False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Cohomologous symplectic forms on a noncompact manifold are always isotopic

Statement refuted

Cohomologous symplectic forms on a noncompact manifold are always related by a Moser isotopy.

Facts & Assumptions

Given: The proposed universal claim.

[F1]

Compact Moser stability requires compact $M$ ; its noncompact replacement requires primitives with one common compact support. Moser stability theorem, Compact-support Moser stability on a noncompact manifold.

[F2]

Change of variables for an injective $C^{1}$ map on a compact Jordan set gives the Riemann substitution formula for an injective $C^{1}$ map with invertible derivative on a compact Jordan set. Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ gives monotonicity and coordinate-slice additivity on rectangles.

[F3]

For $u \geq 0$ , the nonnegative power-series terms give $\exp (u) \geq 1 + u$ (The real exponential function and the number $e$ by a power series). The addition formula gives $\exp (- u) = 1 / \exp (u)$ (The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ ). Thus $0 < \exp (- t^{2}) \leq 1$ for every real $t$ , and $\exp (- t^{2}) \leq t^{- 2}$ when $∣ t ∣ \geq 1$ .

[F4]

Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable gives iterated integrals of continuous functions on compact rectangles. Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative integrates $t^{- 2}$ on $[1, N]$ using the primitive $- 1 / t$ .

Refutation

technique · direct

1.1givenalgebra

On $R^{2}$ let $ω_{0} = d x \land d y = d (x d y)$ and $ω_{1} = e^{- (x^{2} + y^{2})} d x \land d y = d (F d y)$ , where $F (x, y) = \int_{0}^{x} e^{- (s^{2} + y^{2})} d s$ . Both are symplectic and exact, hence cohomologous.

2.1F2F3F4step 1.1

First obtain a uniform finite bound using only Riemann integrals of compact rectangles. For every integer $N > 1$ , [F3] and [F4] give $\int_{- N}^{N} e^{- t^{2}} d t \leq 2 + 2 \int_{1}^{N} t^{- 2} d t = 4 - 2 / N < 4$ ; for $N = 1$ the bound is at most $2 < 4$ directly. The addition formula separates $e^{- (x^{2} + y^{2})} = e^{- x^{2}} e^{- y^{2}}$ , so Jordan–Fubini on $Q_{N} = [- N, N]^{2}$ gives $\int_{Q_{N}} e^{- (x^{2} + y^{2})} d x d y = {(\int_{- N}^{N} e^{- t^{2}} d t)}^{2} < 16$ . This bound applies to every $Q_{N}$ , without defining a total integral over the noncompact plane.

3.1F1F2step 1.1step 2.1∎

Suppose a diffeomorphism $f : R^{2} \to R^{2}$ pulls $ω_{1}$ back to $ω_{0}$ . Its Jacobian is positive and satisfies $e^{- ∣ f (z) ∣^{2}} \det D f (z) = 1$ . Let $K = [- 10, 10]^{2}$ . The compact-Jordan change-of-variables theorem [F2] applies to $f$ on $K$ and gives $\int_{f (K)} e^{- ∣ y ∣^{2}} d y = \int_{K} 1 d z = 400$ . The image $f (K)$ is compact Jordan and lies in some $Q_{N}$ , so monotonicity and step 2.1 bound its integral by $16$ , a contradiction. Thus no symplectomorphism, hence no Moser isotopy, exists. This uses compact-set substitution only; no total-area integral or noncompact change-of-variables theorem is needed. The common-support/global-flow hypothesis in [F1] is substantive.

Depends on

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Sources

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