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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