Arnold conjecture explained
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
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
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
.
, namely
. The
weak Arnold conjecture says that
\#\{fixedpointsof\varphi\}\geq
\dimHi(M;{F})
for
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
in terms of the Betti numbers of
, given that
intersects transversally and
is Hamiltonian isotopic to .
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
\#(\varphi1(L)\capL)\geq
\dimHi(L;Z/2Z)
.
Status
The Arnold–Givental conjecture has been proved for several special cases.
.
- 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
semi-positive.
- Urs Frauenfelder proved it in the case when
is a certain symplectic reduction, using
gauged Floer theory.
See also
References
Bibliography
Notes and References
- Asselle . L. . Izydorek . M. . Starostka . M. . The Arnold conjecture in
and the Conley index . 2022 . 2202.00422 . math.DS .
- Rizell . Georgios Dimitroglou . The number of Hamiltonian fixed points on symplectically aspherical manifolds . 2017-01-05 . 1609.04776 . Golovko . Roman. math.SG .
- Book: Arnold's Problems . 2005 . Springer Berlin, Heidelberg . 284–288 . en . 10.1007/b138219 . 978-3-540-20748-1 . Arnold . Vladimir I. .