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.^[1]

Statement

Let ${\displaystyle (M,\omega )}$ be a compact symplectic manifold. For any smooth function ${\displaystyle H:M\to {\mathbb {R} }}$, the symplectic form ${\displaystyle \omega }$ induces a Hamiltonian vector field ${\displaystyle X_{H}}$ on ${\displaystyle M}$, defined by the identity:

${\displaystyle \omega (X_{H},\cdot )=dH.}$

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

Suppose there is a 1-parameter family of Hamiltonian functions ${\displaystyle H_{t}:M\to {\mathbb {R} },0\leq t\leq 1}$, inducing a 1-parameter family of Hamiltonian vector fields ${\displaystyle X_{H_{t}}}$ on ${\displaystyle M}$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms ${\displaystyle \varphi _{t}:M\to M}$. Each individual ${\displaystyle \varphi _{t}}$ is a Hamiltonian diffeomorphism of ${\displaystyle M}$.

The Arnold conjecture says that for each Hamiltonian diffeomorphism of ${\displaystyle M}$, it possesses at least as many fixed points as a smooth function on ${\displaystyle M}$ possesses critical points.^[2]

Nondegenerate Hamiltonian and weak Arnold conjecture

A Hamiltonian diffeomorphism ${\displaystyle \varphi :M\to M}$ is called nondegenerate if its graph intersects the diagonal of ${\displaystyle M\times M}$ transversely. For nondegenerate Hamiltonian diffeomorphisms, a 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 ${\displaystyle M}$, called the Morse number of ${\displaystyle M}$.

In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of ${\displaystyle M}$, for example, the sum of Betti numbers over a field ${\displaystyle {\mathbb {F} }}$:

${\displaystyle \sum _{i=0}^{2n}{\rm {dim}}H_{i}(M;{\mathbb {F} }).}$

The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on ${\displaystyle M}$ the above integer is a lower bound of its number of fixed points.

See also

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

References

This page was last edited on 15 April 2024, at 18:39