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I am looking for a reference for the proof of the nearby Lagrangian conjecture of $T^*S^1$, that is, that every exact and compact Lagrangian submanifold of the cylinder is Hamiltonianly isotopic to the zero section in $T^*S^1$.

The corresponding result for $T^*S^2$ can be found here https://projecteuclid.org/download/pdf_1/euclid.ajm/1331663450

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There are only two smooth isotopy classes of closed curves in $T^*S^1$. Any inessential (i.e. bounding a disk) curve is non-exact (since the disk has positive area). An essential curve is smoothly isotopic to the zero section. If the original curve is exact, the isotopy may be made exact by translating it in the vertical direction (as a function of time) by the appropriate amount.

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