TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Compact-support Moser stability on a noncompact manifold

Statement

Assume ${A C}_{ω}$ . Let $(ω_{t})_{0 \leq t \leq 1}$ be a smooth path of symplectic forms on a possibly noncompact manifold $M$ . Suppose ${\dot{ω}}_{t} = d σ_{t}$ for a smooth family of one-forms whose supports all lie in one compact set $K$ . Then a compactly supported isotopy $ϕ_{t}$ exists for all $t \in [0, 1]$ and satisfies $ϕ_{t}^{*} ω_{t} = ω_{0}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ and the path, primitives, and common compact support in the statement.

[F1]

The Moser equation has a unique smooth solution and forces pullback constancy. Moser pullback differentiation equation.

[F2]

A smooth time-dependent vector field with common compact support has a global evolution over a compact time interval. Compactly supported time-dependent vector fields have global evolution on a compact time interval.

Proof

technique · direct

1.1

Solve $ι_{X_{t}} ω_{t} = - σ_{t}$ by [F1]. At every point outside $K$ the right side vanishes, and nondegeneracy gives $X_{t} = 0$ ; hence all $X_{t}$ have support in $K$ .

F1given

2.1

By [F2], $X_{t}$ has a global evolution $ϕ_{t}$ on $[0, 1]$ . It is the identity off $K$ , so the isotopy is compactly supported. By [F1], $\frac{d}{d t} (ϕ_{t}^{*} ω_{t}) = 0$ , and evaluation at zero gives $ϕ_{t}^{*} ω_{t} = ω_{0}$ .

F1F2step 1.1∎

Depends on

Used by

Cohomologous symplectic forms on a noncompact manifold are always isotopic False statement

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)