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 formula_1 be a closed (compact without boundary) symplectic manifold. For any smooth function formula_2, the symplectic form formula_3 induces a Hamiltonian vector field formula_4 on formula_5 defined by the formula
formula_6
The function formula_7 is called a Hamiltonian function.
Suppose there is a smooth 1-parameter family of Hamiltonian functions formula_8, formula_9. This family induces a 1-parameter family of Hamiltonian vector fields formula_10 on formula_5. The family of vector fields integrates to a 1-parameter family of diffeomorphisms formula_12. Each individual formula_13 is a called a Hamiltonian diffeomorphism of formula_5.
The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of formula_5 is greater than or equal to the number of critical points of a smooth function on formula_5.
Weak Arnold conjecture.
Let formula_1 be a closed symplectic manifold. A Hamiltonian diffeomorphism formula_18 is called nondegenerate if its graph intersects the diagonal of formula_19 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 formula_5, called the Morse number of formula_5.
In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field formula_22, namely formula_23. The weak Arnold conjecture says that
formula_24
for formula_25 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 formula_26 in terms of the Betti numbers of formula_27, given that formula_26 intersects L transversally and formula_26 is Hamiltonian isotopic to L.
Let formula_1 be a compact formula_31-dimensional symplectic manifold, let formula_32 be a compact Lagrangian submanifold of formula_5, and let formula_34 be an anti-symplectic involution, that is, a diffeomorphism formula_34 such that formula_36 and formula_37, whose fixed point set is formula_27.
Let formula_39, formula_9 be a smooth family of Hamiltonian functions on formula_5. This family generates a 1-parameter family of diffeomorphisms formula_12 by flowing along the Hamiltonian vector field associated to formula_43. The Arnold–Givental conjecture states that if formula_44 intersects transversely with formula_27, then
formula_46.
Status.
The Arnold–Givental conjecture has been proved for several special cases.
References.
Citations.
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