Arnold 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 $(M, \omega)$ be a closed (compact without boundary) symplectic manifold. For any smooth function $H: M \to$, the symplectic form $\omega$ induces a Hamiltonian vector field $X_H$ on $M$ defined by the formula

:$\omega( X_H, \cdot) = dH.$

The function $H$ is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions $H_t \in C^\infty(M)$, t \in ,1/math>. This family induces a 1-parameter family of Hamiltonian vector fields $X_$ on $M$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $\varphi_t: M \to M$. Each individual $\varphi_t$ is a called a Hamiltonian diffeomorphism of $M$.

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of $M$ is greater than or equal to the number of critical points of a smooth function on $M$. See also comments, pp. 284–288.

Weak Arnold conjecture

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

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field $$, namely $\sum_^ \dim H_i (M; )$. The weak Arnold conjecture says that

:$\# \ \geq \sum_^ \dim H_i (M; )$

for $\varphi : M \to M$ 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 and $L'$ in terms of the Betti numbers of $L$, given that $L'$ intersects transversally and $L'$ is Hamiltonian isotopic to .

Let $(M, \omega)$ be a compact $2n$-dimensional symplectic manifold, let $L \subset M$ be a compact Lagrangian submanifold of $M$, and let $\tau : M \to M$ be an anti-symplectic involution, that is, a diffeomorphism $\tau : M \to M$ such that $\tau^* \omega = -\omega$ and $\tau^2 = \text_M$, whose fixed point set is $L$.

Let $H_t\in C^\infty(M)$, t \in ,1/math> be a smooth family of Hamiltonian functions on $M$. This family generates a 1-parameter family of diffeomorphisms $\varphi_t: M \to M$ by flowing along the Hamiltonian vector field associated to $H_t$. The Arnold–Givental conjecture states that if $\varphi_1(L)$ intersects transversely with $L$, then

:$\# (\varphi_1(L) \cap L) \geq \sum_^n \dim H_i(L; \mathbb Z / 2 \mathbb Z)$.

Status

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

* Alexander Givental proved it for $(M, L) = (\mathbb^n, \mathbb^n)$.
* 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 $(M, \omega)$ semi-positive.
* Urs Frauenfelder proved it in the case when $(M, \omega)$ is a certain symplectic reduction, using gauged Floer theory.

See also

* Symplectomorphism#Arnold conjecture
* Floer homology
* Spectral invariants
* Conley–Zehnder theorem

References

Citations

Bibliography

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*{{citation, doi=10.1007/978-3-0348-9217-9_23 , chapter=Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental Conjecture , title=The Floer Memorial Volume , date=1995 , last1=Oh , first1=Yong-Geun , pages=555–573 , isbn=978-3-0348-9948-2

Conjectures
Symplectic geometry
Hamiltonian mechanics
Unsolved problems in mathematics