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.{{cite arXiv | last1=Asselle | first1=L. | last2=Izydorek | first2=M. | last3=Starostka | first3=M. | title=The Arnold conjecture in $\mathbb C\mathbb P^n$ and the Conley index | year=2022 | eprint=2202.00422 | page=| class=math.DS }}

Strong Arnold conjecture

Let $(M, \omega)$ be a closed (compact without boundary) symplectic manifold. For any smooth function $H: M \to {\mathbb R}$ , 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 $H_t \in C^\infty(M)$ , $t \in [0,1]$ . This family induces a 1-parameter family of Hamiltonian vector fields $X_{H_t}$ 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 $M$ .{{cite arXiv|last1=Rizell |first1=Georgios Dimitroglou |title=The number of Hamiltonian fixed points on symplectically aspherical manifolds |date=2017-01-05 |eprint=1609.04776 |last2=Golovko |first2=Roman|class=math.SG }}{{cite book

| contribution = 1972-33

| doi = 10.1007/b138219

| isbn = 3-540-20614-0

| mr = 2078115

| page = 15

| publisher = Springer-Verlag | location = Berlin

| title = Arnold's Problems

| title-link = Arnold's Problems | year = 2004}} 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 ${\mathbb F}$ , namely $\sum_{i=0}^{2n} \dim H_i (M; {\mathbb F})$ . The weak Arnold conjecture says that

: $\# \{ \text{fixed points of } \varphi \} \geq \sum_{i=0}^{2n} \dim H_i (M; {\mathbb F})$

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 {{mvar|L}} and $L'$ in terms of the Betti numbers of $L$ , given that $L'$ intersects {{mvar|L}} transversally and $L'$ is Hamiltonian isotopic to {{mvar|L}}.

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 $\tau^2 = \text{id}_M$ , whose fixed point set is $L$ .

Let $H_t\in C^\infty(M)$ , $t \in [0,1]$ 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

: $\# (\varphi_1(L) \cap L) \geq \sum_{i=0}^n \dim H_i(L; \mathbb Z / 2 \mathbb Z)$ .{{harv|Frauenfelder|2004}}

=Status=

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

Alexander Givental proved it for $(M, L) = (\mathbb{CP}^n, \mathbb{RP}^n)$ .{{harv|Givental|1989b}}
Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.{{harv|Oh|1995}}
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.{{harv|Fukaya|Oh|Ohta|Ono|2009}}
Urs Frauenfelder proved it in the case when $(M, \omega)$ is a certain symplectic reduction, using gauged Floer theory.

References

=Citations=

=Bibliography=

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| issue = 42

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| year = 2004| volume = 2004

| arxiv = math/0309373

| doi-access =

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| journal = Comptes Rendus de l'Académie des Sciences

| mr = 1179726

| pages = 309–314

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| volume = 315

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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 }}

Category:Conjectures

Category:Symplectic geometry

Category:Hamiltonian mechanics

Category:Unsolved problems in mathematics