Moser's Stability Theorem

TheoremMoser Stability for Symplectic Forms

Let $M$ be a compact manifold and $\omega_t$ , $0 \leq t \leq 1$ , a smooth family of symplectic forms on $M$ with $[\omega_t] = [\omega_0] \in H^2_{dR}(M)$ for all $t$ .

Then there exists a family of diffeomorphisms $\phi_t: M \to M$ with $\phi_0 = \text{id}$ such that:

$\phi_t^* \omega_t = \omega_0$

That is, all symplectic forms in the same cohomology class are equivalent via diffeomorphism.

This profound result, proved by Jürgen Moser in 1965, shows that symplectic structures are stable under deformation within a fixed cohomology class. It extends Darboux's theorem from local to global settings (on compact manifolds).

Geometric Interpretation: If we smoothly deform a symplectic form $\omega$ without changing its cohomology class, we can "track" this deformation by a family of diffeomorphisms. The symplectic structure itself doesn't change — only our coordinate description.

Remark

The compactness assumption is essential. On non-compact manifolds, diffeomorphisms may "escape to infinity" and the theorem fails. For example, on $\mathbb{R}^{2n}$ , scaling $\omega_t = (1+t)\omega_0$ changes cohomology in a relative sense.

The proof uses Moser's path method: solve the equation $\mathcal{L}_{X_t} \omega_t = -\dot{\omega}_t$ for a time-dependent vector field $X_t$ , then integrate to obtain $\phi_t$ .

DefinitionCohomological Constraint

The condition $[\omega_t]$ constant in $H^2_{dR}(M)$ is necessary: if two symplectic forms define different cohomology classes, they cannot be related by a diffeomorphism, since:

$[\phi^* \omega] = \phi^*[\omega] = [\omega]$

for any diffeomorphism $\phi$ .

ExampleDeformations of Kähler Manifolds

On a compact Kähler manifold $(M, \omega, J)$ , the Kähler form $\omega$ lies in a specific Kähler cohomology class. Moser's theorem implies that small deformations of $\omega$ within this class are related by diffeomorphisms, even though the complex structure $J$ may change.

This is crucial in complex geometry: the space of Kähler structures modulo diffeomorphism is finite-dimensional (determined by Hodge theory).

DefinitionSymplectic Diffeomorphism Group

Moser's theorem implies that for a compact symplectic manifold $(M, \omega)$ , the space of symplectic structures in $[\omega]$ is:

$\{\omega' : [\omega'] = [\omega]\}/\text{Diff}(M) \cong \{[\omega]\}$

The quotient is a single point! This shows remarkable rigidity: fixing cohomology class determines the symplectic structure up to diffeomorphism.

ExampleTorus with Different Area Forms

Consider the 2-torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ with symplectic forms:

$\omega_A = A \, dx \wedge dy$

for $A > 0$ . These have different cohomology classes: $[\omega_A] = A[dx \wedge dy] \in H^2(T^2; \mathbb{R}) \cong \mathbb{R}$ .

By Moser, $\omega_A$ and $\omega_B$ are diffeomorphic if and only if $A = B$ . The area (total symplectic volume) is the only invariant.

Remark

Moser's theorem contrasts with Riemannian geometry: Riemannian metrics in the same conformal class are NOT generally related by diffeomorphisms. The Yamabe problem asks when metrics of constant scalar curvature exist in a conformal class — a non-trivial question.

Symplectic geometry is simpler: cohomology class alone determines the structure.

DefinitionApplications to Symplectic Reduction

Moser's theorem is crucial for symplectic reduction (Marsden-Weinstein): when a Lie group acts on $(M, \omega)$ with moment map $\mu$ , the reduced space:

$M_{//}G = \mu^{-1}(0)/G$

inherits a symplectic structure. Moser's method proves that this structure is well-defined independent of choices.

ExampleStability of Hamiltonian Systems

In Hamiltonian mechanics, perturbations of the symplectic structure $\omega \leadsto \omega + \delta\omega$ with $[\delta\omega] = 0$ can be "absorbed" by coordinate changes. This means:

Physical perturbations (changing $H$ or $\omega$ in a cohomologically trivial way) are equivalent to coordinate changes in the unperturbed system.

Only cohomologically non-trivial perturbations genuinely change the dynamics.

DefinitionMoser's Trick for Non-Compact Manifolds

On non-compact manifolds, variants of Moser's theorem hold:

Proper support: If $\omega_t$ differ only on a compact set, Moser applies
Bounded geometry: Control on derivatives allows Moser on complete manifolds
Relative version: Fix behavior at infinity, apply Moser to interior

These extensions are crucial for local-to-global principles in symplectic topology.

Remark

Moser's stability has analogues in other geometries:

Contact geometry: Gray's stability theorem for contact forms
Complex geometry: Kodaira-Spencer deformation theory
Riemannian geometry: No direct analogue (metrics not determined by cohomology)

The symplectic case is distinguished by its simplicity.