Knowlpedia

Theorem

Moser stability theorem

A cohomologically constant smooth family of symplectic forms on a compact manifold is trivialized by an isotopy.

Statement

Let MM be a compact without boundary, and let ωt\omega_t, 0≤t≤10\leq t\leq1, be a smooth family of whose de Rham cohomology class [ωt][\omega_t] is independent of tt. The Moser stability theorem states that there is a smooth φt:M→M\varphi_t:M\to M, with φ0=id⁡M\varphi_0=\operatorname{id}_M, such that

φt∗ωt=ω0\varphi_t^*\omega_t=\omega_0

for every tt. In particular, φ1\varphi_1 is a from (M,ω0)(M,\omega_0) to (M,ω1)(M,\omega_1). Compactness ensures that the time-dependent used in the proof has a global flow through the whole interval.

Proof mechanism

Choose a smooth family of one-forms σt\sigma_t such that

ω˙t=dσt.\dot\omega_t=d\sigma_t.

Define the time-dependent vector field XtX_t uniquely by

ιXtωt=−σt.\iota_{X_t}\omega_t=-\sigma_t.

If φt\varphi_t is its flow, then

ddt(φt∗ωt)=φt∗(ω˙t+LXtωt)=φt∗(dσt−dσt)=0.\frac{d}{dt}\bigl(\varphi_t^*\omega_t\bigr) =\varphi_t^*\bigl(\dot\omega_t+\mathcal L_{X_t}\omega_t\bigr) =\varphi_t^*\bigl(d\sigma_t-d\sigma_t\bigr)=0.

This conversion of a deformation equation into an equation for a flow is the Moser trick Cannas da Silva, “Moser Theorems,” pp. 49–53.

Relative and local forms

If the family and the chosen primitives satisfy suitable vanishing conditions along a submanifold, XtX_t can be arranged to vanish there, and the resulting isotopy fixes that submanifold. Local and relative versions underlie the Darboux theorem and symplectic neighborhood theorems. The exact vanishing order matters when one also requires the derivative of the isotopy to be the identity along the submanifold McDuff–Salamon, corrected Lemma 3.2.1.

Hypotheses and limitations
References
  1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Chapter DOI record. Relevant: “Moser Theorems,” pp. 49–53.
  2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Oxford DOI record. Relevant: §3.2, Moser isotopy arguments.
  3. Jürgen Moser, “On the Volume Elements on a Manifold,” Transactions of the American Mathematical Society 120 (1965), 286–294. AMS DOI record. Relevant: the original deformation-by-isotopy method.