problem in [6, 36]. 4.1. The Maslov index. Maslov index for paths of Lagrangian subspaces has been studied by several authors (see =-=[1, 3, 34, 33]-=-). We will follow the notation and terminology of [34] and [33]. Denote by Λ(k) the Lagrangian Grassmanian, i.e., the manifold of Lagrangian subspaces in C k . The Maslov index assigns to every pair o... |

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- Oh
- 1999
(Show Context)
Citation Context ...otangent bundle T ∗M carries a natural symplectic structure associated to a Liouville form θ = pdq. For a given compactly supported Hamiltonian function H : T ∗M → R and a closed submanifold N ⊂ M Oh =-=[30, 27]-=- defined a symplectic invariants of certain Lagrangian submanifolds in T ∗M in a following way. Let ν∗N ⊂ T ∗M be a conormal bundle of N. Denote by HFλ ∗ (H, N; M) the Floer homology groups generated ... |

34 |
Convex symplectic manifolds
- Eliashberg, Gromov
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Citation Context ... neighborhood U of ∆ and transversal to ∆ such that d(X⌋ω) =ω in U. Such vector field is called conformal. It is easy to see that ϱ := X⌋ω defines a contact structure ζ := Ker (ϱ) on ∆. Definition 3 (=-=[7]-=-). Let ∆ be an oriented hypersurface in an almost complex manifold (V,J) and ζq the maximal J-invariant subspace of Tq∆. Then ∆ is called J-convex if for some (and hence any) defining 1-form ϱ for ζq ... |

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Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms
- Bialy, Polterovich
- 1994
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Citation Context ... of the energy functional, i.e., E(ψ) =0iffψ =id has been proved by Hofer [16] (see also [17]) in the case P = C n and by Lalonde and McDuff [20] in general. In the case P = C n Bialy and Polterovich =-=[2]-=- proved that (63) c(µ, Γψ) − c(1, Γψ) ≤ E(ψ) where c(µ, Γψ) − c(1, Γψ) is Viterbo’s norm (see [38]). Moreover, they proved that Viterbo’s and Hofer’s metrics coincide locally in the sense of C1-Whitne... |

24 |
Elliptic methods in symplectic geometry
- McDuff
- 1990
(Show Context)
Citation Context ...,Hα,Sα)(xj ,xj−1),where xj are the solutions of Equation (12) and xl = xα . The complementary concept to the compactness property of Proposition 10 is the gluing construction. It is now standard (see =-=[12, 22]-=-) and can be summarized in the following Proposition 11. For any pair of trajectories (U α ,U αβ ) ∈M (J α ,H α ,S α )(x α ,y α ) ×M (J αβ ,H αβ ,S αβ )(y α ,z β )384 DARKO MILINKOVI Ć there exists a... |

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- Eliashberg, Hofer, et al.
- 1995
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Citation Context ...∣ dtdτ ∫ +∞ ∫ 1 −∞ 0 2 J αβ τ ) dtdτ ∂H αβ τ ∂τ dtdτ 3.3. The image ofthe evaluation map. In this section we prove the C0 estimate necessary for defining Floer homology on a non-compact manifold (see =-=[8, 15, 30]-=- for similar propositions). In fact, we will prove that the image of the evaluation map ev : MJ αβ ,Hαβ,Sαβ(N : E) × [0, 1] × R → T ∗ E defined by ev(U, τ, t) :=U(τ,t) is bounded. Proposition 9. Consi... |

19 |
Persistence d’intersection avec la section nulle au cours d’une isotopie Hamiltonienne dans un fibré cotangent, Invent
- Laudenbach, Sikorav
- 1985
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Citation Context ...details. This construction can be considered as an infinite dimensional version of a construction given earlier by Viterbo [38]. Let L be a Hamiltonian deformation of the zero section oM. It is known =-=[21]-=- that L can be realized as {( L = x, ∂S ) ∣∣∣(x, ( ) } −1 ∂S ζ) ∈ (0) , ∂x ∂ζ where S : M × R m → R is a smooth function fiberwise quadratic outside a compact set. Using that result, Viterbo [38] defi... |

12 |
fixed points and holomorphic spheres
- Symplectic
- 1989
(Show Context)
Citation Context ...H,S)(x, z) (in sense of Propositions 10), then n(U1)n(V1)+n(U2)n(V2) =0. □ Similar statement is true in parameterized version. The proof follows the same lines as the proof of analogous statements in =-=[12, 14, 15]-=-. 5. Floer homology. 5.1. Construction. For x ∈ CFp(H, S : N) and y ∈ CFp−1(H, S : N) we define n(x, y) tobe the number of points in (zero dimensional) manifold ̂M (J,H,S)(N : E) :=M (J,H,S)(N : E)/R ... |

