Moser's Deformation Method

DefinitionMoser's Path Method

Moser's method is a technique for proving that two geometric structures are equivalent by constructing a smooth path connecting them and integrating a time-dependent vector field.

For symplectic forms $\omega_0$ and $\omega_1$ on $M$ with $[\omega_0] = [\omega_1] \in H^2_{dR}(M)$ , Moser's method constructs a diffeomorphism $\phi: M \to M$ with $\phi^* \omega_1 = \omega_0$ .

Moser's method, introduced by Jürgen Moser in 1965, revolutionized symplectic geometry by providing a systematic way to prove equivalence results. It reduces geometric problems to solving differential equations, making many theorems constructive.

DefinitionMoser's Equation

Given a path of closed forms $\omega_t = \omega_0 + t(\omega_1 - \omega_0)$ with $\omega_0 \sim \omega_1$ in cohomology, we seek a vector field $X_t$ satisfying:

$\mathcal{L}_{X_t} \omega_t = -\frac{d\omega_t}{dt} = -(\omega_1 - \omega_0)$

If $X_t$ exists and integrates to a flow $\phi_t$ , then $\phi_1^* \omega_1 = \omega_0$ .

ExampleApplying Moser's Method

Suppose $\omega_0$ and $\omega_1$ are cohomologous: $\omega_1 - \omega_0 = d\alpha$ for some 1-form $\alpha$ . Define:

$\omega_t = \omega_0 + t(\omega_1 - \omega_0) = \omega_0 + t \, d\alpha$

We need $X_t$ with:

$\mathcal{L}_{X_t} \omega_t = -d\alpha$

By Cartan's formula $\mathcal{L}_{X_t} = d \circ \iota_{X_t} + \iota_{X_t} \circ d$ :

$d(\iota_{X_t} \omega_t) = -d\alpha$

So we can take $\iota_{X_t} \omega_t = -\alpha$ . Since $\omega_t$ is non-degenerate (for small $t$ ), this uniquely determines $X_t$ .

Remark

The key insight is that non-degeneracy of $\omega_t$ allows us to solve for $X_t$ uniquely from $\iota_{X_t} \omega_t = -\alpha$ . This is analogous to solving for velocity from momentum in mechanics.

Moser's method applies beyond Darboux's theorem. It's used to prove:

Symplectic neighborhood theorems
Stability of symplectic structures
Equivariant Darboux theorems with group actions

DefinitionMoser Stability

A family $\omega_t$ of symplectic forms on a compact manifold $M$ is stable if there exists a family of diffeomorphisms $\phi_t$ with $\phi_t^* \omega_t = \omega_0$ .

Moser's stability theorem: If $[\omega_t]$ is constant in $H^2_{dR}(M)$ , then the family is stable.

ExampleDeformations of Kähler Structures

On a Kähler manifold, the Kähler form $\omega$ satisfies $d\omega = 0$ and determines a complex structure. Moser's method shows that small deformations of $\omega$ (within the same cohomology class) are related by diffeomorphisms, preserving the symplectic structure.

DefinitionRelative Version

Moser's method extends to relative settings: if we have a submanifold $S \subseteq M$ and vector fields $X_t$ that vanish on $S$ , the resulting diffeomorphisms fix $S$ pointwise.

This is crucial for:

Weinstein's Lagrangian neighborhood theorem
Symplectic normal form theorems

Remark

Moser's method has analogues in other geometries:

Riemannian: Ebin-Palais for metrics in fixed conformal class
Complex: Newlander-Nirenberg for integrable almost-complex structures
Contact: Gray's stability for contact structures

Each uses time-dependent flows to relate geometric structures.

ExampleDarboux via Moser

To prove Darboux's theorem using Moser: given $\omega$ on $M^{2n}$ and $p \in M$ , choose coordinates so $\omega_p = \omega_0|_p$ (possible since tangent spaces are symplectomorphic). Then $\omega = \omega_0 + d\alpha$ near $p$ for some $\alpha$ . Apply Moser's method to get a local diffeomorphism $\phi$ with $\phi^* \omega = \omega_0$ .

DefinitionMoser's Trick

A common application is Moser's trick: to show two structures are equivalent, construct a path between them in a space of "good" structures, then apply the path method to produce the equivalence.

This converts a static comparison problem into a dynamic flow problem.