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Arnold conjecture

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The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.[1]

Strong Arnold conjecture

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Let be a closed (compact without boundary) symplectic manifold. For any smooth function , the symplectic form induces a Hamiltonian vector field on defined by the formula

The function is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions , . This family induces a 1-parameter family of Hamiltonian vector fields on . The family of vector fields integrates to a 1-parameter family of diffeomorphisms . Each individual is a called a Hamiltonian diffeomorphism of .

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of is greater than or equal to the number of critical points of a smooth function on .[2][3]

Weak Arnold conjecture

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Let be a closed symplectic manifold. A Hamiltonian diffeomorphism is called nondegenerate if its graph intersects the diagonal of transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on , called the Morse number of .

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field , namely . The weak Arnold conjecture says that

for a nondegenerate Hamiltonian diffeomorphism.[2][3]

Arnold–Givental conjecture

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The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds L and in terms of the Betti numbers of , given that intersects L transversally and is Hamiltonian isotopic to L.

Let be a compact -dimensional symplectic manifold, let be a compact Lagrangian submanifold of , and let be an anti-symplectic involution, that is, a diffeomorphism such that and , whose fixed point set is .

Let , be a smooth family of Hamiltonian functions on . This family generates a 1-parameter family of diffeomorphisms by flowing along the Hamiltonian vector field associated to . The Arnold–Givental conjecture states that if intersects transversely with , then

.[4]

Status

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The Arnold–Givental conjecture has been proved for several special cases.

  • Givental proved it for .[5]
  • Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.[6]
  • Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.
  • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for semi-positive.[7]
  • Urs Frauenfelder proved it in the case when is a certain symplectic reduction, using gauged Floer theory.[4]

See also

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References

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Citations

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  1. ^ Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in and the Conley index". arXiv:2202.00422 [math.DS].
  2. ^ a b Rizell, Georgios Dimitroglou; Golovko, Roman (2017-01-05). "The number of Hamiltonian fixed points on symplectically aspherical manifolds". arXiv:1609.04776 [math.SG].
  3. ^ a b Arnold, Vladimir I. (2004). "1972-33". Arnold's Problems. Berlin: Springer-Verlag. p. 15. doi:10.1007/b138219. ISBN 3-540-20614-0. MR 2078115. See also comments, pp. 284–288.
  4. ^ a b (Frauenfelder 2004)
  5. ^ (Givental 1989b)
  6. ^ (Oh 1995)
  7. ^ (Fukaya et al. 2009)

Bibliography

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