9 |
complex and infinite dimensional Morse Theory
- Witten’s
- 1989
(Show Context)
Citation Context ...Φ αβ ◦ ∂ =(hαβ) 1 ♯ − (hαβ) 2 ♯ , i.e., Φαβ is a chain homotopy ([12, 15]). Therefore, h1 αβ = h2 αβ . Statement 2 (ii) now follows by choosing the constant homotopy Hαα ≡ Hα . □ 5.2. Computation. In =-=[13]-=- Floer proved that if h : M → R is a C2 Morse function, then HF∗(J, h ◦ π, M) ∼ = H Morse ∗ (h). We incorporate this result and the generalization [31] in our framework. Consider the tubular neighborh... |

9 |
Floer homology for open subsets and a relative version of Arnold’s conjecture
- Kasturirangan, Oh
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Citation Context ...of the above invariants on the subset N ⊂ M, in particular the continuity with respect to the C1-topology of submanifolds is an interesting question, which was further studied by Kasturirangan and Oh =-=[18, 19]-=-. Some applications to wave fronts and Hofer’s geometry are given in [30]. At the end, we give an application of our result to Hofer’s geometry of Lagrangian submanifolds. 2. Preliminaries and notatio... |

8 |
Generating functions versus action functional
- Milinković, Oh
- 1998
(Show Context)
Citation Context ... [30] that ρ is a well defined invariant which (after a suitable normalization of H) depends only on a Lagrangian submanifold L := φ H (OM) and not on a particular choice of H. We refer the reader to =-=[26, 29, 30, 27]-=- for more details. This construction can be considered as an infinite dimensional version of a construction given earlier by Viterbo [38]. Let L be a Hamiltonian deformation of the zero section oM. It... |

5 | Morse homology for generating functions of Lagrangian submanifolds
- Milinković
- 1999
(Show Context)
Citation Context ...n (i.e., replacing S : E → R by S ⊕ Q : E ⊕ F → R), it is enough to consider the case E = M × Rm . We refer the reader to [38] for more details. For an alternative construction via Morse homology see =-=[25]-=-. The natural question of the equality between the two invariants is raised in [30]. In [26] we outlined a proof, constructing the invariants which interpolate the above two. The main technical tool, ... |

5 |
cohomology of Lagrangian intersections and pseudo-holomorphic discs
- Floer
- 1993
(Show Context)
Citation Context ...cular to ∂ρ , i.e., tangent to S2m−1 . Since S2m−1 is i-convex (see Example 4), this again contradicts Lemma 6. □ Once we have established C 0 estimates, the standard compactness result follows as in =-=[12, 11, 28, 35]-=-: Proposition 10. For any sequence Uk ∈M (J αβ ,Hαβ,Sαβ)(xα,xβ) there exist a subsequence (denoted by Uk again), sequences τ j k ∈ R (0 ≤ j ≤ l) and an integer s (0 ≤ s ≤ l) such that 1) for 0 ≤ j ≤ s... |

3 |
Hofer’s symplectic energy and invariant metrics on the space of Lagrangian embeddings, ETH-preprint
- Chekanov
- 1996
(Show Context)
Citation Context ...ined in the following: Definition 43. For L1,L2 ∈LM(P ) we define d(L1,L2) :=inf{E(φ) | φ ∈D c (64) ω(P ), φ(L1) =L2}. The non-degeneracy of d has been proved by Oh [30] for P = T ∗ M and by Chekanov =-=[4, 5]-=- in general case. Moreover, for P = T ∗ M (65) ρ(µ, L) − ρ(1,L) ≤ d(oM,L) (see [30] or apply Lemma 39 to the inequalities at the end of the proof of Theorem 33 with Sα = Sβ = Q, Hα = 0, setting first ... |

2 |
intersections, symplectic energy and areas of holomorphic curves
- Lagrangian
- 1995
(Show Context)
Citation Context ...ined in the following: Definition 43. For L1,L2 ∈LM(P ) we define d(L1,L2) :=inf{E(φ) | φ ∈D c (64) ω(P ), φ(L1) =L2}. The non-degeneracy of d has been proved by Oh [30] for P = T ∗ M and by Chekanov =-=[4, 5]-=- in general case. Moreover, for P = T ∗ M (65) ρ(µ, L) − ρ(1,L) ≤ d(oM,L) (see [30] or apply Lemma 39 to the inequalities at the end of the proof of Theorem 33 with Sα = Sβ = Q, Hα = 0, setting first ... |

1 |
Floer homology of standard pairs
- Kasturirangan
- 1998
(Show Context)
Citation Context ...of the above invariants on the subset N ⊂ M, in particular the continuity with respect to the C1-topology of submanifolds is an interesting question, which was further studied by Kasturirangan and Oh =-=[18, 19]-=-. Some applications to wave fronts and Hofer’s geometry are given in [30]. At the end, we give an application of our result to Hofer’s geometry of Lagrangian submanifolds. 2. Preliminaries and notatio... |