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Arnold conjecture

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(Redirected from Arnold–Givental conjecture) Mathematical conjecture

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

Strong Arnold conjecture

Let ( M , ω ) {\displaystyle (M,\omega )} be a closed (compact without boundary) symplectic manifold. For any smooth function H : M → R {\displaystyle H:M\to {\mathbb {R} }} , the symplectic form ω {\displaystyle \omega } induces a Hamiltonian vector field X H {\displaystyle X_{H}} on M {\displaystyle M} defined by the formula

ω ( X H , ⋅ ) = d H . {\displaystyle \omega (X_{H},\cdot )=dH.}

The function H {\displaystyle H} is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions H t ∈ C ∞ ( M ) {\displaystyle H_{t}\in C^{\infty }(M)} , t ∈ [ 0 , 1 ] {\displaystyle t\in } . This family induces a 1-parameter family of Hamiltonian vector fields X H t {\displaystyle X_{H_{t}}} on M {\displaystyle M} . The family of vector fields integrates to a 1-parameter family of diffeomorphisms φ t : M → M {\displaystyle \varphi _{t}:M\to M} . Each individual φ t {\displaystyle \varphi _{t}} is a called a Hamiltonian diffeomorphism of M {\displaystyle M} .

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M {\displaystyle M} is greater than or equal to the number of critical points of a smooth function on M {\displaystyle M} .

Weak Arnold conjecture

Let ( M , ω ) {\displaystyle (M,\omega )} be a closed symplectic manifold. A Hamiltonian diffeomorphism φ : M → M {\displaystyle \varphi :M\to M} is called nondegenerate if its graph intersects the diagonal of M × M {\displaystyle M\times M} 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 M {\displaystyle M} , called the Morse number of M {\displaystyle M} .

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field F {\displaystyle {\mathbb {F} }} , namely ∑ i = 0 2 n dim ⁡ H i ( M ; F ) {\textstyle \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })} . The weak Arnold conjecture says that

# { fixed points of  φ } ≥ ∑ i = 0 2 n dim ⁡ H i ( M ; F ) {\displaystyle \#\{{\text{fixed points of }}\varphi \}\geq \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })}

for φ : M → M {\displaystyle \varphi :M\to M} a nondegenerate Hamiltonian diffeomorphism.

Arnold–Givental conjecture

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 L ′ {\displaystyle L'} in terms of the Betti numbers of L {\displaystyle L} , given that L ′ {\displaystyle L'} intersects L transversally and L ′ {\displaystyle L'} is Hamiltonian isotopic to L.

Let ( M , ω ) {\displaystyle (M,\omega )} be a compact 2 n {\displaystyle 2n} -dimensional symplectic manifold, let L ⊂ M {\displaystyle L\subset M} be a compact Lagrangian submanifold of M {\displaystyle M} , and let τ : M → M {\displaystyle \tau :M\to M} be an anti-symplectic involution, that is, a diffeomorphism τ : M → M {\displaystyle \tau :M\to M} such that τ ∗ ω = − ω {\displaystyle \tau ^{*}\omega =-\omega } and τ 2 = id M {\displaystyle \tau ^{2}={\text{id}}_{M}} , whose fixed point set is L {\displaystyle L} .

Let H t ∈ C ∞ ( M ) {\displaystyle H_{t}\in C^{\infty }(M)} , t ∈ [ 0 , 1 ] {\displaystyle t\in } be a smooth family of Hamiltonian functions on M {\displaystyle M} . This family generates a 1-parameter family of diffeomorphisms φ t : M → M {\displaystyle \varphi _{t}:M\to M} by flowing along the Hamiltonian vector field associated to H t {\displaystyle H_{t}} . The Arnold–Givental conjecture states that if φ 1 ( L ) {\displaystyle \varphi _{1}(L)} intersects transversely with L {\displaystyle L} , then

# ( φ 1 ( L ) ∩ L ) ≥ ∑ i = 0 n dim ⁡ H i ( L ; Z / 2 Z ) {\displaystyle \#(\varphi _{1}(L)\cap L)\geq \sum _{i=0}^{n}\dim H_{i}(L;\mathbb {Z} /2\mathbb {Z} )} .

Status

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

  • Givental proved it for ( M , L ) = ( C P n , R P n ) {\displaystyle (M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})} .
  • Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.
  • 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 ( M , ω ) {\displaystyle (M,\omega )} semi-positive.
  • Urs Frauenfelder proved it in the case when ( M , ω ) {\displaystyle (M,\omega )} is a certain symplectic reduction, using gauged Floer theory.

See also

References

Citations

  1. Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in C P n {\displaystyle \mathbb {C} \mathbb {P} ^{n}} and the Conley index". arXiv:2202.00422 .
  2. ^ Rizell, Georgios Dimitroglou; Golovko, Roman (2017-01-05). "The number of Hamiltonian fixed points on symplectically aspherical manifolds". arXiv:1609.04776 .
  3. ^ Arnold, Vladimir I. (2004). "1972-33". In Arnold, Vladimir I. (ed.). 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. ^ (Frauenfelder 2004)
  5. (Givental 1989b)
  6. (Oh 1995)
  7. (Fukaya et al. 2009)

Bibliography

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